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 Cambridge Tracts in Mathematics
and Mathematical Physics

General Editors
P.HALL, F.R.S.and F. SMITHIES, Ph.D.

No. 2

THE
INTEGRATION OF FUNCTIONS
OF A SINGLE VARIABLE
BY THE LATE
G. H. HARDY

Cornell University Library

QA 311. H26 1916

The integration of functions of a single

3 1924 001 539 570

CAMBRIDGE UNIVERSITY PRESS

Cambridge Tracts in Mathematics
and Mathematical Physics
General Editors
P. HALL, F.R.S. and F. SMITHIES, Ph.D.

No. 2

The Integration of Functions of a

Single Variable
 THE
INTEGRATION OF FUNCTIONS
OF A SINGLE VARIABLE

BY THE LATE

G. H. HARDY

SECOND EDITION

CAMBRIDGE
AT THE UNIVERSITY PRESS
1966
 PUBLISHED BY
THE SYNDICS OF THE CAMBRIDGE UNIVERSITY PRESS
Bentley House, 200 Euston Koad, London, N.W. 1
American Branch: 32 East 57th Street, New York, N.Y. 10022

\ Yr\ o\ IC S

First Edition 1905

Second Edition 1916
Reprinted 1928
1958
1966

V*

First printed in Cheat Britain at the University Press, Cambridge

Reprinted by offset-litho by Jarrold & Sons Ltd., Norwich
 PREFACE
fTlHIS tract has been long out of print, and there is still some
-*- demand for it. I did not publish a second edition before,

because I intended to incorporate its contents in a larger treatise on

the subject which I had arranged to write in collaboration with

Dr Bromwich. Four or five years have passed, and it seems very

doubtful whether either of us will ever find the time to carry out

our intention. I have therefore decided to republish the tract.

The new edition differs from the first in one important point
only. In the first edition I reproduced a proof of Abel's which
Mr J. E. Littlewood afterwards discovered to be invalid. The
correction of this error has led me to rewrite a few sections (pp. 36-41
of the present edition) completely. The proof which I give now is

due to Mr H. T. J. Norton. I am also indebted to Mr Norton,

and to Mr S. Pollard, for many other criticisms of a less important

character.

G. H. H.

January 1916.
 CONTENTS
PAGE
I. Introduction ... 1

II. Elementary functions and their classification ... 3

III. The integration of elementary functions. Summary of results . 8

IV. The integration of rational functions 12

1-3. The method of partial fractions . .12
4. Hermite's method of integration ... 15
5. Particular problems of integration . . 17

6. The limitations of the methods of integration . . 19

7. Conclusion . . 21

V. The integration of algebraical functions 22

1. Algebraical functions . . ... .22
2. Integration by rationalisation. Integrals associated with
conies . 23
3-6. The integral \R{x, Jiax* + %hx + c)}dx . 24
7. Unicursal plane curves 30
...
. .

33
8.

9.

10.
Particular cases
Unicursal curves in space
. .

Integrals of algebraical functions in general

..... . .
35
35
11-14. The general form of the integral of an algebraical function.
Integrals which are themselves algebraical ... 36

15. Discussion of a particular case . 42

16. The transcendence of e* and log# 44.,

17. Laplace's principle •

44

18. The general form of the integral of an algebraical function

(continued). Integrals expressible by algebraical functions
and logarithms .... . . . 45
 .

Vlll CONTENTS

19. Elliptic and pseudo-elliptic integrals. Binomial integrals

20. Curves of deficiency 1. The plane cubic
....
.

21. Degenerate Abelian integrals

22. The classification of elliptic integrals .

VI. The integration of transcendental functions

1. Preliminary . . ...
2. The integral J R{e ax , e6* ..., <**)dx .

3. The integral j P(x, e™>, e

te
, ...)dx
4. The integral \^R(x)dx. The logarithm-integral
5. Liouville's general theorem
6. The integral J log x R (x) dx
7. Conclusion

Appendix I. Bibliography
Appendix II. On Abel's proof of the theorem of v., §.11
 THE INTEGRATION OF FUNCTIONS OF
A SINGLE VARIABLE

I. Introduction

The problem considered in the following pages is what is sometimes

called the problem of '
indefinite integration ' or of '
finding a function
is a given function
whose differential coefficient These descriptions '.

are vague and in some ways misleading and it is necessary to define ;

our problem more precisely before we proceed further.

Let us suppose for the moment that f{x) is a real continuous
function of the real variable x. We wish to determine a function y
whose differential coefficient is fix), or to solve the equation

!=/<*> »
A little reflection shows that this problem may be analysed into a
number of parts.
We wish, first, to know whether such a function as y necessarily
exists, whether the equation (l) has always a solution ; whether the
solution, if it exists, is and what
unique ; relations hold between
different solutions, if there are more than one. The answers to these
questions are contained in that part of the theory of functions of a
real variable which deals with 'definite integrals'. The definite
integral

)dt (2),
Ja

which is defined as the limit of a certain sum, is a solution of the

equation (1). Further
y+ c (3),

where G is an arbitrary constant, is also a solution, and all solutions of

(1) are of the form (3).

h. 1
Z INTRODUCTION L1

These results we shall take for granted. The questions with which
we be concerned are of a quite different character. They are
shall
questions as to the functional form of y when f{x) is a function of
some stated form. It is sometimes said that the problem of indefinite
integration is that of '
finding an actual expression for y when fix) is
given'. This statement is however still lacking in precision. The theory
of definite integrals provides us not only with a proof of the existence
of a solution, but also with an expression forit, an expression in the

form of a The problem of indefinite integration can be stated

limit.

precisely only when we introduce sweeping restrictions as to the classes

of functions and the modes of expression which we are considering.
Let us suppose that f(x) belongs to some special class of functions
df. Then we may ask whether y is itself a member of Jf, or can be
expressed, according to some simple standard mode of expression, in
terms of functions which are members of jp. To take a trivial
example, we might suppose that $ is the class of polynomials with
rational coefficients : the answer would then be that y is in all cases
itself a member of Jf.
The range and difficulty of our problem will depend upon our
choice of (1) a class of functions and (2) a standard 'mode of ex-
pression '.
We the purposes of this tract, take if to be the
shall, for

class of elementary functions, a class which will be defined precisely in

the next section, and our mode of expression to be that of explicit
expression in finite terms, i.e. by formulae which do not involve passages
to a limit.
One or two more preliminary remarks are needed. The subject-
matter of the tract forms a chapter in the 'integral calculus'*, but
does not depend in any way on any direct theory of integration. Such
an equation as

y = jf(x)dx (4)

is to be regarded as merely another way of writing (1) the integral :

sign isused merely on grounds of technical convenience, and might

be eliminated throughout without any substantial change in the
argument.
* Euler, the first systematic writer on the 'integral calculus', denned it in
a manner which with the theory of differential equations
identifies it calculus :
'

integralis est methodus, ex data differentialium relatione inveniendi relationem

ipsarum quantitatum' (Imtitutionea calculi integralis, p. 1). We are concerned
only with the special equation (1), but all the remarks we have made may be
generalised so as to apply to the wider theory.
Il] ELEMENTARY FUNCTIONS AND THEIR CLASSIFICATION 3

The variable x is in general supposed to be complex. But the tract

should be intelligible to a reader who is not acquainted with the theory
of analytic functions and who regards x as real and the functions of x
which occur as real or complex functions of a real variable.
The functions with which we shall be dealing will always be such
as are regular except for certain special values of x. These values of
x we shall simply igaore. The meaning of such an equation as

fdx ,
/' = log X
x

is in no way affected by the fact that l/x and log a; have infinities for

II. Elementary functions and their classification

An elementary function is a member of the class of functions which

comprises

(i) rational functions,

(ii) algebraical functions, explicit or implicit,

-

(iii) the exponential function e*,

(iv) the logarithmic function log x,

(v) all functions which can be denned by means of any finite

combination of the symbols proper to the preceding four classes of

functions.
A few remarks and examples may help to elucidate this definition.

1. A rational function is a function defined by means of any finite

combination of the elementary operations of addition, multiplication,
and division, operating on the variable x.
shown in elementary algebra that any rational function of x
It is
may be expressed in the form
a a xm + a xm
-1
... 1
+ ... +am
f(x) = n n~ 1
'

b x +b x i
+...+&„ '

where m and n are positive integers, the a's and b's are constants, and
the numerator and denominator have no common factor. We shall
adopt this expression as the standard form of a rational function. It

is hardly necessary to remark that it is in no way involved in the

4 ELEMENTARY FUNCTIONS AND THEIR CLASSIFICATION [il

definition of a rational function that these constants should be rational

or algebraical* or real numbers. Thus

ar +x+ i J2
xj2-e
is a rational function.

2. An explicit algebraical function is a function defined by means

of any finite combination of the four elementary operations and any
finite number of operations of root extraction. Thus

J(l+ *)-#(!-*) n x+J(x+ jx)]

V^+V^+n/^K
& + * + iJ* \*
J(l + x)+y(l-x)' \ x j 2 -e J

are explicit algebraical functions. And so is x mln (i.e. Z/x

m) for any
integral values of m and n. On the other hand

1*1
x-J", x
are not algebraical functions at all, but transcendental functions, as

irrational or complex powers are defined by the aid of exponentials

and logarithms.
Any explicit algebraical function of x satisfies an equation
n~l
P yn + P y
a i + ... +P„ =
whose coefficients are polynomials in x. Thus, for example, the
function
y= Jx + J(x + Jx)
satisfies the equation ,

2
y-(42/ + iy +1)^ = 0.

The converse is not true, since it has been proved that in general
equations of degree higher than the fourth have no roots which are
explicit algebraical functions of their coefficients. A simple example
is given by the equation
y*-y-x = 0.
We are thus led to consider a more general class of functions, implicit
algebraical functions, which includes the class of explicit algebraical
functions.

* An algebraical number is a number which is the root of an algebraical equa-

tion whose coefficients are integral. It is known that there are numbers (such as
e and ir) which are not roots of any such equation. See, for example, Hobson's
Squaring the circle (Cambridge, 1913).
1-3] ELEMENTARY FUNCTIONS AND THEIR CLASSIFICATION 5

3. An algebraical function of a; is a function which satisfies an

equation
P yn + P,yn - l
+ ...+Pn = Q (1)

whose coefficients are polynomials in x.

Let us denote by P (x, y) a polynomial such as occurs on the left-

hand side of (1). Then there are two possibilities as regards any
particular polynomial P (x, y). Either it is possible to express P (x, y)
as the product of two polynomials of the same type, neither of which
is a mere constant, or it is not. In the first case P (x, y) is said to
be reducible, in the second irreducible. Thus
y -a? = (tf + x)(J/ -x)
i i

is reducible, while both y' + x and tf — x are irreducible.

The equation (1) is said to be reducible or irreducible according as
its left-hand side is reducible or irreducible. A reducible equation can
always be replaced by the logical alternative of a number of irreducible
equations. Reducible equations are therefore of subsidiary importance
only ; and we shall always suppose that the equation (1) is irreducible.
An algebraical function of x is regular except at a finite number
of points which are poles or branch points of the function.
Let be D
any closed simply connected domain in the plane of x which does
not include any branch point. Then there are n and only n distinct
functions which are one-valued in D
and satisfy the equation (1).
These n functions will be called the roots of (1) in 1). Thus if we
write
x = r (cos 6 + i sin 6),
where - ir < 6 $ ir, then the roots of

y*-x = 0,
in the domain

•< ^$ 7" $ r2 ,
— ir< — ir + S^O ^tt — 8<ir,
are Jx and - Jx, where
Jx = Jr (cos \6 + i sin \ 6).
The relations which hold between the different roots of (1) are of
the greatest importance in the theory of functions*. For our present
purposes we require only the two which follow.

(i) Any symmetric polynomial in the roots y u y 2 , ,y n of (1) is

a rational function of x.

* For fuller information the reader may be referred to Appell and Goursat's
THorie den fonctions algibriques.
 . — , . . ,

6 ELEMENTARY FUNCTIONS AND THEIR CLASSIFICATION [II

(ii) Any symmetric polynomial in y2 y3 , , ...,yn is a polynomial in

#! with coefficients which are rational functions oix.
The first proposition follows directly from the equations

2y1 y2 -^=(-i)'( Pn -./A) J («=1, 2, •..*)

To prove the second weobserve that
2 y2 ys-..y.= 2 yiy-L-y.-i-yi 2 ya y,...y,-i.
2, 3, . . 1, 2, . . 2, 3, . .

so that the theorem is true for 2y2 y3 ...y, if it is true for 2y^y% ...y,_i-
It is certainly true for

yi+y3+ +y
n =d/i+y2+ +yn) -yi-
It is therefore true for Sy2 y3 ...y„ and so for any symmetric polynomial io
y2,y3,--,y»-

4. Elementary functions which are not rational or algebraical are

called elementary transcendental functions or elementary transcendents.
They include all the remaining functions which are of ordinary occur-
rence in elementary analysis.
The trigonometrical (or circular) and hyperbolic functions, direct
and inverse, may all be expressed in terms of exponential or logarithmic
functions by means of the ordinary formulae of elementary trigonometry.
Thus, for example,

sm#=
e
ix_ e
—^ —
-ix
t

sinh# = ——
<?-e-
-
x

arc tan x= I jog arg tanh x = \ log

(£g) ,
(£|) .

There was therefore no need to specify them particularly in our

definition.
The elementary transcendents have been further classified in a
manner first indicated by Liouville* According to him a function is
a transcendent of the first order if the signs of exponentiation or of
the taking of logarithms which occur in the formula which defines
it apply only to rational or algebraical functions. For example
«-*, **+*%/(log«r)
are of the first order ; and so is

arc tan y
V(i+*y
'*
'Memoire sur la classification des transcendantes, et sur Pimpossibilit6
d'exprimer les racines de certainea Equations en fonction finie explicite dea
coefficients', Journal de matMmatiques, ser. 1, vol. 2, 1837, pp. 56-104 ;
'
Suite du
memoire...', ibid. vol. 3, 1838, pp. 523-546.
3-4] ELEMENTARY FUNCTIONS AND THEIR CLASSIFICATION 7

where y is defined by the equation

f-y-x=0;
and so is the function y defined by the equation
x
y — y-e \ogx = 0.
5

An elementary transcendent of the second order is one defined by

a formula in which the exponentiations and takings of logarithms are
applied to rational or algebraical functions or to transcendents of the
first order. This class of functions includes many of great interest and
importance, of which the simplest are
e
e , log log x.

It also includes irrational and complex powers of x, since, e.g.,

the function a? = &***;

and the logarithms of the circular functions.
It is of course presupposed in the definition of a transcendent of the
second kind that the function in question is incapable of expression as
one of the first kind or as a rational or algebraical function. The
function
glogKM

where R (x) is rational, is not a transcendent of the second kind, since

it can be expressed in the simpler form R (x).
It is obvious that we can in this way proceed to define transcendents
of the wth order for all values of n. Thus
log log log x, log log log log x,

are of the third, fourth, orders.

Of course a similar classification of algebraical functions can be and
has been made. Thus we may say that
Jx, J{x + Jx), J{x + J(x + Jx)},
are algebraical functions of the first, second, third, orders. But
the fact that there is a general theory of algebraical equations and
therefore of implicit algebraical functions has deprived this classifica-
tion of most of its importance. There is no such general theory

of elementary transcendental equations*, and therefore we shall not

* The natural generalisations of the theory of algebraical equations are to

be found in parts of the theory of differential equations.
See Kouigsberger,
'
Bemerkunger. zu Liouville's Classificirung der Transoendenten ', Math. Annalen,
vol. 28, 1886, pp. 483-492.
8 THE INTEGRATION OF ELEMENTARY FUNCTIONS L
nl

rank as 'elementary' functions denned by transcendental equations

such as
y = x logy,
but incapable (as Liouville has shown that in this case y is incapable)

of explicit expression in finite terms.

5. The preceding analysis of elementary transcendental functions

rests on the following theorems :

(a) e* is not an algebraical function of % ;

(b) log x is not an algebraical function of x ;

(c) log x is not expressible in finite terms by means of signs of

exponentiation and of algebraical operations, explicit or implicit* ;

(d) transcendental functions of the first, second, third, orders

actually exist.

A proof of the first two theorems will be given later, but limitations
of space will prevent us from giving detailed proofs of the third and
fourth. Liouville has given interesting extensions of some of these
theorems : he has proved, for example, that no equation of the form
Ae** + Befiv + ... + ReM> = S,

where p, A, B, ..., R, ^are algebraical functions of x, and a, y3, ..., p

different constants, can hold for all values of x.

III. The integration of elementary functions.

Summary of results
In the following pages we shall be concerned exclusively with the
problem of the integration of elementary functions. We shall endeavour
to give as complete an account as the space at our disposal permits of
the progress which has been made by mathematicians towards the
solution of the two following problems :

(i) if f (x) is an elementary function, how can we determine

whether its integral is also an elementary function 1
(ii) if the integral is an elementary function, how can we find it I

It would be unreasonable to expect complete answers to these

questions. But sufficient has been done to give us a tolerably com-
plete insight into the nature of the answers, and to ensure that it

* For example, log x cannot be equal to e", where y is an algebraical function

of x.
1-2] THE INTEGRATION OF ELEMENTARY FUNCTIONS 9

shall not be difficult to find the complete answers in any particular

case which is at all likely to occur in elementary analysis or in its
applications.
It will probably be well for us at this point to summarise the
principal results which have been obtained.

1. The integral of a rational function (iv.) is always an elementary

function. It is either rational or the sum of a rational function and
of a finite number of constant multiples of logarithms of rational
functions (iv., 1).
If certain constants which are the roots of an algebraical equation
are treated as known then the form of the integral can always be
determined completely. But as the roots of such equations are not in
general capable of explicit expression in finite terms, it is not in
general possible to express the integral in an absolutely explicit form
(iv. ; 2, 3).

We can always determine, by means of a finite number of

the elementary operations of addition, multiplication, and division,
whether the integral is rational or not. If it is rational, we can
determine it completely by means of such operations ; if not, we
can determine its rational part (iv. ; 4, 5).

The solution of the problem in the case of rational functions may

therefore be said to be complete ; for the difficulty with regard to the
explicit solution of algebraical equations is one not of inadequate
knowledge but of proved impossibility (rv., 6).

2. The integral of an algebraical function (v.), explicit or implicit,

may or may not be elementary.
If y is an algebraical function of x then the integral Jydx, or, more
generally, the integral

\R(x,y)dx,
I'
where B denotes a rational function, is, if an elementary function,
either algebraical or the sum of an algebraical function and of a finite
number of constant multiples of logarithms of algebraical functions.
All algebrai cal functions w hich occur in the integ ral ar e rational
functions qfx~anoTy(y . ; 11-14, 18).
These theorems give a precise statement of a general principle
enunciated by Laplace* : ' lintegrale d'une fonction differentielle

* ThSorie analytique des probability, p. 7.

10 THE INTEGRATION OF ELEMENTARY FUNCTIONS [ill

(algebrique) ne pent contenir cFautres quantites radicaux que celles

qui entrent dans cette fonction '
; and, we may add, cannot contain
exponentials at all. Thus it is impossible that

dx
/;V(i +
'
o
should contain e* or the appearance of these functions in
J(l—x) :

the integral could only be apparent, and they could be eliminated

before differentiation. Laplace's principle really rests on the fact, of
which it is easy enough to convince oneself by a little reflection

and the consideration of a few particular cases (though to give a , s

rigorous proof is of course quite another matter), that differentiation

will not eliminate exponentials or algebraical irrationalities. Nor, we
may add, will it eliminate logarithms except when they occur in the
simple form
A log <t> O),
where A is a constant, and this is why logarithms can only occur
in this form in the integrals of rational or algebraical functions.
We have thus a general knowledge of the form of the integral
of an algebraical function y, when it is itself an elementary
function. Whether this is so or not of course depends on the nature
of the equation /(ar, y)=0 which defines y. If this equation, when
interpreted as that of a curve in the plane (x, y), represents a unicursa l
curve, i.e. a curv e which has the maximum number of double point s
possible for a curve of its degree, or whose deficiency is zero, then
x and y can be expressed simultaneously as rational functions of a third
variable t, and the integral can be reduced by a substitution to that
of a rational function (v. ; 2, 7-9). In this case, therefore, the integral
is always an/elementary function. But this condition, though sufficient,
is not necessary. It is in general true that, when f(x, y) = is not
unicursal, the integral is not an elementary function but a new
'

transcendent ; and we are able to classify these transcendents according

to the deficiency of the curve. If, for example, the deficiency is unity,
then the integral is kind known as
in general a tianscendent of the
whose characteristic is that they can be transformed
elliptic integrals,

into integrals containing no other irrationality than the square root of

a polynomial of the third or fourth degree (v., 20). But there are in-
finitely many cases in which the integral can be expressed by algebraical
functions and logarithms. Similarly there are infinitely many cases
in which integrals associated with curves whose deficiency is greater
2-3] THE INTEGRATION OF ELEMENTARY FUNCTIONS 11

than unity are in reality reducible to elliptic integrals.Such ab-

normal cases have formed the subject of many exceedingly interesting
researches, but no general method has been devised by which we can
always tell, after a finite series of operations, whether any given
integral is really elementary, or elliptic, or belongs to a higher order
of transcendents.
When f(x, y) = is unicursal we can carry out the integration
completely in exactly the same sense as in the case of rational functions.
In particular, if the integral is algebraical then it can be found by
means of elementary operations which are always practicable. And
it has been shown, more generally, that we can always determine by
means of such operations whether the integral of any given algebraical
function is algebraical or not, and evaluate the integral when it is

algebraical. And although the general problem of determining whether

any given integral is an elementary function, and calculating it if it
is one, has not been solved, the solution in the particular case in which

the deficiency of the curve f(x, y) =Q is unity is as complete as it is

reasonable to expect any possible solution to be.

3. The theory of the integration of transcendental functions

(vi.) is naturally much less complete, and the number of classes
of such functions for which general methods of integration exist is
very small. These few classes are, however, of extreme importance
in applications (vi. ; 2, 3).

There is a general theorem concerning the form of an integral of

a transcendental function, when it is itself an elementary function,
which is quite analogous to those already stated for rational and
algebraical functions. The general statement of this theorem will be
found in vi., § 5 ; it shows, for instance, that the integral of a rational
function of x, e° and log# is either a rational function of those
functions or the sum and of a finite
of such a rational function
number of constant multiples of logarithms of similar functions.
From this general theorem may be deduced a number of more precise
results concerning integrals of more special forms, such as

I yex dx, I y log x dx,

where y is an algebraical function of x (vi. ; 4, 6).

 .

12 RATIONAL FUNCTIONS [
IV

IV. Rational functions

1. It is proved in treatises on algebra* that any polynomial

Q (x) = b xn + b 1 af- l +...+ b n

can be expressed in the form

b (x - Oa)»> (x - a2 > . . . (x - a r )»r ,

where Wjj n2 , are positive integers whose sum is n, and a^a^, ... are
constants ; and that any rational function R (x), whose denominator
is Q (x), may be expressed in the form

A x* + A^ + ... + A p+ i j-^^+r
J
^i + - + ^lV»}'
where Ad, A lt ... ,
/J
M ,
... are also constants. It follows that

\B{x)dx = A -+Ai— + ... + Ap x+C

°p+l p
_ A^
J

+ i{^og(x- as) -^- ..._

( I rf }

From this we conclude that the integral of any rational function is an

elementary junction which is rational save for the possible presence
of logarithms of rational functions. In particular the integral will be
rational if each of the numbers /?„, ! is zero : this condition is evidently
necessary and sufficient. A necessary but not sufficient condition is

that Q (x) should contain no simple factors.

The integral of the general rational function may be expressed in
a very simple and elegant form by means of symbols of differentiation.
We may suppose for simplicity that the degree of P (x) is less than
that of Q (x) ; this can of course always be ensured by subtracting
a polynomial from B(x). Then

Q(x)

~
1 3^ P(x)
(fij - 1) ! (b, - 1) ! • (nr - 1) ! Sa/." 1 3a,"," 1 . . . BO-i Q (x) '

where Qo(#) = b (x - °-i) (x - a,) . . . (x - a^).

Now t^-W = sr„ (x) + 2 -.

>'' ,
,

* See, e.g., Weber's Trixiti d'algebre sup£rieure (French translation by J. Griess,

Paris, 1898), vol. 1, pp. 61-64, 143-149, 350-353 ; or Chrystal's Algebra, vol. 1,
pp. 151-162.
1] HATIONAL FUNCTIONS 13

where nr (x) is a polynomial ; and so

/ B (x) dx

= ^
^.-l)^^^-l)^^^ [^^W ^l^l(O ]0g(^ ,0]'
^
i

where n (x) = j
vr (x) dx.

is also a polynomial, and the integral contains no polynomial term,

since the degree of P (x) is less than that of Q (x). Thus II (x) must
vanish identically, so that

I B (x) dx
1 d^~ r [ r P (a„) 108
=
K-l)!...(Wr -l) !
3<i-... 3 <W Li^oS ^"^. '

For example
/"
d# o
2
J
1 , /^-aX]
, ~ g \x~-b)\ "

J{f>-a)(*-&)} dadb \a~^b

That Ho (x) is annihilated by the partial differentiations performed on it

may be verified directly as follows. We obtain Ho(x) by picking out from

the expansion

P(x)
XT \ X X* / \ X X* J
the terms which involve positive powers of x. Any such term is of the form

Axr-r-s,-*t -... ai
*.
a /2 .. m
where *i + sa + ... ^v-r^TO — r,
m being the degree of P. It follows that

Si+s2 + ...<n-r=(m 1 — l) + (m,i -l) + ...;

so that at least one of *j, *2 > •• must be less than the corresponding one of
%-l, m2 -l, ....

It has been assumed above that if

F(x,a)=jf(x,a)dx,

then ~- = | ^- d#.
'a J da'
14 RATIONAL FUNCTIONS [l v

The first equation means that /= =— and the second that £ = 5—5- -^ i*

follows from the first that ^

da
= *-=-
oaox
, what has really been assumed is that

&F = &F
dadx dxda'
It is known that this equation is always true for x=x , a=a if a circle
can be drawu in the plane of (x, a) whose centre is (x , ao) and within which
the differential coefficients are continuous.

2. It appears from § 1 that the integral of a rational function is

in generalcomposed of two parts, one of which is a rational function
and the other a function of the form
2.4 logO-a) (1).

We may call these two functions the rational part and the transcen-
dental part of the integral. It is evidently of great importance to
show that the transcendental part of the integral is really transcen-
'
'

dental and cannot he expressed, wholly or in part, as a rational or

algebraical function.
We are not yet in a position to prove this completely*; but we can
take the first step in this direction by showing that no sum of the
form (1) can be rational, unless every A is zero-

Suppose, if possible, that

S4 1og(*-«)=||g (2),

where P and Q are polynomials without common factor. Then

, a _PQ-pq
(8).

T
Suppose now that {x-p) is a factor of Q. Then PQ-PQ' is
divisible by (x-pf- 1 and by no higher power of x-p. Thus the
right-hand side of (3), when expressed in its lowest terms, has a factor
1
(x-pY* in its denominator. On the other hand the left-hand side,
when expressed as a rational fraction in its lowest terms, has no
repeated factor in its denominator. Hence r = 0, and so Q is a con-
stant. We may therefore replace (2) by

2^1ogO-a) = P(»,
and (3) by 2 ^ = P'(^>

Multiplying by x- a, and making x tend to a, we see that .4=0.

* The proof will be completed in v., 16.
I-4 ] RATIONAL FUNCTIONS 15

3. The method of gives a complete solution of the problem

§ 1 if
the roots of Q(x) = can be determined; and in practice this is
usually the case. But this case, though it is the one which occurs
most frequently in practice, is from a theoretical point of view an
exceedingly special case. The roots of Q (x) = are not in general
explicit algebraical functions of the coefficients,
and cannot as a rule
be determined in any explicit form. The method of partial fractions
is therefore subject to serious limitations. For example, we cannot
determine, by the method of decomposition into partial fractions, such
an integral as
4^+21^ + 2^-3^-3
— ,

P (a?-x+ l)
a
ax,

or even determine whether the integral is rational or not, although it

is in reality a very simple function. A high degree of importance

therefore attaches to the further problem of determining the integral
of a given rational function so far as possible in an absolutely explicit
form and by means of operations which are always practicable.
It is easy to see that a complete solution of this problem cannot be
looked for.

Suppose for example that P(x) reduces to unity, and that Q(x)—0 is
an equation of the fifth degree, whose roots a,, a^, ...as are all distinct and
not capable of explicit algebraical expression.

Then [RWdx-l W*-?*

J l V {<h)

=lo g n{(^- a ,) 1/Q '^>},

and it is only if at least two of the numbers Q* (a,) are commensurable that
any two or more of the factors (x-a,) 1 ^^
can be associated so as to give
a single term of the type A log iS (x), where S (x) is rational. In general this
will not be the case, and so it will not be possible to express the integral in

any finite form which does not explicitly involve the roots. A more precise
result in this connection will be proved later (§ 6).

4. The and most important part of the problem has been

first

solved by Hermite, who has shown that the rational part of the
integral can always be determined without a knowledge of the roots of

Q (x), and indeed without the performance of any operations other

than those of elementary algebra*.

* The following account of Hermite's method is taken in substance from

Goursat's Court d'avalyse mathimatique (first edition), t. 1, pp. 238-241.

 : '

16 RATIONAL FUNCTIONS [
IV

Hermite's method depends upon a fundamental theorem in

elementary algebra* which is also of great importance in the ordinary

theory of partial fractions, viz.
'
Ij X andx X
2 are two polynomials in x
which have no common
factor, X
and 3 any third polynomial, then we can determine two poly-
nomials Au At, such that
A-lJTi + A^Xi = JT3 .'
Suppose that Q (*) = $,&»&•. ..Qf,
Q1} denoting polynomials which have only simple roots and of
...

which no two have any common factor. We can always determine

Qi, by elementary methods, as is shown in the elements of the
theory of equations t-
We can determine B and A-,, so that

BQ + A1 1 Q?Q?...Q? = P,
and therefore so that
'
t
Q Qi WQf...Q t

By a repetition of this process we can express R (x) in the form

4} , 4* 4. " j. 4j
Qf
"
'Qi Q t

and the problem of the integration of B {x) is reduced to that of the

integration of a function
A
<T
where Q is a polynomial whose roots are all distinct. Since this is so,

Q and its derived function Q' have no common factor : we can therefore
determine C and D so that
CQ + DQ = A.
Hence
\*dx=\ G ^ D(tdx
}q" 1
v-ij dxKQ"- 1 ;
D 1 + f E dx,
i

(v -i)Qr-
1
Jqr-

where E=C+ v — -.
1
—
* See ChrystaPs Algebra, vol. 1, pp. 119 et seq.

t See, for example, Hardy, A course of pure mathematics (2nd edition), p. 208.
 B X .

4-5] RATIONAL FUNCTIONS 17

Proceeding in this way, and reducing by unity at each step the

power
of 1/Q which figures under the sign of integration, we ultimately
arrive at an equation

j— dx = R„ 0) + j- dx,

where R v is a rational function and Sa polynomial.

The
integral on the right-hand side has no rational part, since all
the roots of Q are simple (§ 2). Thus the rational part of JR (x) dec is
R 2 (x)+R3 (x) + ... + R,(x),
and has been determined without the need of any calculations other
it

than those involved in the addition, multiplication and division of

polynomials*.

5. (i) Let us consider, for example, the integral

fix*
4 + 21x* + 2^ - 3a- 2 -3
dx.
I {xi-x+iy
mentioned above (§ 3). We require polynomials A x , A2 such that

A 1 X + A X =X
1 2 2 3 (1),
where
X^x'-x + l, X = 1afi-\, 2 X = 4a +21s + 2a? -3a* -3.
3
9 8

In general, if the degrees of X x

and X are m and m and that of X
2 1 2, 3
does not exceed m + m 2 - 1, we can suppose that the degrees of A and A do
1
x 2
not exceed m -\ and m -l respectively. For we know that polynomials
2 1

Bx and B2 exist such that

B x Xx + 2 X 2 = 3.

If B x
is of degree not exceeding mj - 1, we take A 1 = B 1 , and if it is of higher
degree we write
B =L X2 +A 1 l l ,

where A x
is of degree not exceeding m 2 — 1. Similarly we write

B2 = L 2 X + A g x
We have then
(L x +L 2 ) X X + A X +A 2 X2 =X3
1 2 1 1 .

In this identity x L L2 or both

vanish identically, and in any case we
or may
see, by equating to zero the coefficients of the powers of x higher than the
(m 1 +m 2 - l)th, that L + L2
x
vanishes identically. Thus X 3 is expressed in
the form required.
The actual determination of the coefficients in Ai and A2 is most easily
performed by equating coefficients. We have then m + m2 l
linear equations

* The operation of formiDg the derived function of a given polynomial can of

course be effected by a combination of these operations.
18 RATIONAL FUNCTIONS [
Iv

in the same number of unknowns. These equations must be consistent,

since we know that a solution exists*.
If X3 is of degree higher than m,i + m 2 -l, we must divide it by r X2 and X
express the remainder in the form required.
In this case we may suppose A x of degree 5 and A.2 of degree 6, and we
find that
A =-3x\x
A 2 =x3 + 3.
Thus the rational part of the integral is
3? + 3
x 7 -x+l'
and, since — 3x2 + (x3 + 3)'=0, there is no transcendental part.
(ii) The following problem is instructive : to find the conditions that
ax 2 + 2Bx+y ,

/<(Ax +2Bx+C)
2 2

may be rational, and to determine the integral when it is rational.

We shall suppose that Ax2 + 2Bx + C is not a perfect square, as if it were
the integral would certainly be rational. We can determine p, q and r
so that
p(Ax2 +2Bx+C) + 2{qx+r){Ax + B) = ax2 + 2Px+y,
and the integral becomes

\ ambx+c / ^
p ~ +r) dx
(
d\ (i^rhsxTc)
qx + r f dx
~ Ax2 + 2Bx+C +kP + q> J Ax2 + 2Bx+C
The condition that the integral should be rational is therefore p + q = 0.
Equating coefficients we find

A(p + 2q) = a, B(p + q) + Ar = P, Cp + 2Br = y.

Hence we deduce
a a
P = ~A> q = A> r=
A'
and Ay + Ca = 22?j9. The condition required is therefore that the two quadratics
ax2 + 2^x+y and Ax2 + 2Bx+C should be harmonically related, and in this
case
ax 2 + 2^x+y , ax + fi
/<{Ax + 2Bx+Gf
2 A{Axi + 2Bx+Cy
(iii) Another method of solution of this problem is as follows. If we write
Ax2 + 2Bx + C= A (x - X) {x - p%
and use the bilinear substitution
Xyj-^
y+i '

then the integral is reduced to one of the form

"
ay 2 + 2by+c
/' **•
».

* It is easy to show that the solution is also unique.

5-6] RATIONAL FUNCTIONS 19

and is rational if and only if 6 = 0. But this is the condition that the
quadratic ay 2 + 2by + c, corresponding to ax 2 + 2ftx+y, should be harmonically
related to the degenerate quadratic y, corresponding to Ax2 + 2Bx+C. The
result now follows from the fact that harmonic relations are not changed by
bilinear transformation.
It is not difficult to show, by an adaptation of this method, that

(aX* + 2l3x+y)(a xi + W x + y ...(anx* + 2l3nx;+yn

l 1 1) )

P {Ax* + 2Bx + Cy + *

is rational if all the quadratics are harmonically related to any one of those
in the numerator. This condition is sufficient but not necessary.

(iv) As a further example of the use of the method (ii) the reader may
show that the necessary and sufficient condition that

/(*)
, dx.
1 \F{x)f
where f and F are polynomials with no common factor, and F has no repeated
factor, should be rational, is that f'F'—fF" should be divisible by F.

6. It appears from the preceding paragraphs that we can always

and can find the complete integral
find the rational part of the integral,
if we can find Q = 0. The question is naturally
the roots of (x)
suggested as to the maximum of information which can be obtained
about the logarithmic part of the integral in the general case in which
the factors of the denominator cannot be determined explicitly. For
there are polynomials which, although they cannot be completely resolved
into such factors, can nevertheless be partially resolved. For example

x -2xli s
-2x -x -2x* + 2x+l = {x
7 i 7 l 7
+ x -i)(x -si?- 2x- 1),
xli -2a*- 2x7 -2x4 - 4x3 -x2 + 2x + 1
= {x7 + a? J2 + x(j2 -\)-l\{x7 -a?j2-x (J2 + 1) - 1}.

The factors of the first polynomial have rational coefficients : in the

language of the theory of equations, the polynomial is reducible in the
rational domain. The second polynomial is reducible in the domain
formed by the adjunction of the single irrational ^2 to the rational

domain*
We may suppose that every possible decomposition of Q (x) of this
nature has been made, so that

* See Cajori, An introduction to the modern theory of equations (Macmillan,

1904); Mathews, Algebraic equations [Cambridge tracts in mathematics, no. 6),
pp. 6-7.
20 RATIONAL FUNCTIONS [IV

Then we can resolve R (x) into a sum of partial fractions of the type

X
/ W/ '

and so we need only consider integrals of the type

:
dx >
/Q
where no further resolution of Q is possible or, in technical language,
Q is irreducible by the adjunction of any algebraical irrationality.
Suppose that this integral can be evaluated in a form involving only
constants which can be expressed explicitly in terms of the constants
which occur in PjQ. It must be of the form
vl 1 logX + 1
... + A k logXk (1),

where the A's are constants and the Ts polynomials. We can

suppose that no X
has any repeated factor £"*, where £ is a polynomial.
For such a factor could be determined rationally in terms of the co-
of X, and the expression (1) could then be modified by
efficients

taking out the factor f from X and inserting a new term mA log £.
And for similar reasons we can suppose that no two Xs have any
factor in common.

Now q=A X\
j^- + Aa Xi
^+... + A . X,.
x
,

1 2 k

or PX-lXi ... X k — (JzAyXi ... Ji. „~iX X —X w v +i k .

All the terms under the sign of summation are divisible by X^ save the
first, which is prime to X,. Hence Q must be divisible by X
1 : and
similarly, of course, by X X 2 , 3 , ..., X k. But, since P is prime to Q,
XjX X 2 k is divisible by Q. Thus Q must be a constant multiple of
X X ...X
1 i k . But Q is ex hypothesi not resoluble into factors which
contain only explicit algebraical irrationalities. Hence all the X'a
save one must reduce to constants, and so P must be a constant
multiple of Q', and
fP
Qdx = A\ogQ,
l\
where A is a constant. Unless this is the case the integral cannot be
expressed in a form involving only constants expressed explicitly in
terms of the constants which occur in and Q. P
Thus, for instance, the integral
dx
h f x^ + ax +b
6-7] RATIONAL FUNCTIONS 21

cannot, except in special cases*, be expressed in a form involving only

constants expressed explicitly in terms of a and b ; and the integral
5x* + c ,
ax
h '
xfi + ax+b
can in general be so expressed if and only if c—a. We thus confirm an
inference made before (§ 3) in a less accurate way.
Before quitting this part of our subject we may consider one further
problem : under what circumstances is

R(x)dx=A log R x (x)

I
where A isa constant and /i, rational ? Since the integral has no rational
part, it is clear that Q (x) must have only simple factors, and that the degree
of P (x) must be less than that of Q (x). We may therefore use the formula
'

R (x) dx = log n {(x - a,) P(a,)/Q ( ",)

}.
/
The necessary and sufficient condition is that all the numbers P(a,)/(? (a,)

should be commensurable. If e.g.

then (a - y)j(a - /3) and ($ - y)/(S — a) must be commensurable, i.e. (a — y)l($ - y)

must be a rational number. If the denominator is given we can find all the
values of y which are admissible for y = {aq — f3p)/(q-p), where p and q are
:

integers.

7. Our discussion of the integration of rational functions is now

complete. It has been throughout of a theoretical character. We
have not attempted to consider what are the simplest and quickest
methods for the actual calculation of the types of integral which occur
most commonly in practice. This problem lies outside our present
range : the reader may consult

0. Stolz, Grundziige der Differential- und-integralrechnung, vol. 1,

ch. 7 :

J. Tannery, Lemons d'algebre et d'analyse, vol. 2, ch. 18 :

Ch.-J. de la Vallee-Poussin, Cours d'analyse, ed. 3, vol. 1, ch. 5 :

T. J. I'A. Bromwich, Elementary integrals (Bowes and Bowes,

1911):
6. H. Hardy, A course of pure mathematics, ed. 2, ch. 6.

* The equation x 6 + ax + 6 = is soluble by radicals in certain cases. See

Mathews, I.e., pp. 52 et seq.
 y

22 ALGEBRAICAL FUNCTIONS [
V

V. Algebraical Functions

1. "We shall now consider the integrals of algebraical functions,

explicit or implicit. The theory of the integration of such functions is
far more extensive and difficult than that of rational functions, and
we can give here only a brief account of a few of the most important
results and of the most obvious of their applications.

Vn are algebraical functions of x, then any algebraical

If V\i Vii •••>

function z of x, yu ... yn is an algebraical function of x.

,
This is
obvious if we confine ourselves to explicit algebraical functions. In
the general case we have a number of equations of the type
mv + P.. 1
P*,„ 0*0 y> 1 0) yf- + • • • + Pv,m v 0) = o = i, 2, . .
. , »),

and P m+ + Pm (x, yu = 0,
(x, yu ... ,yn ) z ... ..., yn)
where the P's represent polynomials in their arguments. The elimina-
tion of y1} y2 ... yn between these equations gives an equation in &
, ,

whose coefficients are polynomials in x only.

The importance of this from our present point of view lies in the
fact that we may consider the standard algebraical integral under any
of the forms

Jydx,
/»
where/(#,y)= ;

\R(x,y)dx,

where f(x, y) = and R is rational ; or

\R{x,yi,.--,yn )dx,

where fx (x,y) = 0, ..., /„ {as, n ) = 0. It is, for example, much more

convenient to treat such an irrational as
x-J{x+ \)- J(x-1)
1 + J(x+1) + J{x-l)
as a rational function of x, ylt y2 where yx = J{x +
,
1), y2 = J{x - 1),

y1 = x+ y£ = x-
2
1, 1, than as a rational function of x and y, where
y= J(<c + l)+J{x-l),
y* - 4:xyz + 4 = 0.
To treat it as a simple irrational y, so that our fundamental equation is

(x - yf - ix - yf (1 + yf + 4 (1 + yy =
is evidently the least convenient course of all.
1-2] ALGEBRAICAL FUNCTIONS 23

Before we proceed to consider the general form of the integral of an

algebraical function we shall consider one most important case in which
the integral can be at once reduced to that of a rational function, and
is therefore always an elementary function itself.

2. The class of integrals alluded to immediately above is that

covered by the following theorem.

If there is a variable t connected with x and y (or yit y2 , ... , yn )

by rational relations
x = Rl (i), y = R*(t)
(or yi = -B2 (1)
(*)> Vi - R^ (*)> • •
)> then the integral

hR (x, y) dx

(or jR (x, yx ,
... , ya) dx) is an elementary function.

The truth of this proposition follows immediately from the

equations
R(x,y) = R{R1 (t),Ri (t)) = S(t),

^ =Rat)=T(t),
t

JR 0, y) dx = JS (t) T(t)dt= JU (t) dt,

where all the capital letters denote rational functions.
The most important case of this theorem is that in which x and y
are connected by the general quadratic relation
2
(a, b, c,f g, h\x,y, 1) = 0.
The integral can then be made rational in an infinite number of ways
For suppose that (f, rf) is any point on the conic, and that

(y—o) = t(a;-€)
is any line through the point. If we eliminate y between these
equations, we obtain an equation of the second degree in x, say

where To, 7\, T* are polynomials in t. But one root of this equation
must be £, which is independent of t ; and when we divide by x — £ we
obtain an equation of the first degree for the abscissa of the variable
point of intersection, in which the coefficients are again polynomials
in t. Hence this abscissa is a rational function of t ; the ordinate of
the point is also a rational function of t, and as t varies this point
 ;

24 ALGEBRAICAL FUNCTIONS [V

coincides with every point of the conic in turn. In fact the equation
of the conic may be written in the form
au2 + 2hwv + bv + 2(at + hy+g)u + 2(h£ + by +/) v =
i
0,

where u = x — £, v=y — y, and the other point of intersection of the line

v = tu and the conic is given by

+
~ 2 {a$ + hr) + g + t (k£ + by +/)}
x~ i
a + 2ht + b<?

y~v
~ 2t{a$+hy + g + t (h£ + by +/)}
a+2ht + bf
An alternative method is to write

aaP + 2hxy + by = b(y — px)l

{y - px),
so that y-fuc = Q and y-ix'x = are parallel to the asymptotes of
the conic, and to put
y — nx = t.

Then y -//# = - -*
—^—
and from these two equations we can calculate x and y as rational
functions of t. The principle of this method is of course the same as
that of the former method (£ y) is now at infinity, and the pencil of
:
,

lines through (£, y) is replaced by a pencil parallel to an asymptote.

The most important case is that in which b = — 1, /= h = 0, so that
y = aa? + 2gx + c.
2

The integral is then made rational by the substitution

2 (a£ + g-ty) 2t(a£ + g-ty)
x ~^~ V~ V
a-f ' a-t 2

where $, y are any numbers such that

rf = a? + 2g£, + c.

We may for instance suppose that £ = 0, y = Jc ; or that y = 0, while £

is a root of the equation a? -r 2g£ + c = 0. Or again the integral is

made rational by putting y-x Ja = t, when

X~
f-c _ (i? + c)Ja- 2gt
2{tja-g)' y ~ 2{tja-g) '

3. We shall now consider in more detail the problem of the calculation of

I R (x, y) dx,

where y = *JX= Kl(ax i + %bx+c) %

* We now write 6 for g for the sake of symmetry in notation.
2-4] ALGEBRAICAL FUNCTIONS 25

The most interesting case is that in which a, b, c and the constants which
occur in R are real, and we shall confine our attention to this case.

Let
r<*3)
R{Xiy)

where P and Q are polynomials. Then, by means of the equation

y = ax + 2bx+c,
2 2

R (x, y) may be reduced to the form

A + BJX = (A4-B sfX)(C-DJX)

C+DJX C*-D*X
where A, B, C, D are polynomials in x; and so to the form M+NJX, where
M and N
are rational, or (what is the same thing) the form

where P and Q are rational. The rational part may be integrated by the
methods of section IV., and the integral

kijx dx
may be reduced to the sum of a number of integrals of the forms

( xf j [ dx f tx + rj
ax
Jjx '
Jix-pyjx' ](ax*+wx+ y yjx ( >'

where p, £, ij, a, |3, y are real constants and r a positive integer. The result
is generally required in an explicitly real form : and, as further progress
depends on transformations involving p (or a, /3, y), it is generally not
advisable to break up a quadratic factor ax 2 + 2^x + y into its constituent
linear factors when these factors are complex.
All of the integrals (1) may be reduced, by means of elementary formulae
of reduction*, to dependence upon three fundamental integrals, viz.

dx £x+ v
JJX'
tdx_ f
](x-p)JX'
[
)(ax*+'2,ISx+y)JX
ax w
4. The first of these integrals may be reduced, by a substitution of the
type x= t+k, to one or other of the three standard forms

f dt f dt f dt
V(«2 -m2 )'
where m > 0. These integrals may be rationalised by the substitutions
2mu _ 2mu to(1 + m2
t=
)
<=
I+w2 '
T^tf' 2m
;

but it is simpler to use the transcendental substitutions

t = msm<t>, t=msmh<j>, t=mcosh<f>.

* See, for example, Bromwich, I.e., pp. 16 et seq.

 ,

26 ALGEBRAICAL FUNCTIONS [V

These last substitutions are generally the most convenient for the reduction
of an integral which contains one or other of the irrationalities

»/K-«2 ), V^+ra 8 ). V(*

2 -™ 2
),

though the alternative substitutions

t=mta.nh<t>, t = 7ntan<f>, t=maeo<f>
are often useful.
It has been pointed out by Dr Bromwich that the forms usually given in
text-books for these three standard integrals, viz.

arc sin — , arg sinh — arg cosh —

m m m
are not quite accurate. It is obvious, for example, that the first two of these
functions are odd functions of m, while the corresponding integrals are even
functions. The correct formulae are

arc sin
.

\m\'
—
t
:
, arg sinh
. ,

;
—=
\m\
t ,
log
6
t + J(0->-m?)
p—
\m\

and + arg cosh- —L= log 1 1

\m\ |
m
where the ambiguous sign is the same as that of t. It is in some ways more
convenient to use the equivalent forms

arc tan . „ —-=, argtanh-77-= = , ar^tanh—77-= 5:.

5. The integral | -, r—
}{*-P)s
may be evaluated in a variety of ways. •

If p is a root of the equation X=0, then X may be written in the form

a {x-p) (x—q), and the value of the integral is given by one or other of the
formulae
f dx 2 //x-q\
J (x-p) J{(x -j>) (x - q)}
~ q~^> \x~^>) '

\J
dx 2
hl{x-p)M 3{x-p)W
We may therefore suppose that p is not a root of X*=0.
(i) We may follow the general method described above, taking

i=P, v = J(ap*+2bp+c)*.
Eliminating y from the equations
y*=ax*+2bx+c, y-r, = t(x-£),
and dividing by x - £, we obtain
t*(x-£) + 2 v t-a(x+£)-2b=0,

--**- dx
and so = dJl
t'-a t(x-$) + n y
* Cf. Jordan, Cours d' analyse, ed. 2, vol. 2, p. 21.
 p '

4- 5 ] ALGEBRAICAL FUNCTIONS 27

dx dt
Hence {- g -2 f— -i)(fi- a y

and so

f <& /• dt 1, , » ,
,0

7(a^+26p + c) l0g{ ^
= (ffip2 + 2 ^ +<!) - ffiy " 6} -

If ap2 + 2ftp + c<0 the transformation is imaginary.

Suppose, e.^r., (a) y=V(#+l), p=0, or (6) y=J(x-l\ p=0. We find

(a)
/^T)= l0g(< -^
where <»* + 2*-l = 0,
or f
_ -l+V(*+l) .

and
(6)
litthr)— ***&-*>>
;V(*-1)"
where tfr+2i<- ] =0.
Neither of these results is expressed in the simplest form, the second in
particular being very inconvenient.

(ii) The most straightforward method of procedure is to use the

substitution
1

*-p=r
We then obtain
f dx _ f dt
>vW2 + 2M+Ci)'
}{x-p)y~ Jl
where a lt b lt Cj are certain simple functions of a, b, c, and p. The further
reduction of this integral has been discussed already.

(iii) A third method of integration is that adopted by Sir G. Greenhill*,

who uses the transformation
^'(ax2 + 2bx+c)
x—
It will be found that

f dx _ [ d[
J (x-p)JX~JJ{(ap l + 2bp + c) fl + b*-ac}
which is of one of the three standard forms mentioned in § 4.

* A. G. Greenhill, A chapter in the integral calculus (Francis Hodgson, 1888),

p. 12 : Differential and integral calculus, p. 399.
 + '

28 ALGEBBAICAL FUNCTIONS IT

6. It remains to consider the integral

[ £» + '/ j _ f &+V j
J (SF+ljte+^jT* J *i 7* '

where a* 5*
+ 20a; + y or X t
is a quadratic with complex linear factors. Here
again there is a choice of methods at our disposal.

We may suppose that X t

is not a constant multiple of X. If it is, then

the value of the integral is given by the formula

&c + ij j,{ax+b)-t;(bx+c) * .

/,{aaP + Zbx+cyi* J{(ae-b») (ax*+2bx+c)}

(i) The standard method is to use the substitution

(1) '
-Sf?
where /* and v are so chosen th&t
anv + b(n+v) + c = 0, apv + P(n + v) + y=0 (*)•

The values of /i and v which satisfy these conditions are the roots of the
quadratic
(a,p-ba)iJ?-(ca-ay)ti+(by-C0)=O.
The roots will be real and distinct if

{ca - ay) 2 >4 (a/3 - ba) {by - Cj3),

Ol If (ay ca - 26/3) 2 > 4 (ac-6 2

)
(ay -0 2 ) -....(3).

Now ay-02 >O, so that (3) is certainly satisfied if ac - 62 <0. But if ac- 6*
and ay - 2 are both positive then ay and ca have the same sign, and

(ay+ca- 26j9) 2 > (

| ay+ca - 2
6/3
2
>4 y(acay)- 60 2
| I
|) 1
|}

= 4[(ao-6 2 )(ay-02 + {|6|V(ay)-|0|V(ac)} 2] )

^4(ac-62)(ay-02 ).

Thus the values and v are in any case real and distinct.
of /x

It will be found, on carrying out the substitution (1), that

J
[$x+r,
X 1 sJX
M _~ „ f

] (At + B)
2
tdt
J(At 2 + B)
+*
I dt
J (At + B) y/(Afi +B)
2

where A, B, A, B, IT, and are constants. K Of these two integrals, the first
is rationalised by the substitution

U
J(At* + B) '

and the second by the substitution

V r
-

>J{AP+B)-
It should be observed that this method fails in the special case in which

* Bromwich, I.e., p. 16.

t The method sketched here is that followed by Stolz (see the references given
on p. 21). Dr Bromwich's method is different in detail but the same in principle.
6] ALGEBRAICAL FUNCTIONS 29

a/3— 6o=0. In this case, however, the substitution ax+b=t reduces the
integral to one of the form
Ht+K
-dt.
h
and the reduction may then be completed as before.

(ii) An alternative method is to use Sir G. Greenhill's substitution

%
\J \ax*+Zfjx+y) VW'
If J=(a$-ba)x2 -{ca — ay)x + (by — c@i),
i.1.
A dt J .,,
then
7S = XX, (1) -

The maximum and minimum values of t are given by J=0.

fl _ x
_(a-Xa)*' + 8(6-Xfl)x + (e-Xy ),
Again

and the numerator will be a perfect square if

K= (ay - 2
/3 ) X 2 - (ay + Ca - 26/3) X + (ae - b 2 ) = 0.
It will be found by a little calculation that the discriminant of this
quadratic and that of J differ from one another and from

where <£, <j> are the roots of 1=0

and fa, fa' those of J
= 0, only by X
a constant factor which is always negative. Since fa and fa' are conjugate
complex numbers, this product is positive, and so J=0 and have real ^=0
roots*. We denote the roots of the latter by
Xi, X2 (Xi>X 2 ).

Then \i — t*= {xj(\ 1

a-a) + st(\ y-c)} 2 _
-y
1
—
('.nx+n)'
-g 1//,

<2 _x {^ N/(a-X 2a) + N/( e -X 2y)}

Xx
2 (m'x + n'f
x,
^
say. Further, since - X can vanish for two equal values of x only if X is
t°

equal to \ x or X 2 , i.e. when t is a maximum or a minimum, J can differ from

(mx + n) (m'x+n')
only by a constant factor; and by comparing coefficients and using the
identity
(a/3-Oa) 2
(Xja-a)(a-X 2 a) = -—— £j-,
.

we find that J=J(ay-P2 ) (mx+n) (m'x+n') (3).

Finally, we can write £#+>? i» the form

4 (mar + n) + B (m'x + ri).
* J=0 are roal has been proved already (p. 28) in a different
That the roots of
manner.
 '

30 ALGEBRAICAL FUNCTIONS [V

Using equations ^,1), (2), (2'), and (3), we find that

' A (mx+n)+B(m'x
=
+ n') , „ ,
nJAiat

A f dt B [
dt
+ V(ay -
{ay - p) J V(X t - &) J V(* - * 2)
2
J fi)

and the integral is reduced to a sum of two standard forms.

This method is very elegant, and has the advantage that the whole work
of transformation is performed in one step. On the other hand it is
somewhat artificial, and it is open to the logical objection that it introduces
the root .JXi, which, in virtue qf Laplace's principle (in., 2), cannot really
be involved in the final result*.

7. We may now proceed to consider the general case to which the

theorem of iv., § 2 applies. It will be convenient to recall two well-
known definitions in the theory of algebraical plane curves. A curve
of degree n can have at most £ (n - 1) (w - 2) double points t. If the

actual number of double points is v, then the number

p=i(n-l)(n-2)-v
is called the deficiency \ of the curve.
If the coordinates x, y of the points on a curve can be expressed
rationally in terms of a parameter t by means of equations
x=R 1 (t), y = R,{t),
then we shall say that the curve is unicursal. In this case we have
seen that we can always evaluate

I R (x, y) dx
in terms of elementary functions.
The fundamental theorem in this part of our subject is

'
A curve whose deficiency is zero is unicursal, and vice versa '.

Suppose first that the curve possesses the maximum number of

double points §. Since

10-1) (w-2) + «-3 = J(«-2) + l)-l,

* The superfluous root may be eliminated from the result by a trivial trans-

formation, just as J(l + x 2 may

) be eliminated from

v/(l + i2 )

by writing this function in the form arc tan x.

t Salmon, Higher plane curves, p. 29.
J Salmon, ibid., p. 29. French genre, German Gesehlecht.
§ We suppose in what follows that the singularities of the curve are all ordinary
nodes. The necessary modifications when this is not the case are not difficult to
 J ALGEBRAICAL FUNCTIONS 31
and £ -2) (re+ 1) points are just sufficient to determine a curve
(re
of
degree re- 2*, we can draw, through
the *(»-!)(»- 2) double points
and re - 3 other points chosen arbitrarily on the
curve, a simply infinite
set of curves of degree re
-2, which we may suppose to have the
equation
g(x,y)+th(x,y) = 0,

where t is a variable parameter and g=0, h = are the equations of

two particular members of the one of these curves meets set. Any
the given curve in re (re
-2) points, of which (re -1) (re -2) are ac-
counted for by the i(»-l) (re -2) double points, and
re- 3 by the
other re - 3 arbitrarily chosen points. These
(re - 1) (re - 2) + re -3= re (re - 2) - 1

points are independent of t ; and so there is but one point of inter-

section which depends on t. The coordinates of this point are given by

g(<c,y) + th(x,y) = 0, f(x,y) = 0.

The elimination of y re (re - 2) in x, whose

gives an equation of degree
coefficients are polynomials in t and but one root of this equation ;

varies with t. The eliminant is therefore divisible by a factor of

degree re (re - 2) - 1 which does not contain t. There remains a simple
equation in x whose coefficients are polynomials in t. Thus the
^-coordinate of the variable point is determined as a rational function
of t, and the ^-coordinate may be similarly determined.
We may therefore write

x = R-.it), y = Rs(t).
If we reduce these fractions to the same denominator, we express the
coordinates in the form
&(t) <m*)
x y (1))
~M*r ~ut)
where <f> lt <f> 2 , <j> 3 are polynomials which have no common factor. The
polynomials will in general be of degree re ; none of them can be of

make. ordinary multiple point of order k may be regarded as equivalent to

An
\k (k - ordinary double points.
1) A curve of degree n which has an ordinary
multiple point of order n - 1, equivalent to \ (to - 1) (n - 2) ordinary double points,
is therefore unicursal. The theory of higher plane curves abounds in puzzling
particular cases which have to be fitted into the general theory by more or less
obvious conventions, and to give a satisfactory account of a complicated compound
singularity is sometimes by no means easy. In the investigation which follows we
confine ourselves to the simplest case.
* Salmon, I.e., p. 16.
32 ALGEBRAICAL FUNCTIONS [V

higher degree, and one at least must be actually of that degree, since
an arbitrary straight line
\x + \iy + v =
must cut the curve in exactly n points*.
We can now prove the second part of the theorem. If

x:y:l::fa(t);fa(t):fa(t),
where fa, fa, fa are polynomials of degree n, then the line
ux + vy + w=
will meet the curve in n points whose parameters are given by
Ufa (t) + Vfa (t) + w<f> 3 (t) = 0.

This equation will have a double root t„ if

ufa (t ) + vfa (t ) + to fa (t ) = 0,
ufa' (t ) + vfa' (t ) + wfa' (to) = 0.
Hence the equation of the tangent at the point £„ is

x y 1

fa (to) fa (to) fa (to) (2).

fa' (t«) <f>2 (to) <f>3 (to)

If (x, y) is a fixed point, then the equation (2) may be regarded as

an equation to determine the parameters of the points of contact
of the tangents from (x, y). Now
fa (to) fa' (to) — fa^ (to) fa (to)

*n ~l
is of degree 2w-2 in tc , the coefficient of t obviously vanishing.
Hence in general the number of tangents which can be drawn to a
unicursal curve from a fixed point (the class of the curve) is 2» - 2.
But the class of a curve whose only singular points are 8 nodes is

known t to be n (n — 1) — 28. Hence the number of nodes is

J {» (» - 1) - (2n - 2)} = J (n - 1) (n - 2).

It is perhaps worth pointing out how the proof which precedes requires
modification if some only of the singular points are nodes and the rest
ordinary cusps. The first part of the proof remains unaltered. The equation

* See Niewenglowski's Cnurs de geomitrie analytique, vol. 2, p. 103. By way of

illustration of theremark concerning particular cases in the footnote (§) to page 30,
the reader may consider the example given by Niewenglowski in which
t* t
2
+l

equations which appear to represent the straight line 2x = y +1 (part of the line
only, if we consider only real values of (),
t Salmon, I.e., p. 54.
*?-&]
ALGEBRAICAL FUNCTIONS 33

(2) must now be regarded as giving the values of t which correspond to

(a) points at which the tangent passes through (x, y) and (6) cusps, since any
line through a cusp 'cuts the curve in two coincident points'*. We have
therefore
2n-2 = m + <,
where m is the class of the curve. But
m = n(n-l)-28-3 K ,t
and so 8 + «=4 (n-1) (»-2). \
8. (i) The preceding argument fails if n < 3, but we have already
seen that all conies are unicursal. The case next in importance is
that of a cubic with a double point. If the double point is not at
infinity we can, by a change of origin, reduce the equation of the
curve to the form

(ax + by) (ex + dy) =px 3

+ 3qx2y + 3rxy'2 + sy3 ;

and, by considering the intersections of the curve with the line

y = tx, we find
(a + bt) (c + dt) _ t(a + bt) (c + dt)
°° ~
p + 3qt + 3rf + sf V ~p + Sqt + 'Srt* + sf
'

If the double point is at infinity, the equation of the curve is of the

form
2
(ax + fiy) (yx + By) + tx + £y + = 0,

the curve having a pair of parallel asymptotes ; and, by considering

the intersection of the curve with the line ax + (3y = t, we find

X~
8tr> + & + /3d _ yf + €t + ad
(/3y-a8)f+el3-ai;' V~ - <x8)> + <0 - af
(/3y

(ii) The case next in complexity is that of a quartic with three double
points.

(a) The lemuiscate (x l + ?/ 2 ) 2 = a 2 (x2 - y*)

has three double points, the origin and the circular points at infinity. The
circle
x 2 +y 2 = t(x-y)
* This means of course that the equation obtained by substituting for x and y,
in the equation of the line, their parametric expressions in terms of t, has a
repeated root. This property is possessed by the tangent at an ordinary point and
by any line through a cusp, but not by any line through a node except the two
tangents.
+ Salmon, I.e., p. 65.

J I owe this remark to Mr A. B. Mayne. Dr Bromwich has however pointed

out to me that substantially the same argument is given by Mr W. A. Houston, Note
'

on unicursal plane curves', Messenger of mathematics, vol. 28, 1899, pp. 187-189.
3
34 ALGEBRAICAL FUNCTIONS t V

passes through these poiats and one other fixed point at the origin, as it

touches the curve there. Solving, we find

_a?t(t 2 +a?) a2 t{t 2 -a 2 )

x= y~-
t* + a'
(b) The curve 2ay s -3a 2y 2 = x*-2a 2x2
has the double points (0, 0), (a, a), ( - a, a). Using the auxiliary conic

x 2 — ay = tx (y — a),

we find * = |(2-30, y=|-4 (2-3^2-«2 ).

(iii) (a) The curve y

n =xn + axn ~ 1

has a multiple point of order k-1 at the origin, and is therefore unicursal.
In this case it is sufficient to consider the intersection of the curve with the
line y = tx. This may be harmonised with the general theory by regarding
the curve

as passing through each of the \(yi — 1) in — 2) double points collected at the

origin and through n — 3 other fixed points collected at the point

x= —a, y=0.
n n~
y =x + ax
n
The curves 1
(1),

n =\-\-az
y (2),

are projectively equivalent, as appears on rendering their equations homo-

geneous by the introduction of variables z in (1) and x in (2). We conclude
that (2) is unicursal, having the maximum number of double points at
infinity. In fact we may put
y='t, az = tn -l.

The integral \R {z, ^(1 + az)} dz

is accordingly an elementary function.

(6) The curve jT = A (x - of (x - b) v

is unicursal if and only if either (i) = or
/i (ii) i/ = or (iii) p+v=m.
Hence the integral
!m
JR{x, {x-aY (x-bfl n}dx

is an elementary function, for all forms of R, in these three cases only ; of

course it is integrable for special forms of R in other cases*.

* See Ptaszycki, ' Extrait d'une lettre adreseee a M. Hermite ', Bulletin des
sciences math€matiques, eer. 2, vol. 12, 1888, pp. 262-270: Appell and Goursat,
Theorie des fonctions alggbriques, p. 245.
8-10] ALGEBRAICAL FUNCTIONS 35

9. There is a similar theory connected with unicursal curves

in space of any number of dimensions. Consider for example the
integral

\R{x, J(ax + b), J(cx+d)}dx.

!'

A linear substitution x=lx + m reduces this integral to the form

JRiil/, J(y + 2),J(y-2)}dy,

and this integral can be rationalised by putting

The curve whose Cartesian coordinates £, rj, £ are given by

i:i) : t, : 1 :: f+l : t(?+l) : t(f-l) : t\

is a unicursal twisted quartic, the intersection of the parabolic cylinders

£ = rf-2, £ =p+ 2.

It is easy to deduce that the integral

J [ V \mx + n/ V \mx + nj)

is always an elementary function.

10. When the deficiency of the curve f{x, y) =d is not zero, the
integral

!'R (x, y) dx
I

is in general not an elementary function ; and the consideration of

such integrals has consequently introduced a whole series of classes of
new transcendents into analysis. The simplest case is that in which
the deficiency is unity : in this case, as we shall see later on, the
integrals are expressible in terms of elementary functions and certain
new transcendents known as elliptic integrals. When the deficiency
rises above unity the integration necessitates the introduction of new
transcendents of growing complexity.
But there are infinitely many particular cases in which integrals,
associated with curves whose deficiency is unity or greater than unity,
36 ALGEBRAICAL FUNCTIONS [V

can be expressed in terms of elementary functions, or are even

algebraical themselves. For instance the deficiency of
y* = 1 + a?
is unity. But
x+l dx . (1+aQ 2 - 3,^(1+0
6 g
/ x-2 J(l+a?)~ (1 +«) + 3 7(1 +a?y
2

2 —x 3
dx 2x
r 1 + ^ V(l + ^)
3
V(l + « )
'

And, before we say anything concerning the new transcendents to

which integrals of this class in general give rise, we shall consider what
has been done in the way of formulating rules to enable us to identify
such cases and to assign the form of the integral when it is an
elementary function. It will be as well to say at once that this

problem has not been solved completely.

11. The first general theorem of this character deals with the
case in which the integral is algebraical, and asserts that if

: = I ydx

is an algebraical function of x, then it is a rational function of x and y.

Our proof will be based on the following lemmas.

(1) If f{x, y) and g (x, y) are polynomials, and there is no factor

common to all the coefficients of the various powers ofying (x, y) ; and

where h(x) is a rational function of x ; then h{x) is a polynomial.

Let h = P/Q, where P and Q are polynomials without a common
factor. Then
fQ = gP.
If x-a is a factor of Q, then

9 («. V) =
for all values of y ; and so all the coefficients of powers of y in g {x, y)
are divisible by x — a, which is contrary to our hypotheses. Hence
Q is a constant and h a polynomial.

(2) Suppose that fix, y) is an irreducible polynomial, and that

Vk Vi> •••» y% are the roots of

/(*, V) =
10-11] ALGEBRAICAL FUNCTIONS 37
in a certain domain D.
Suppose further that <f>{x, y) is another
polynomial, and that

$ 0> yd = o.
Then
<K*,y.) = 0,
where y e is any one of the roots of (l) ,-
and
*(*. V) =/(*, y)<K«, y),

where ij/ (x, y) also is a polynomial in x and y.

Let us determine the highest common factor m of and <j>, con- /
sidered as polynomials in y, by the ordinary process for the deter-
mination of the highest common factor of two polynomials. This
process depends only on a series of algebraical divisions, and so -m is a
polynomial in y with coefficients rational in x. We have therefore
n(x,y) = < {x,y)k(x) (1),

f(x, y) = m O, y)p O, y) n(x) = g (x, y)v(x) (2),

4> 0, y) = <*
O, y) q O, y) *(x) = h (x, y) v 0) (3),

•where q, g, and h are polynomials and A, p, and v rational

<u,
p,
functions and evidently we may suppose that neither in g nor in h
;

have the coefficients of all powers of y a common factor. Hence, by

Lemma (1), and v are polynomials. But / is irreducible, and there-
/jl

fore /* and either o> or p must be constants. If were a constant, <*>

st would be a function of x only. But this is impossible. For we can

determine polynomials L, M in y, with coefficients rational in x, such
that
Lf+M$=n (4),

and the left-hand side of (4) vanishes when we write y for y. Hence x

p is a constant, and so u is a constant multiple of /. The truth of

the lemma now follows from (3).
It follows from Lemma (2) that y cannot satisfy any equation of
degree less than n whose coefficients are polynomials in x.

(3) Ify is an algebraical function of x, defined by an equation

f{x,y) = (1)

of degree n, then any rational function R (x, y) of x and y can be

expressed in the form

R(x,y) = Ro + Riy+ + Hn-iy"- 1 (2),

where R , Ri, •••-> &»-i are rational functions of x.

38 ALGEBRAICAL FUNCTIONS [V

The function y is one of the n roots of (1). Let y,y',y", •••be the
complete system of roots. Then

R(x \=*&*1
V

P(x,y)Q(x,y')Q(x,y")... ,.
~Q(x,y)Q(.*,y')Q(*,y")- w '

where P and Q are polynomials. The denominator is a polynomial in

x whose coefficients are symmetric polynomials in y, y', y", ... and is ,

therefore, hy n., § 3, (i), a rational function of x. On the other hand

Q(x,y')Q( x ,y")
is a polynomial in x whose coefficients are symmetric polynomials
in y', y", ..., and therefore, by n., § 3, (ii), polynomials in y with
coefficients rational in x. Thus the numerator of (3) is a polynomial
in y with coefficients rational in x.

It follows that R (x, y) is a polynomial in y with coefficients rational

in x. From this polynomial we can eliminate, by means of (1), all
powers of y as high as or higher than the wth. Hence R (x, y) is of
the form prescribed by the lemma.

12. We proceed now to the proof of our main theorem. We have

\ydx = u
where u is algebraical. Let

f(x,y) = 0, *(*,«)=<> (1)

be the irreducible equations satisfied by y and u, and let us suppose

that they are of degrees n and m respectively. The first stage in the
proof consists in showing that
m = n.
It will be convenient now to write y u ux for y, u, and to denote by

V\i yn • j yn> Uijiht j

um ,

the complete systems of roots of the equations (1).

We have \\i {x, Wj) = 0,

, 3^ dif/ dxi\ d<(/ dxj/

dx Siit dx dx Siii

Nowlet Ofc-O-AS^g*)-
Then Q is a polynomial inuu with coefficients symmetric in yy yt
, , ..., yn
and therefore rational in x.
 H-12] ALGEBRAICAL FUNCTIONS 39

The equations i/r = and fi = have a root «i in common, and the

first equation is irreducible. It follows, by Lemma (2) of § 11, that

fi
O, m.) =
for s=l, 2, ... , m* And from this it follows that, when s is given,
we have

l + ^v° &
for some value of the suffix r.
But we have also

j* +
ox ££
du, dx
= (3);
K "
and from (2) and (3) it follows t that

du„ .
= y- (4) >
N

d*-

i.e. that every u is the integral of some y.

In the same way we can show that every y is the derivative of some u.
Let
-

<3f 3^
*»>-i®-»£)-
Then <o is a polynomial in y with coefficients symmetric inu1 ,ui ,...,um
t ,

and therefore rational in x. The equations /= and <o = have a

root y1 in common, and so

<° 0> yr) =

for r= 1, 2, ... , n. From
we deduce that, when r is given, (2) must
this
be true for some value of and so that the same is true of (4).
s,

Now it is impossible that, in (4), two different values of s should

correspond to the same value of r. For this would involve
ue -u = c t

where s =t= t and c is a constant. Hence we should have

\l>
(x, ua) = 0, >j/ (x, ua -c)= 0.

* If
p (x) is the least common multiple of the denominators of the coefficients
of powers of u in 0, then
ii(x, u)p{x) = x(x, «),

where x is a polynomial. Applying Lemma (2), we see that x x u ») = °.

( <
ar>d so

fi(x, it 8 = 0. )

impossible that and £- should both vanish for u=u,, since is

t It is <j/ \f>

irreducible.
 ,

40 ALGEBRAICAL FUNCTIONS [V

Subtracting these equations, we should obtain an equation of degree

m - 1 in u3 with coefficients which are polynomials in x ; and this is
,

impossible. In the same way we can prove that two different values of
r cannot correspond to the same value of s.

The equation (4) therefore establishes a one-one correspondence

between the values of r and s. It follows that

It is moreover evident that, by arranging the suffixes properly, we can

make
dur ,,-.

s-* (5)

for r= 1, 2, ... , n.

13. We have

^T *<-!£/£
dx Sxi du
-*c.*). r

where R is a rational function which may, in virtue of Lemma (3) of

§11, be expressed as a polynomial of degree n-\ in ur , with co-

efficients rational in x.

The product
II(*-y.)
s=t=r

is a polynomial of degree n - 1 in z, with coefficients which are sym-

metric polynomials in y lt y2 ..., yr ^i, y r +i, Vn and therefore,
, ,

by n., § 3, (ii), polynomials in y T with coefficients rational in x.

Replacing yr by its expression as a polynomial in ur obtained above,
and eliminating urn and all higher powers of u r , we obtain an equation

s=t=r j=0 & =

where the <S"s are rational functions of x which are, from the method
of their formation, independent of the particular value of r selected.
We may therefore write
u (z-y,)=J, (x, z,ur),

where P is a polynomial in z and ur with coefficients rational in x. It

is evident that
P 0, y s, ur) =
for every value of s other than r. In particular

P(?,yu = (»- = 2, 3, ...

12-14] ALGEBRAICAL FUNCTIONS 41

It follows that the n —1 roots of the equation in u

P(?,y x , w) =
are m 2 m 3
, , ..., un . We have therefore
n
P 0> Vi,u)= T„ l>, y±) n (u - u r)
2

= T (x, 2/l) {W""

1
- W"" 2 («*2 + M3 + •• + «„) + •• }

= r (, )
„)[^ + «»-{ Ml+ ||]} + ...],
where T (x, yt ) is the coefficient of 2<"
_1
in /', and i? (#) and Bx (x)
n -1
are the coefficients of u and m" in ij/. Equating the coefficients of
un ~- on the two sides of this equation, we obtain
B 1 (x)_ flfott)
Ul
B (x) T (x, yi )'

where T
x (x, 3/1) is the coefficient of u n ~* in P. Thus the theorem is

proved.

14. We can now apply Lemma (3) of § 11 ; and we arrive at the

final conclusion that if

jydx
is algebraical then it can be expressed in the form

R !l + R y+...+Bn ^yn -\
1

w/tere R„, R it ... are rational functions of x.

The most important case is that in which

y = "J{R(x)\,
where R (x) is rational. In this case

r=R(?) (i),

dy R'(x) - s
= '1 " 2
dx ny 11

But
y= R '
+ Ri'y + + R'n-iy
71 '1

<
&
+ {R1 + 2R y+... 2 + (n-l)Rn _ y n -*} 1 (3).

Eliminating -4- between these equations, we obtain an equation

*r(x,y) = (4),

where & y) (x, is a polynomial. It follows from Lemma (2) of § 11

that this equation must be satisfied by all the roots of (1). Thus
(4) is still true if we replace y by any other root y' of (1); and as
42 L
ALGEBRAICAL FUNCTIONS

when we effect this substitution, it follows

that (3) is
(2) is still true
also still true. Integrating, we see that the equation

jydx = B + R,y + ...+ Rn -iyn ~ l

is when y is replaced by y. We may therefore replace y by <ay,

true
<obeing any primitive rath root of unity. Making this substitution,
and multiplying by to"- 1 we obtain ,

f
ydx = o)"-
1
^ + R y + wR^y + ...+ w -*R - y
x
n
n 1
n
„ra-l ,

and on adding the n equations of this type we obtain

/'
ix = R y. t

Thus in this case the functions R , jK 2 , •••, Rn -\ all disappear.

been shown by Liouville* that the preceding results enable
It has
us to obtain in all cases, by a finite number of elementary algebraical
operations, a solution of the problem to determine whether jydx is '

algebraical, and to find the integral when it is algebraical '.

15. It would take too long to attempt to trace in detail the steps of the
general argument. We shall confine ourselves to a solution of a particular
problem which will give a sufficient illustration of the general nature of the
arguments which must be employed.
We shall determine under what circumstances the integral
dx

This question
h\(x-p)might
J (ax2 + 2bx + c)
of course be answered by actually
is algebraical.
evaluating the integral in the general case and finding when the integral
function reduces to an algebraical function. We are now, however, in a
position to answer it without any such integration.
We shall suppose first that ax2 + 2bx+c is not a perfect square. In this
case
1

where
X= (x - pf (ax 2
+ 2bx + c),
and if jydx is algebraical it must be of the form
14-15] ALGEBRAICAL FUNCTIONS 43

We can now show that R is a polynomial in x. For if R=U/V, where V

and V are polynomials, then V, if not a mere constant, must contain a factor
(*-<•)* 0*>o),

and we can put R= ,

W(x-af
where W
U and do not contain the factor x - a. Substituting this expression
for R, and reducing, we obtain
2 ll UWX =2U WX2UW x _ u wx>_ 2W iX(x-aT.
, ,
-

x—a
Hence X must be divisible by x- a. Suppose then that
X={x- afY,
where Y is prime to x - a. Substituting in the equation last obtained we
deduce
(2 M +6) ^ W ^=2 U'WF-2UW'Y- UWY'SWLYix-af,
x—a
which is obviously impossible, since neither U, W, nor Y is divisible by x - a.
Thus V must be a constant. Hence
dx U(x)
h (x -p) ^(ax2 + 2bx + c) (x-p) ^(ax 2 + 2bx + c)
where U(x) is a polynomial.
Differentiating and clearing of radicals we obtain
{(x-p) (U'-l)- U) (ax2 + 2bx + c) = U(x-p) {ax + b).
+2
Suppose that the first term inU is Axm Equating
. the coefficients of x™ ,
we find at once that m=2. We may therefore take
U=Ax2 +2Bx+C,
so that
{(x-p) (2Ax+2B- l)-Axi -2Bx-C}(ax2 + 2bx+c)
= (x-p)(ax+b)(Ax* + 2Bx+C) (1).
From (1) it follows that
(x-p)(ax + b)(Ax i + 2Bx + C)
is divisible But ax + b is not a factor of ax2 + 2bx+c, as
by aa? + 2bx + c.
the latter is Hence either (i) ax i + 2bx + c and
not a perfect square.
Ax + 2Bx+C differ only by a constant factor or (ii) the two quadratics have
2

one and only one factor in common, and x -p is also a factor of ax'i +2bx+c.
In the latter case we may write

ax2 + 2bx + c = a(x-p)(x-q), Ax2 + 2Bx+C=A(x-q)(x-r),

where p 4= q, p^r. It then follows from (1) that

a(x-p)(2Ax + 2B-\)-aA(x-q)(£-r) = A (ax + b)(x-r).

Hence 2 Ax + 2B — 1 is divisible by x - r. Dividing by a A (x - r) we obtain

2(x-p)-(x-q) = x+- = x-$(p + q),

and sop =q, which is untrue.
 . ' :

44 ALGEBRAICAL FUNCTIONS [V

Hence case (ii) is impossible, and so ax 2 +2bx + c ana Ax 2 + %Bx+C differ

only by a constant factor. It then follows from (1) that x — p is a factor
of axi + '2bx-t-c; and the result becomes
f dx _ _ J(ax + 2bz+c)
2

J (x-p) »J(ax + 2bx+c)

2 x—p
where K is a constant. It is easily verified that this equation is actually

true when ap* + 2bp + c = 0, and that

l
K- 2
V(6 -«c)'
The formula is equivalent to
f dx 2 //x-q\
J (* -p) »J{(x-p)(x-q)} ~ q-pV K^p)
There remains for consideration
:rati the case in which ax 2 +2bx+c is a
perfect square, say a (x - qf. Th<
Then
dx
J^-p)(x-q)
must be rational, and so p = q.
As a further example, the reader may verify that if

y -3y + 2x = Q
3

then I ydx = '-(2xy-y 1 ).*

16. The theorem of § 11 enables us to complete the proof of the

two fundamental theorems stated without proof in n., § 5, viz.
(a) e° is not an algebraical function of x,
(b) log x is not an algebraical function of x.

We shall prove (b) as a special case of a more general theorem, viz.

'
no sum of the form
A log (w - a) + B log (x - @) + . .
,

in which the coefficients A, B, ... are not all zero, can be an algebraical
function of x' . To prove this we have only to observe that the turn
in question is the integral of a rational function of x. If then it is

algebraicalit must, by the theorem of § 11, be rational, and this we

have already seen to be impossible (iv., 2).
That e* is not algebraical now follows at once from the fact that it

is the inverse function of log x.

17. The general theorem of § 11 gives the first step in the rigid
proof of 'Laplace's principle' stated in in., § 2. On account of the
immense importance of this principle we repeat Laplace's words

* Raffy, '
Sur les quadratures alg^briques et logarithmiques ', Annalei de VEcole
Normale, ser. 3, vol. 2, 1885, pp. 185-206.
15-18] ALGEBRAICAL FUNCTIONS 45
'Vintegrale d'urn fonction differentiate ne pent
contenir d'autres quan-
tity radicaux que celles qui entrmt dans
cette fonction '. This general
principle,combined with arguments similar to those used above (§ 15) in
a particular case, enables us to prove without difficulty that a great
many integrals cannot be algebraical, notably the standard elliptic
integrals
<&
f [ I ( \-a? \ j
dx { dx
)J{(1 - *) (1 - *V)} ' j V KT^Wtf) '
) J^- 9i x-g )
3

which give rise by inversion to the elliptic functions.

18. We must now consider in a very summary manner the more

difficult question of the nature of those integrals of algebraical func-
tions which are expressible in finite terms by means of the elementary
transcendental functions. In the
place no integral of any alge-
first

braical Junction can contain any exponential. Of this theorem it is, as

we remarked before, easy to become convinced by a little reflection,
as doubtless did Laplace, who certainly possessed uo rigorous proof.
The reader will find little difficulty in coming to the conclusion that
exponentials cannot be eliminated from an elementary function by
differentiation. But we would strongly recommend him to study the
exceedingly beautiful and ingenious proof of this proposition given by
Liouville*. We have unfortunately no space to insert it here.

It is instructive to consider particular cases of this theorem. Suppose for

example that \ydx, where y is algebraical, were a polynomial in x and e*, say
2Sa m ,,.r m e~ (1).

When must disappear from it otherwise

this expression is differentiated, e* :

we should have an algebraical relation between x and e*. Expressing the con-
ditions that the coefficient of every power of e* in the differential coefficient
of (1) vanishes identically, we find that the same must be true of (1), so that
after all the integral does not really contain e*. Liouville's proof is in reality

a development of this idea.

The integral of an algebraical function, if expressible in terms

of elementary functions, can therefore only contain algebraical or

logarithmic functions. The next step is to show that the logarithms
must be simple logarithms of algebraical functions and can only
the type
enter linearly, so that the general integral must be of

jydx = u + A log v + B log w + ...,

consid^es comme fonctions de
•
Memoire sur les transcendantes eUiptiques
'

Journal de V±coU Polytechnique, vol. 14, cahier 3iI, 1834,

lenr amplitude',
Calcul integral, p. 99.
pp. 37-83. The proof may also be found in Bertrand's
 46 ALGEBRAICAL FUNCTIONS [V

where A, B, ... are constants and u, v, w, ... algebraical functions.

Only when the logarithms occur in this simple form will differentiation
eliminate them.
Lastly it can be shown by arguments similar to those of §§ 11-14
that u, v, w, ... are rational functions of x and y. Thus jydx, if

au elementary function, is the sum of a rational function of x and

y and of certain constant multiples of logarithms of such functions.
We can suppose that no two of A, B, ... are commensurable, or indeed,
more generally, that no linear relation

Aa + B/3+...=0,
with rational coefficients, holds between them. For if such a relation
held then we could eliminate A from the integral, writing it in the
form

jydx- u + B log (wv *) +

It is instructive to verify the truth of this theorem in the special case in

which the curve f(x,y) = is unicursal. In this case x and y are rational
functions R(t), S(t) of a parameter t, and the integral, being the integral of
a rational function of t, is of the form
u + A \ogv + Blogw+ ...,
where u, v, w, ... t. But t may be expressed, by
are rational functions of
means of elementary algebraical operations, as a rational function of x and y.
Thus u, v, w, ... are rational functions of x and y.

The case of greatest interest is that in which y is a rational function

of x and JX, where X is a polynomial. As we have already seen,
y can in this case be expressed in the form

^ JX'
where P and Q are rational functions of x. We shall suppress the
rational part and suppose that y = Q/JX. In this case the general
theorem gives

\-%-dx = S+ A log (a + pJX) -f B log (y + 8JX) + ...,

-JJ.+
where S, T, a, /?, y, S, . . . are rational. If we differentiate this equation
we obtain an algebraical identity in which we can change the sign of
JX. Thus we may change the sign of JX in the integral equation.
If we do this and subtract, and write 2 A, ... for A we obtain

Q j T . . a+ PJX D . y+ 8JX
/.
 ;

18-19] ALGEBRAICAL FUNCTIONS 47

which is the standard form for such an integral. It is evident that we
may suppose a, /?,
y ,
... to be polynomials.

19. (i) By means of this theorem it ia possible to prove that a number

of important integrals, and notably the integrals

dx
J v{(i -*2 )
(i - ***» '
) Vtrr^y **, /;
are not expressible in terms of elementary functions, and so represent genuinely
new transcendents. The formal proof of this was worked out by Liouville*
it rests merely on a consideration of the possible forms of the differential
coefficients of expressions of the form

and the arguments used are purely algebraical and of no great theoretical
The proof is however too detailed to be inserted here. It is not
difficulty.

but these are of a less elementary character,

difficult to find shorter proofs,
being based on ideas drawn from the theory of functions t.
The general questions of this nature which arise in connection with
integrals of the form

,dx,
I
more generally,
f Q
or, / ^fydx,

are of extreme interest and difficulty. The case which has received most
attention is that in which »i = 2 and X is of the third or fourth degree, in
which case the integral is An integral of this kind is
said to be elliptic.
terms of algebraical and logarithmic
called pseudo-elliptic if it is expressible in
functions. Two examples were given above (§ 10). General methods have
been given for the construction of such integrals, and it has been shown that
certain interesting forms are pseudo-elliptic. In Goursat's Cows d'analyse J,
for instance, it is shown that if f(x) is a rational function such that

/« +/(i)= '

then
\
J J{x{\-x){\-Wx)}
\
>

is pseudo-elliptic. But no method has been devised as yet by which we can

always determine in a finite number of steps whether a given elliptic integral

• See Liouville's memoir quoted on p. 45 (pp. 45 et seq.).

appears at
proof given by Laurent (Traite d'analyse, vol. 4, pp. 153
et seq.)
t The
first sight to combine the advantages of
both methods of proof, but unfortunately
will not bear a closer examination.
+ Second edition, vol. 1, pp. 267-269.
 48 ALGEBRAICAL FUNCTIONS [V

and integrate it if it is, and there is reason to suppose that

is pseudo-elliptic,

no such method can be given. And up to the present it has not, so far as
we know, been proved rigorously and explicitly that {e.g.) the function
dx
-/.
iVUi-^Xi-* * 2 2
)}

isnot a root of an elementary transcendental equation all that has been ;

shown is that it is not explicitly expressible in terms of elementary trans-

cendents. The processes of reasoning employed here, and in the memoirs
to which we have referred, do not therefore suffice to prove that the inverse
function x=snu is not an elementary function of Such a proof must rest
u.

on the known properties of the function sn u, and would lie altogether outside
the province of this tract.
The reader who desires to pursue the subject further will find references
to the original authorities in Appendix I.

(ii) One particular class of integrals which is of especial interest is

that of the binomial integrals

x m (axn + b)»dx,
s
where m, n, p are rational. Putting axn =bt, and neglecting a constant
factor, we obtain an integral of the form

fi(l + i)Pdt,
/
where p and q are rational. If p is an integer, and q a fraction r/s, this

integral can be evaluated at once by putting t — u", a substitution which

rationalises the integrand. If q is an integer, and p = rjs, we put l + t=%K
If p+q is an integer, and p = r/s, we put 1 + t = tu'.
It follows from Tschebyschef 's researches (to which references are given
in Appendix I) that these three cases are the only ones in which the integral
can be evaluated in finite form.

20. In §§ 7-9 we considered in some detail the integrals con-

nected with curves whose deficiency is zero. We shall now consider
in a more summary way the case next in simplicity, that in which
the deficiency is unity, so that the number of double points is

i (w - 1) - 2) - 1 = \n («- 3).

It has been shown by Clebsch* that in this case the coordinates of

the points of the curve can be expressed as rational functions of
a parameter t and of the squai-e root of a polynomial in t of the third
or fourth degree.
* '
Uber diejenigen Curven, deren Coordinaten sich als elliptische Functionen
eines Parameters darstellen lassen ', Journal fiir Jlatliemutik, vol. 64, 1865,
pp. 210-270.
19-20] ALGEBRAICAL FUNCTIONS 49

The fact is that the curves

I/
2 = a + bx + ex2 + dx 3
,

y
i
= a + bx + cx +dx + exi
i i
,

are the simplest curves of deficiency 1. The first is the typical cubic
without a double point. The second is a quartic with two double points,
in this case coinciding in a 'tacnode' at infinity, as we see by making the
equation homogeneous with z, writing and then comparing the
1 for y,
resulting equation with the. form treated by Salmon on p. 215 of his Higher
plane curves. The reader who is familiar with the theory of algebraical plane
curves will remember that the is unaltered by any
deficiency of a curve
birational transformation of coordinates,and that any curve can be biration-
ally transformed into any other curve of the same deficiency, so that auy
curve of deficiency 1 can be birationally transformed into the cubic whose
equation is written above.

The argument by which this general theorem is proved is very

much like that by which we proved the corresponding theorem for
unicursal curves. The simplest case is that of the general cubic curve.
We take a point on the curve as origin, so that the equation of the
curve is of the form

ax? + &bx*y + Sexy2 + dy* + ex + 2/xy + gif + hx + ky =

2
0.

Let us consider the intersections of this curve with the secant y = tx.
Eliminating y, and solving the resulting quadratic in x, we see that the

only irrationality which enters into the expression of x is

where T x
= h + U, T 2
= e + 2ft + gf, T
3 =a+ 3bt + 3ct2 + dt?.

A more elegant method has been given by Clebsch*. If we

write the cubic in the form
LMN=P,
where L, M, N, P are linear functions of x and y, so that L, M, N are
the asymptotes, then the hyperbolas LM=t will meet the cubic in

four fixed points at infinity, and therefore in two points only which
depend on t. For these points
LM=t, P=tN.
Eliminating y from these equations, we obtain an equation of the form
Ax> + 2Bx+C=0,
where A,B,C are quadratics in t. Hence
B + JiB^AO) = Ji{Un
x A
A
* See Hermite, Cows d'analyse, pp. 422-425.

4
 50 ALGEBRAICAL FUNCTIONS [V

where T = B* - AC is a polynomial in t of degree not higher than the

fourth.
Thus if the curve is

<& + y - 3a#y + 1 = 0,
so that

L = <ox + u?y + a, M=w x + a>y + a, 2

N=x + y + a, P = d -1, t

<u heing an imaginary cube root of unity, then we find that the line
a 3 -I
3 + a=
x +y
t

meets the curve in the points given by

where b = a 3 — 1 and
T=4P-9aiP + Gabt- b
2
.

In particular, for the curve

we have

2tj3 ' y - 2tj3 ~

21. It will be plain from what precedes that

/ R {x, $(a + bx + cx* + dx3) } dx

can always be reduced to an elliptic integral, the deficiency of the cubic

y = a+bx + cx? + dx
3 3

being unity.
In general integrals associated with curves whose deficiency is

greater than unity cannot be so reduced. But associated with every

curve of, let us say, deficiency 2 there will be an infinity of integrals

JR(x,y) dx
reducible to elliptic integrals or even to elementary functions ; and
there are curves of deficiency 2 for which all such integrals are
reducible.
For example, the integral

JR {x, J(xe + ax* +bx* + c)}dx

 ;

20-22] ALGEBRAICAL FUNCTIONS 51

may be split up into the sum of the integral of a rational function and
two integrals of the types

f B (V) dx r X R Q2 ) dx
JJ(a* + ax* + bx> + cy JJ(x + axi +bx> +
s
c)'

and each of these integrals becomes elliptic on putting x* = t. But

the deficiency of
y = x* + ax* + bx* + c
is 2. Another example is given by the integral

IB {x, y(x* + an? + ba? + ex + d) ) dx*

22. It would be beside our present purpose to enter into any

details as to the general theory of elliptic integrals, still less of the
integrals (usually called Abelian) associated with curves of deficiency
greater than unity. We have seen that if the deficiency is unity then
the integral can be transformed into the form

I R (x, JX) dx
where JT=x* + ax3 + bxi + cx + d.\
It can be shown that, by a transformation of the type
at + P
X
~yt + h'
this integral can be transformed into an integral

(B
\> (t, JT) dt

^here T^f + Af+B.

We can then, as when T is of the second degree (§ 3), decompose
this integral into two integrals of the forms
/„,., [R (t) dt
jb (o *,
j yT
Of these integrals the first is elementary, and the second can be

vol., 1, ohs. 26-27, 32-33

*
See Legendre, TraiU des fonctions elliptiques,
Bertrand, Calcul integral, pp. 67 et seq. and Enueper, Elbptisclie Funktionen,
;

note 1,where abundant references are given.

for curves of deficiency 2, in which X is of the sixth
t There is a similar theory
degree.
4*
52 TRANSCENDENTAL FUNCTIONS [VI

decomposed* into the sum of an algebraical term, of certain multiples

of the integrals
Cdt_ [t'dt
JJT' )JT>
and of a number of integrals of the type
dt

k
These integrals cannot in general be reduced to elementary functions,
and are therefore new transcendents.
We will only add, before leaving this part of our subject, that the
algebraical part of these integrals can be found by means of the
elementary algebraical operations, as was the case with the rational
part of the integral of a rational function, and with the algebraical part
of the simple integrals considered in §§ 14-15.

VI. Transcendental functions

1. The theory of the integration of transcendental functions is

naturally much less complete than that of the integration of rational

or even of algebraical functions. It is obvious from the nature of the
case that this must be so, as there is no general theorem concerning
transcendental functions which in any way corresponds to the theorem
that any algebraical combination of algebraical functions may be
regarded as a simple algebraical function, the root of an equation of

a simple standard type.

It is indeed almost true to say that there is no general theory, or

that the theory reduces to an enumeration of the few cases in which

the integral may be transformed by an appropriate substitution into an
integral of a rational or algebraical function. These few cases are
however of great importance in applications.

2. (i) The integral

(F(eT,e^, ...,t^)dx
I'
where F
is an algebraical function, and a,b,...,k commensurable

numbers, can always be reduced to that of an algebraical function.

In particular the integral

\R(<r,ehx , ...,e**)dw,
I'
See, e.g., Goursat, Court, d'analyse, ed. 2, vol. 1, pp. 257 et seq.
 —
1- 2] TRANSCENDENTAL FUNCTIONS 53
where R
is rational, is always an
elementary function. In the first
place a substitution of the type x = ay will reduce
it to the form

JB{e»)dy,

and then the substitution e" =z will reduce this integral to the integral
of a rational function.
In particular, since cosh x and sinh x are rational functions of
4?, and cos x and sin x are rational functions of e to, the integrals

R (cosh x, sinh x) dx, I R (cos x, sin x) dx

J

are always elementary functions. In the second place the substitution

just indicated is imaginary, and it is generally more convenient
to use the substitution

tan \x = t,
which reduces the integral to that of a rational function, since

cos x-
1-f
-

1+f
— -=
,
.

sina;= -
1
2*
+ f-. , dx =
2dt
1+f
(ii) The integrals

!R (cosh x, sinh x, cosh 2x, sinh mx) dx,

R (cos x, sin x, cos 2x, sin mx) dx,

!
are included in the two standard integrals above.
Let us consider some further developments concerning the integral

R (cos x, sin x) dx.*

/
If we make the substitution z=eix the subject of integration becomes a
,

rational function H(z), which we may suppose split up into

(a) a constant and certain positive and negative powers of z,

(b) groups of terms of the type

Ap_ Ai An
z-a (z-a) 2 (z-a) n + 1 v '

The terms (i), when expressed in terms of x, give rise to a term

2 (ct cos kx + dk sin kx).

In the group (1) we put z=e", a = eia and, using the equation

J—
—
= !<.-«» {-1 -icoH(a--a)},
z Ob

* See Hermite, Cours d'analyse, pp. 320 et seq.

 . —+ — , '

54 TRANSCENDENTAL FUNCTIONS [VI

we obtain a polynomial of degree n+1 in cot$(x-a). Since

COt 2 A' = -1 , dcotx

; ,
.,
COt 3
Id..
# = -COt^---r
,

(COfr^), ....
dx '
1dx s

this polynomial may be transformed into the form

d dn
C+C ™i%{x-a) + C 1 -^cot\{x- a )+... + Cn -^cot\,(x-a).

The function iJ(cosx, sin x) is now expressed as a sum of a number of

terms each of which is immediately integrable. The integral is a rational
function of cos x and sin x if all the constants (7 vanish ; otherwise it includes
a number of terms of the type

2C logsin£(.2.' — a).
Let us suppose for simplicity that H{z), when split up into partial fractions,
contains no terms of the types

C, zm , z~ m ,
(z-a)-» (/>>!).
Then
Rlcosx, sin x) — C cot \ (x — a)+D cot ^ (x — ji) + ...

and the constants C D ... may be determined by multiplying each side of

, ,

the equation by sin^(x— a), sin^(^ — /3), ... and making x tend to a, ft....
It is often convenient to use the equation

cot^(x — a) = cot (x — a) cosec {x — a)

which enables us to decompose the function R into two parts U(x) and
V(x) such that
U(x+n)=U(x), V(x + w)=-V(x).
If R has the period n, then V must vanish identically if it changes sign ;

when x is increased by it, then U must vanish identically. Thus we find

without difficulty that, if m<n,
sin mx _ 1 2n ~ 1 — l) l sin«ia_ 1 "- 1 — l)*sin ma
( (

sinner 2re sin(# — a) n sin(x-a)

or — = - 2 ( - 1)* sin ma cot (x - a),

sin MX' n o

where a=Jcnjn, according as rn+n is odd or even.

Similarly
1 '
-2-
sin (x — a) sin (x — b) sin {x — c)
t
sin (a — b) sin (as — c) sin (# - a)
sin (x-d)
(a-d) sin
— — ;
rr^ 7
8m(x-a)sin(x-b)Bm(x-c)\.
= s — i rr-^
. i
sin (a -6) sin (a -c)
c cottx
v —— a).

(iii) One of the most important integrals in applications is

dx
ha + bcosx'
'

where a and b are real. This integral may be evaluated in the manner
explained above, or by the transformation tan ^x = t. more elegant method A
 3J TRANSCENDENTAL FUNCTIONS 55
is the following. If |a| > |6|, we suppose a positive, and use the trans-
formation
(a + b cos x) (a - b cos y) = a2 - 6 2 ,

which leads to dx _ dy
a + bcosx „/(a2 -62 )"
If |
a |
< |
6 1 ,
we suppose b positive, and use the transformation
(b cos x + a) (b cosh y - a) = ft
2 - a2 .

The integral ._
dx
j

+&cos#-t-csin#
maybe reduced to this form by the substitution x + a=y, where cota = 6/c.
The forms of the integrals

f dx r dx
J (a+b cos x) n '
J(a~-
i + bcosx + csmx) n
may be deduced by the use of formulae of reduction, or by differentiation
with respect to a. The integral

dx
h{A cos 2 x + 2B cos x sin x + C sin 2 x) n
is really of the same type, since

A cos*x+2B cos x sin x + C sin 2 # = £ (A + G) +\ (A - C) cos 2x + B sin 2x.

And similar methods may be applied to the corresponding integrals which
contain hyperbolic functions, so that this type includes a large variety of
integrals of common occurrence.

(iv) The same substitutions may of course be used when the subject of
integration an irrational function of cos a; and sin#, though sometimes
is

it is better to use the substitutions cosx = t, sinx = t, or ta.nx=t. Thus

the integral

!R (cos x, sin x, JX) dx,

where X=(a, b, c,f, g, h\cosx, sins, l) 2,

is reduced to an elliptic integral by the substitution tsm%x=t.

The most
important integrals of this type are
[ R(cosx, sinx)dx f R (cos x, sin x) dx

J ^(l-^sin 2 ^) '
Js/ia + pcosx+ysinx)'

3. The integral
to
[P{x, <T, e ,
...,<?*) dx,

where a b k are any numbers (commensurable or not), and P is

... ,

an elementary function. For it is obvious

a polynomial, is always
 ; ;

56 TRANSCENDENTAL FUNCTIONS [VI

that the integral can be reduced to the sum of a finite number of

integrals of the type
Ax dx;
fx"e

and j"^* = (u)'l^*-(£d'£-

This type of integral includes a large variety of integrals, such as

\x m (cospxY v
(sin qx) dx, jxm (cosh jixf (sinh qxf dx,

m ax v
fare'" (eospxf dx, j x e~ (sin qx) dx,
(m, /*, v, being positive integers) for which formulae of reduction are
given in text-books on the integral calculus.
Such integrals as

I P (x, log x) dx, J

P (x, arc sin x)dx, ...,

where P
is a polynomial, may be reduced to particular cases of the

above general integral by the obvious substitutions

x=ey , ar = siny

4. Except for the two classes of functions considered in the three

preceding paragraphs, there are no really general classes of transcen-
dental functions which we can always integrate in finite terms, although
of course there are innumerable particular forms which may be
integrated by particular devices. There are however many classes

of such integrals for which a systematic reduction theory may be given,

analogous to the reduction theory for elliptic integrals. Such a reduction
theory endeavours in each case
(i) to split up any integral of the class under consideration into
the sum of a number of parts of which some are elementary and
the others not
(ii) to reduce the number of the latter terms to the least possible

(iii) to prove that these terms are incapable of further reduction,

and are genuinely new and independent transcendents.
As an example of this process we shall consider the integral

\<?R{x)dw
/«
where R (x) is a rational function of x. * The theory of partial

* See Hermite, Gours d'analyse, pp. 352 et eeq.

 ; h

" 4] TRANSCENDENTAL FUNCTIONS 57

fractions enables us to decompose

this integral into the sum of a
number of terms

Since
( g
}{x - a)™ **
j„_
— m{_J°-aT X
+
1 /•

m ){x~^af
e*
^
,

the integral may be further reduced so as to contain only

(i) a term ef°S(x)

where 8 (x) is a rational function

(ii) a number of terms of the type

(e°dx
a. .
\
J x—a
If all the constants a vanish, then the integral can be calculated in the
finite form e° S(x). If they do not we can at any rate assert that the
integral cannot be calculated in this form*- For no such relation as

where T is rational, can hold for all values of x. To see this it is

only necessary to put x = a + h and to expand in ascending powers

of h. Then
« / =ae"j T dh
Jx-a J

= <u a (\ogh + h+...),

and no logarithm can occur in any of the other terms t.

Consider, for example, the integral

and since 3 dx= -— +SJ -dx,

jj t

and
[e* , e* 1 /> , e* _,_
e* 1 fe* ,

* See the remarks at the end of this paragraph.

+ It is not difficult to give a purely algebraical proof on the lines of rv., § 2.

4^5
 58 TRANSCENDENTAL FUNCTIONS [VI

we obtain finally

/'('-3'*-*('-B + Bi)-{/J' k
Similarly it will be found that

this integral being an elementary function.

Since / dx = «• I - %,
J x — ct J y
if x=y + a, all integrals of this kind may be made to depend on known
functions and on the single transcendent

which is usually denoted by Li e" and is of great importance in the

theory of numbers. The question of course arises as to whether this
integral is not itself an elementary function.
Now
Liouville* has proved the following theorem: 'if y fa any
algebraical /miction of x, and

\e"ydx
I'
is an elementary function, then
x
\<?ydx = e
/• (a + Py+...+ Ay 1-1 ),

a, /J, ... ,X being rational functions of x and n the degree of the

algebraical equation which determines y as a function of x\
Liouville's proof rests on the same general principles as do those of
the corresponding theorems concerning the integral fydx. It will
be observed that no logarithmic terms can occur, and that the theorem
is therefore very similar to that which holds for fydx in the simple
case in which the integral is algebraical. The argument which shows
that no logarithmic terms occur is substantially the same as that which
shows that, when they occur in the integral of an algebraical function,
they must occur linearly. In this case the occurrence of the ex-
ponential factor precludes even this possibility, since differentiation
will not eliminate logarithms when they occur in the form

^log/O).
* '
Memoire sur l'integration d'une classe de fonctions transcendantes ', Journal
fiir Mathematik, vol. 13, 1835, pp. 93-118. Liouville shows how the integral, when
of this form, may always be calculated by elementary methods.
 4—51J
TRANSCENDENTAL FUNCTIONS 59
In particular, if 3, is a rational function, then the integral must
be of the form

e°R{x)
and this we have already seen to be impossible. Hence the logarithm-
&
<

integral

J x J logy
is really a new transcendent, which cannot be expressed
in finite terms
by means of elementary functions and the same
; is true of all integrals
of the type

jee R(x)dx

which cannot be calculated in finite terms by means of the process of

reduction sketched above.
The integrals

/ sin x R (x) dx, JcosxR (x) dx

may be treated in a similar manner. Either the integral is of the form

cos xR x (x) + sin x R2 (x)

or it consists of a term of this kind together with a number of terms
which involve the transcendents

\—
/"cos x
dx
,
>
] —
/sin x 7
**>

which are called the cosine-integral and sine-integral of x, and denoted

by Gi x and Si x. These transcendents are of course not fundament-
ally distinct from the logarithm-integral.

5. Liouville has gone further and shown that it is always possible

to determine whether the integral

t
UP(P+Q,0+ ...+Te )dx,

where P Q T p,q, ,t are algebraical functions, is an elementary

function, and to obtain the integral in case it is one*. The most
general theorem which has been proved in this region of mathematics,
due to Liouville, is the following.
and which is also

is that the 'error function' /«-«• dx is not an

* An interesting particular result
elementary function.
 '' ,

60 TRANSCENDENTAL FUNCTIONS [ V1

are functions of x whose differential coefficients are

'If y,z, ...

algebraical functions of x, y, z, ..., and F denotes an algebraical

function, and if

P\F{x,y, z, ...)dx

is an elementary function, then it is of the form

t + A log w +2? log 0+ ...

where t, u, v, ... are algebraical functions of x, y, z, ... Ij the

differential coefficients are rational in x, y, z, and F is rational,

then t, are rational in x, y, z,

u,v, ...
— ... ,

Thus for example the theorem applies to

F(x, e", e*, log x, log log x, cos x, sin#),

since, if the various arguments of F are denoted by x, y, z, £, t], £, 0,

we have
dy _ dz _ d£ _
- ^'
- ^'
1^

dx das dx~ x'

The proof of the theorem does not involve ideas different in principle
from those which have been employed continually throughout the
preceding pages.

6. As a final example of the manner in which these ideas may be applied,

we shall consider the following question :

'
in what circumstances is

R (x) log x dx,

I
where R is rational, an elementary function t
In the first place the integral must be of the form
R (x, logx)+A logS 1 l
(x, logx) + A 2 \ogR2{x, log #) + ....

A general consideration of the form of the differential coefficient of this

expression, in which log x must only occur linearly and multiplied by a
rational function, leads us to anticipate that (i) R (x, log.r) must be of the
form
S {x) (log xf+ T (x) log x + U (x),
where S, T, and U are rational, and (ii) Ru R%, ... must be rational functions
of x only ; so that the integral can be expressed in the form

S (x) (log xf + T (x) log x + U (x) + %Bk log (x - ak ).

5-6] TRANSCENDENTAL FUNCTIONS 61

Differentiating, and comparing the result with the subject of integration,

we obtain the equations

S'=0, —+T'=R, -
x
+ *7'+2-^-=0.
x
x -at
Hence S is a constant, say £C, and

We can always determine by means of elementary operations, as in iv., § 4,

whether this integral is rational for any value of C or not. If not, then the
given integral is not an elementary function. If T is rational, then we must
calculate its value, and substitute it in the integral

which must be rational for some value of the arbitrary constant implied in
T. We can calculate the rational part of

- dx:
/x
the transcendental part must be cancelled by the logarithmic terms

2Bk log(x-ak ).

The necessary and sufficient condition that the original integral should be
an elementary function is therefore that R should be of the form

where C is a constant and R x is rational. That the integral is in this case

such a function becomes obvious if we integrate by parts, for

({£ + rA log xdx = \C (log xY +R 1 logx-f —' dx.

In particular

w JA*£ f '<**
(i) rf,, (ii ) dx,'
x-a v
){x-a)(x — b)

are not elementary functions unless in (i) a = and in (ii) b=a. If the
integral is elementary then the integration can always be carried out, with
the same reservation as was necessary in the case of rational functions.
It is evident that the problem considered in this paragraph is but one of
a whole class of similar problems. The reader will find it instructive to
formulate and consider such problems for himself.
62 TRANSCENDENTAL FUNCTIONS [
VI T

7. It will be obvious by now that the number of classes of

transcendental functions whose integrals are always elementary is very
small, and that such integrals as

J
f (*, e°) dx, \f{x, log x) dx,

I f(x, cos x, sin x) dx. I f(d°, cos x, sin x) dx,

where/ is algebraical, or even rational, are generally new transcendents.

These new transcendents, like the transcendents (such as the elliptic
integrals) which arise from the integration of algebraical functions,
are in many cases of great interest and importance. They may often
be expressed by means of infinite series or definite integrals, or their
properties may be studied by means of the integral expressions which
define them. The very fact that such a function is not an elementary
function in so far enhances its importance. And when such functions
have been introduced into analysis new problems of integration arise
in connection with them. We may enquire, for example, under what
circumstances an elliptic integral or elliptic function, or a combination
of such functions with elementary functions, can be integrated in finite
terms by means of elementary and elliptic functions. But before we
can be in a position to restate the fundamental problem of the Integral
Calculus in any such more general form, it is essential that we should
have disposed of the particular problem formulated in Section in.
 63

APPENDIX I

BIBLIOGRAPHY

The following is a list of the memoirs by Abel, Liouville and Tschebyschef

which have reference to the subject matter of this tract.

N. H. Abel
1. 't)ber die Integration der Differential-Formel ^-ni, wenn R und p ganze
*J Jtt
Funktionen sind', Journal fur Mathematik, vol. 1, 1826, pp. 185-221
{(Euvres, vol. 1, pp. 104-144).
2. ' Precis d'une theorie des fonctions elliptiques ', Journal fur Mathematik,
vol. 4, 1829, pp. 236-277, 309-348 ((Euvres, vol. 1, pp. 518-617).

3. ' Theorie des transcendantes elliptiques', (Euvres, vol. 2, pp. 87-188.

J. Liouville
1. ' Memoire sur la classification des transcendantes, et sur l'impossibilite
d'exprimer les racines de certaines equations en fonction finie explicite

des coefficients', Journal de mathematiques, ser. 1, vol. 2, 1837,

pp. 56-104.
2. '
Nouvelles recherches sur la determination des integrates dont la valeur
est algebrique', ibid., vol. 3, 1838, pp. 20-24 (previously published in
the Comptes Rendus, 28 Aug. 1837).

3. '
Suite du memoire sur la classification des transcendantes, et sur
l'impossibilite d'exprimer les racines de certaines Equations en fonction
finie explicite des coefficients ', ibid., pp. 523-546

4. '
Note transcendantes elliptiques considerees
sur les comme fonctions de

leur module', ibid., vol. 5, 1840, pp. 34-37.

5. '
Memoire sur les transcendantes elliptiques considers comme fonctions

de leur module', ibid., pp. 441-464.

6 '
Premier memoire sur la determination des integrates dont la valeur est
algebrique Journal de VEcole Polytechnique, vol. 14, cahier 22, 1833,
',

pp. 124-148 (also published in the Me'moires presented par divers

savants a VAcademie des Sciences, vol. 5, 1838, pp. 76-151).

7 '
Second memoire sur la determination des integrates dont la valeur est

algebrique', ibid., pp. 149-193 (also published as above).

 —

64 APPENDIX I

8. '
Memoire sur les transcendantes elliptiques considerees comme fonctions
de leur amplitude', ibid., eahier 23, 1834, pp. 37-83.

9. 'Memoire sur l'integration d'une classe de fonctions transcendantes',

Journal fiir Mathematik, vol. 13, 1835, pp. 93-118.

P. Tschebyschef
1. 'Sur l'integration des differentielles irrationnelles ', Journal de mathi-
matiques, ser. 1, vol. 18, 1853, pp. 87-111 ((Euvres, vol. 1, pp. 147-168).
2. ' Sur l'integration des differentielles qui contiennent une racine carree
d'une polynome du troisieme ou du quatrieme degre', ibid., ser. 2,

vol. 2, 1857, pp. 1-42 ((Euvres, vol. 1, pp. 171-200; also published
in the Memoires de I'Academie Impdriale des Sciences de St-Petersbourg,
ser. 6, vol. 6, 1857, pp. 203-232).

3. ' Sur l'integration de la differentielle .. . '

v-^-A
. „ — -=r dx ', ibid.,

ser. 2, vol.
9, 1864, pp. 225-241 ((Euvres, vol. 1, pp. 517-530;
previously published in the Bulletin de I'Academie Imperiale des
Sciences de St-Petersbourg, vol. 3, 1861, pp. 1-12).

4. '
Sur l'integration des differentielles irrationnelles ', ibid., pp. 242-246
((Euvres, vol. 1, pp. 511-514 ;
previously published in the Gomptes
Rendus, 9 July 1860).
5. 'Sur ^integration des differentielles qui contiennent une racine cubique'
((Euvres, vol. 1, pp. 563-608 ;
previously published only in Eussian).

Other memoirs which may be consulted are :

A. Clebsch
'
Uber diejenigen Curven, deren Coordinaten sich als elliptische Functionen
eines Parameters darstellen lassen Journal fiir Mathematik, vol. 64,
',

1865, pp. 210-270.

J. Dolbnia
'
Sur les integrates pseudo-elliptiques d'Abel ', Journal de mathematiques,
ser. 4, vol. 6, 1890, pp. 293-311.

Sir A. G. Greenhill
'
Pseudo-elliptic iutegrals and their dynamical applications ', Proc. London
Math. Soc, ser. 1, vol. 25, 1894, pp. 195-304.

G. H. Hardy
'
Properties of logarithmico-exponential functions ', Proc. London Math.
Soc, ser. 2, vol. 10, 1910, pp. 54-90.

L. Konigsberger
' Bemerkungen zu Liouville's Classificirung der Transcendenten ', Mathe-
matische Annalen, vol. 28, 1886, pp. 483-492.
 APPENDIX I 65

L. Raflfy
ur les
quadratures algebriques et logarithmiques ', Annates de VEcok
Normale, ser. 3, vol.
2, 1885, pp. 185-206.

K. Weierstrass
'Uber die Integration algebraischer
Differentiate vermittelst Logarith-
men', Monatsberickte der Akademie der
Wissenschaften zu Berlin, 1857,
pp. 148-157 ( Werke, vol. 1, pp. 227-232).

G. Zolotareff
' Sur la methode d'integration de M. Tsohebyschef ', Journal de mathi-
matiques, ser. 2, vol. 19, 1874, pp. 161-188.

Further information concerning pseudo-elliptic and degenerate

integrals,
cases of Abelian integrals generally, will be found in a number of short notes
by Dolbnia, Kapteyn and Ptaszycki in the Bulletin des sciences mathematiques,
and by Goursat, Gunther, Picard, Poincare, and Raffy in the Bulletin de la
Societe Mathematique de France, in Legendre's Traite des fonctions elliptiques
(vol. 1, ch. 26), in Halphen's Traite' des fonctions elliptiques (vol. 2, ch. 14),

and in Euneper's Elliptische Funktionen. The literature concerning the

general theory of algebraical functions and their integrals is too extensive to
be summarised here : the reader may be referred to Appell and Goursat's
Theorie des fonctions algebriques, and Wirtinger's article Algebraische Funk-
tionen und ihre Integrate in the Encyclopddie der Mathematischen Wissen-
schaften, II b 2.
 '

66

APPENDIX II

ON ABEL'S PROOF OF THE THEOREM OF V., § 11

Abel's proof {(Euvres, vol. 1, p. 545) is as follows* :

We have
*(*, «)=0 (1),

where ijr is an irreducible polynomial of degree m in u. If we make use of the

equation f(x, y) = 0, we can introduce y into this equation, and write it in the
form
<j>(x,y, «)=0 (2),

where <f> is a polynomial in the three variables .r, y, and wt; and we can
suppose like \^, of degree m in m and irreducible, that is to say not
<f>,

divisible by any polynomial of the same form which is not a constant

multiple of <£ or itself a constant.
From/=0, iji = Owe deduce
¥ + <!£ty=o d± o±dy .tydu
dx
+ dx
dx dy ' dx dy du dx '

and, eliminating -f , we obtain an equation of the form

du X (x, y, u)

dx y. (x, y, u)

where X and p are polynomials in x, y, and u. And in order that u should

be an integral of y it is necessary and sufficient that

X-y/* = (3).

Abel now applies Lemma (2) of § 11, or rather its analogue for polynomials
in u whose coefficients are polynomials in x and y, to the two polynomials (f>

and X-^/u, and infers that all the roots u, u\... of 0=^0 satisfy (3). From
this he deduces that u, u',... are all integrals of y, and so that

u+u' + ...
m+1 •(4)

* The theorem with which Abel is engaged is a very much more general
theorem.
t Or, au lieu de supposer ees coefficiens rationnels en x, nous les supposerons
'

rationnels en x, y ; car cette supposition permise simplifiera beaucoup le

raisonnement '.
 APPENDIX II 67

is an integral of y. As (4) is a symmetric function of the roots of (2), it is a

rational function of x and y, whence his conclusion follows*.
It will be observed that the hypothesis that (2) does actually involve
y is essential, if we are to avoid the absurd conclusion that u is necessarily
a rational function of x only. On the other hand it is not obvious how
the presence of y in <j> affects the other steps in the argument.
The crucial inference is that which asserts that because the equations
<f>
= and \-yp = 0, considered as equations in u, have a root in common,
and <f>
is irreducible, therefore X-y/i is divisible by <fi.
This inference is

invalid.
We could only apply the lemma in this way if the equation (3) were
satisfiedby one of the roots of (2) identically, that is to say for all values of
x and y. But this is not the case. The equations are satisfied by the same
value of u only when x and y are connected by the equation (1).
Suppose, for example, that

«= 2 ^+*>-
'~W+*)>
Then we may take
f=(l+x)y*-l,

and = uy-2. <j>

Differentiating the equations /=0 and = 0, (f>

and eliminating^, we find

du u X
dx 2(l+#) p'

Thus <j> = ity-2, \-yii=u-2y(\+x);

and these polynomials have a common factor only iu virtue of the equation

f=0-

* Bertrand (Calcul inteffral, ch. 5) replaces the last step in Abel's argument by

the observation that if u and u' are both integrals of y then u - u' is constant (cf.
p. 39, bottom). It follows that the degree of the equation which defines u can be

decreased, which contradicts the hypothesis that it is irreducible.

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