Mathematical Formula Handbook

1. Series

Arithmetic and Geometric progressions

n
A.P. Sn = a + ( a + d) + ( a + 2d) + · · · + [ a + (n − 1)d] = [2a + (n − 1)d]
2  
2 n−1 1 − rn a
G.P. Sn = a + ar + ar + · · · + ar =a , S∞ = for |r| < 1
1−r 1−r
(These results also hold for complex series.)

Convergence of series: the ratio test

 
 un+1 
Sn = u1 + u2 + u3 + · · · + un converges as n→∞ if lim  <1
n→∞ un 

Convergence of series: the comparison test

If each term in a series of positive terms is less than the corresponding term in a series known to be convergent,
then the given series is also convergent.

Binomial expansion
n(n − 1) 2 n(n − 1)(n − 2) 3
(1 + x)n = 1 + nx + x + x + ···
2! 3!
n
If n is a positive integer the series terminates and is valid for all x: the term in x r is n Cr xr or where n Cr ≡
n! r
is the number of different ways in which an unordered sample of r objects can be selected from a set of
r!(n − r)!
n objects without replacement. When n is not a positive integer, the series does not terminate: the infinite series is
convergent for | x| < 1.

Taylor and Maclaurin Series

If y( x) is well-behaved in the vicinity of x = a then it has a Taylor series,
dy u2 d2 y u3 d3 y
y( x) = y( a + u) = y( a) + u + + + ···
dx 2! dx2 3! dx3
where u = x − a and the differential coefficients are evaluated at x = a. A Maclaurin series is a Taylor series with
a = 0,
dy x2 d2 y x3 d3 y
y( x) = y(0) + x + 2
+ +···
dx 2! dx 3! dx3

Power series with real variables

x2 xn
ex =1+x+ + ···+ +··· valid for all x
2! n!
2 3
x x xn
ln(1 + x) = x− + + · · · + (−1)n+1 + · · · valid for −1 < x ≤ 1
2 3 n
eix + e−ix x2 x4 x6
cos x = = 1− + − + ··· valid for all values of x
2 2! 4! 6!
eix − e−ix x3 x5
sin x = = x− + + ··· valid for all values of x
2i 3! 5!
1 2 5 π π
tan x = x + x3 + x +··· valid for − <x<
3 15 2 2
x3 x5
tan−1 x = x− + − ··· valid for −1 ≤ x ≤ 1
3 5
1 x3 1.3 x5
sin−1 x = x+ + + ··· valid for −1 < x < 1
2 3 2.4 5

2
Integer series
N
N ( N + 1)
∑n = 1+ 2+ 3+ ···+ N =
2
1
N
N ( N + 1)(2N + 1)
∑ n2 = 12 + 22 + 32 + · · · + N 2 = 6
1
N
N 2 ( N + 1)2
∑ n3 = 13 + 23 + 33 + · · · + N 3 = [1 + 2 + 3 + · · · N ] 2 = 4
1

∞
(−1)n+1 1 1 1
∑ n
= 1 − + − + · · · = ln 2
2 3 4
[see expansion of ln (1 + x)]
1
∞
(−1)n+1 1 1 1 π
∑ 2n − 1
= 1− + − + ··· =
3 5 7 4
[see expansion of tan−1 x]
1
∞
1 1 1 1 π2
∑ n2 =1+
4
+ +
9 16
+··· =
6
1
N
N ( N + 1)( N + 2)( N + 3)
∑ n(n + 1)(n + 2) = 1.2.3 + 2.3.4 + · · · + N ( N + 1)( N + 2) = 4
1

This last result is a special case of the more general formula,

N
N ( N + 1)( N + 2) . . . ( N + r)( N + r + 1)
∑ n(n + 1)(n + 2) . . . (n + r) = r+2
.
1

Plane wave expansion

∞
exp(ikz) = exp(ikr cos θ ) = ∑ (2l + 1)il jl (kr) Pl (cos θ),
l =0

where Pl (cos θ ) are Legendre polynomials (see section 11) and j l (kr) are spherical Bessel functions, defined by
r
π
jl (ρ) = J 1 (ρ), with Jl ( x) the Bessel function of order l (see section 11).
2ρ l + /2

2. Vector Algebra

2
If i, j, k are orthonormal vectors and A = A x i + A y j + A z k then | A| = A2x + A2y + A2z . [Orthonormal vectors ≡
orthogonal unit vectors.]

Scalar product

A · B = | A| | B| cos θ where θ is the angle between the vectors

 
Bx
= A x Bx + A y B y + A z Bz = [ A x A y A z ]  B y 
Bz

Scalar multiplication is commutative: A · B = B · A.

Equation of a line
A point r ≡ ( x, y, z) lies on a line passing through a point a and parallel to vector b if
r = a + λb
with λ a real number.

3
Equation of a plane
A point r ≡ ( x, y, z) is on a plane if either
(a) r · b
d = |d|, where d is the normal from the origin to the plane, or
x y z
(b) + + = 1 where X, Y, Z are the intercepts on the axes.
X Y Z

Vector product
A × B = n | A| | B| sin θ, where θ is the angle between the vectors and n is a unit vector normal to the plane containing
A and B in the direction for which A, B, n form a right-handed set of axes.

A × B in determinant form A × B in matrix form

    
 i j k  0 − Az A y Bx

 Ax A y Az   Az 0 − Ax   By 
 
 Bx B y Bz  − A y Ax 0 Bz

Vector multiplication is not commutative: A × B = − B × A.

Scalar triple product

 
 Ax Ay A z 

A × B · C = A · B × C =  Bx By Bz  = − A × C · B, etc.
 Cx Cy Cz 

Vector triple product

A × ( B × C ) = ( A · C ) B − ( A · B)C, ( A × B) × C = ( A · C ) B − ( B · C ) A

Non-orthogonal basis

A = A1 e1 + A2 e2 + A3 e3
e2 × e3
A1 = 0 · A where 0 =
e1 · (e2 × e3 )
Similarly for A2 and A3 .

Summation convention
a = ai ei implies summation over i = 1 . . . 3
a·b = ai bi
( a × b)i = εi jk a j bk where ε123 = 1; εi jk = −εik j
εi jkεklm = δil δ jm − δimδ jl

4
 3. Matrix Algebra

Unit matrices
The unit matrix I of order n is a square matrix with all diagonal elements equal to one and all off-diagonal elements
zero, i.e., ( I ) i j = δi j . If A is a square matrix of order n, then AI = I A = A. Also I = I −1 .
I is sometimes written as In if the order needs to be stated explicitly.

Products
If A is a (n × l ) matrix and B is a (l × m) then the product AB is defined by
l
( AB)i j = ∑ Aik Bk j
k=1

In general AB 6= BA.

Transpose matrices
If A is a matrix, then transpose matrix A T is such that ( A T )i j = ( A) ji .

Inverse matrices
If A is a square matrix with non-zero determinant, then its inverse A −1 is such that AA−1 = A−1 A = I.
transpose of cofactor of A i j
( A−1 )i j =
| A|
where the cofactor of A i j is (−1)i+ j times the determinant of the matrix A with the j-th row and i-th column deleted.

Determinants
If A is a square matrix then the determinant of A, | A| (≡ det A) is defined by
| A| = ∑ i jk... A1i A2 j A3k . . .
i, j,k,...

where the number of the suffixes is equal to the order of the matrix.

2×2 matrices
 
a b
If A = then,
c d
   
a c 1 d −b
| A| = ad − bc AT = A−1 =
b d | A| −c a

Product rules
( AB . . . N ) T = N T . . . B T A T
( AB . . . N )−1 = N −1 . . . B−1 A−1 (if individual inverses exist)
| AB . . . N | = | A| | B| . . . | N | (if individual matrices are square)

Orthogonal matrices
An orthogonal matrix Q is a square matrix whose columns q i form a set of orthonormal vectors. For any orthogonal
matrix Q,
Q−1 = Q T , | Q| = ±1, Q T is also orthogonal.

5
Solving sets of linear simultaneous equations
If A is square then Ax = b has a unique solution x = A −1 b if A−1 exists, i.e., if | A| 6= 0.
If A is square then Ax = 0 has a non-trivial solution if and only if | A| = 0.
An over-constrained set of equations Ax = b is one in which A has m rows and n columns, where m (the number
of equations) is greater than n (the number of variables). The best solution x (in the sense that it minimizes the
error | Ax − b|) is the solution of the n equations A T Ax = A T b. If the columns of A are orthonormal vectors then
x = A T b.

Hermitian matrices
The Hermitian conjugate of A is A † = ( A∗ ) T , where A∗ is a matrix each of whose components is the complex
conjugate of the corresponding components of A. If A = A † then A is called a Hermitian matrix.

Eigenvalues and eigenvectors

The n eigenvalues λ i and eigenvectors u i of an n × n matrix A are the solutions of the equation Au = λ u. The
eigenvalues are the zeros of the polynomial of degree n, Pn (λ ) = | A − λ I |. If A is Hermitian then the eigenvalues
λi are real and the eigenvectors u i are mutually orthogonal. | A − λ I | = 0 is called the characteristic equation of the
matrix A.
Tr A = ∑ λi , also | A| = ∏ λi .
i i

If S is a symmetric matrix, Λ is the diagonal matrix whose diagonal elements are the eigenvalues of S, and U is the
matrix whose columns are the normalized eigenvectors of A, then
U T SU = Λ and S = UΛU T.
If x is an approximation to an eigenvector of A then x T Ax/( x T x) (Rayleigh’s quotient) is an approximation to the
corresponding eigenvalue.

Commutators
[ A, B] ≡ AB − BA
[ A, B] = −[ B, A]
[ A, B]† = [ B† , A† ]
[ A + B, C ] = [ A, C ] + [ B, C ]
[ AB, C ] = A[ B, C ] + [ A, C ] B
[ A, [ B, C ]] + [ B, [C, A]] + [C, [ A, B]] = 0

Hermitian algebra

b† = (b∗1 , b∗2 , . . .)

Matrix form Operator form Bra-ket form

Z Z
Hermiticity b∗ · A · c = ( A · b)∗ · c ψ∗ Oφ = (Oψ)∗φ hψ|O|φi

Eigenvalues, λ real Au i = λ(i) ui Oψi = λ(i)ψi O |i i = λ i | i i

Z
Orthogonality ui · u j = 0 ψ∗i ψ j = 0 hi | j i = 0 (i 6 = j )
Z 
Completeness b = ∑ ui (ui · b) φ = ∑ ψi ψ∗i φ φ = ∑ |i i hi |φi
i i i

Rayleigh–Ritz
Z
b∗ · A · b ψ∗ Oψ hψ|O|ψi
Lowest eigenvalue λ0 ≤ λ0 ≤ Z
b∗ · b ψ ψ ∗ hψ|ψi

6
 6. Trigonometric Formulae

cos2 A + sin 2 A = 1 sec2 A − tan2 A = 1 cosec2 A − cot2 A = 1

2 tan A
sin 2A = 2 sin A cos A cos 2A = cos 2 A − sin 2 A tan 2A = .
1 − tan2 A

cos( A + B) + cos( A − B)
sin ( A ± B) = sin A cos B ± cos A sin B cos A cos B =
2

cos( A − B) − cos( A + B)
cos( A ± B) = cos A cos B ∓ sin A sin B sin A sin B =
2

tan A ± tan B sin( A + B) + sin ( A − B)

tan( A ± B) = sin A cos B =
1 ∓ tan A tan B 2

A+B A−B 1 + cos 2A

sin A + sin B = 2 sin cos cos2 A =
2 2 2
A+B A−B 1 − cos 2A
sin A − sin B = 2 cos sin sin 2 A =
2 2 2
A+B A−B 3 cos A + cos 3A
cos A + cos B = 2 cos cos cos3 A =
2 2 4
A+B A−B 3 sin A − sin 3A
cos A − cos B = −2 sin sin sin 3 A =
2 2 4

Relations between sides and angles of any plane triangle

In a plane triangle with angles A, B, and C and sides opposite a, b, and c respectively,
a b c
= = = diameter of circumscribed circle.
sin A sin B sin C
a2 = b2 + c2 − 2bc cos A
a = b cos C + c cos B
b2 + c2 − a2
cos A =
2bc
A−B a−b C
tan = cot
2 a+b 2
q
1 1 1 1
area = ab sin C = bc sin A = ca sin B = s(s − a)(s − b)(s − c), where s = ( a + b + c)
2 2 2 2

Relations between sides and angles of any spherical triangle

In a spherical triangle with angles A, B, and C and sides opposite a, b, and c respectively,
sin a sin b sin c
= =
sin A sin B sin C
cos a = cos b cos c + sin b sin c cos A
cos A = − cos B cos C + sin B sin C cos a

10
 7. Hyperbolic Functions

1 x x2 x4
cosh x = ( e + e− x ) = 1 + + + ··· valid for all x
2 2! 4!
1 x3 x5
sinh x = ( ex − e− x ) = x + + + ··· valid for all x
2 3! 5!
cosh ix = cos x cos ix = cosh x
sinh ix = i sin x sin ix = i sinh x
sinh x 1
tanh x = sech x =
cosh x cosh x
cosh x 1
coth x = cosech x =
sinh x sinh x
cosh 2 x − sinh 2 x = 1

For large positive x:

ex
cosh x ≈ sinh x →
2
tanh x → 1
For large negative x:
e− x
cosh x ≈ − sinh x →
2
tanh x → −1

Relations of the functions

sinh x = − sinh (− x) sech x = sech(− x)
cosh x = cosh (− x) cosech x = − cosech(− x)

tanh x = − tanh(− x) coth x = − coth (− x)

2 tanh ( x/2) tanh x 1 + tanh2 ( x/2) 1

sinh x = 2
=q cosh x = 2
= q
1 − tanh ( x/2) 1 − tanh ( x/2)
1 − tanh2 x 1 − tanh2 x
q q
2
tanh x = 1 − sech x sech x = 1 − tanh2 x
q q
coth x = cosech 2 x + 1 cosech x = coth 2 x − 1
r r
cosh x − 1 cosh x + 1
sinh ( x/2) = cosh ( x/2) =
2 2
cosh x − 1 sinh x
tanh( x/2) = =
sinh x cosh x + 1
2 tanh x
sinh (2x) = 2 sinh x cosh x tanh(2x) =
1 + tanh 2 x
cosh (2x) = cosh 2 x + sinh 2 x = 2 cosh2 x − 1 = 1 + 2 sinh 2 x

sinh (3x) = 3 sinh x + 4 sinh 3 x cosh 3x = 4 cosh 3 x − 3 cosh x

3 tanh x + tanh3 x
tanh(3x) =
1 + 3 tanh2 x

11
 sinh ( x ± y) = sinh x cosh y ± cosh x sinh y

cosh( x ± y) = cosh x cosh y ± sinh x sinh y

tanh x ± tanh y
tanh( x ± y) =
1 ± tanh x tanh y
1 1 1 1
sinh x + sinh y = 2 sinh ( x + y) cosh ( x − y) cosh x + cosh y = 2 cosh ( x + y) cosh ( x − y)
2 2 2 2
1 1 1 1
sinh x − sinh y = 2 cosh ( x + y) sinh ( x − y) cosh x − cosh y = 2 sinh ( x + y) sinh ( x − y)
2 2 2 2
1 ± tanh ( x/2)
sinh x ± cosh x = = e± x
1 ∓ tanh( x/2)
sinh ( x ± y)
tanh x ± tanh y =
cosh x cosh y
sinh ( x ± y)
coth x ± coth y = ±
sinh x sinh y

Inverse functions
p !
−1 x x+ x2 + a2
sinh = ln for −∞ < x < ∞
a a
p !
−1 x x + x2 − a2
cosh = ln for x ≥ a
a a
 
−1 x 1 a+x
tanh = ln for x2 < a2
a 2 a−x
 
−1 x 1 x +a
coth = ln for x2 > a2
a 2 x−a
 s 
2
x a a
sech−1 = ln  + − 1 for 0 < x ≤ a
a x x2
 s 
2
x a a
cosech−1 = ln  + + 1 for x 6= 0
a x x2

8. Limits

nc xn → 0 as n → ∞ if | x| < 1 (any fixed c)

xn /n! → 0 as n → ∞ (any fixed x)

(1 + x/n)n → ex as n → ∞, x ln x → 0 as x → 0
f ( x) f 0 ( a)
If f ( a) = g( a) = 0 then lim = 0 (l’Hôpital’s rule)
x→ a g( x) g ( a)

12
 9. Differentiation
 u 0 u0 v − uv0
(uv)0 = u0 v + uv0 , =
v v2

(uv)(n) = u(n) v + nu(n−1) v(1) + · · · + n Cr u(n−r) v(r) + · · · + uv(n) Leibniz Theorem

 
n n n!
where Cr ≡ =
r r!(n − r)!

d d
(sin x) = cos x (sinh x) = cosh x
dx dx
d d
(cos x) = − sin x (cosh x) = sinh x
dx dx
d d
(tan x) = sec2 x (tanh x) = sech2 x
dx dx
d d
(sec x) = sec x tan x (sech x) = − sech x tanh x
dx dx
d d
(cot x) = − cosec2 x (coth x) = − cosech2 x
dx dx
d d
(cosec x) = − cosec x cot x (cosech x) = − cosech x coth x
dx dx

10. Integration

Standard forms
xn+1
Z
xn dx = +c for n 6= −1
n+1
1
Z Z
dx = ln x + c ln x dx = x(ln x − 1) + c
x  
1 ax x 1
Z Z
eax dx = e +c ax
x e dx = e ax
− 2 +c
a a a
 
x2 1
Z
x ln x dx = ln x − +c
2 2
Z
1 1 x
2 2
dx = tan−1 +c
a +x a a
   
1 1 −1 x 1 a+x
Z
dx = tanh + c = ln +c for x2 < a2
a2 − x2 a a 2a a−x
   
1 1 −1 x 1 x−a
Z
dx = − coth +c= ln +c for x2 > a2
x2 − a2 a a 2a x+a
x −1 1
Z
dx = +c for n 6= 1
( x2 ± a2 )n 2(n − 1) ( x2 ± a2 )n−1
x 1
Z
dx = ln( x2 ± a2 ) + c
x2 ± a2 2
Z
1 x
p dx = sin−1 +c
a2 − x2 a
Z
1  p 
p dx = ln x + x2 ± a2 + c
x2 ± a2
Z
x p
p dx = x2 ± a2 + c
x2 ± a2
p 1h p 2  x i
Z
a2 − x2 dx = x a − x2 + a2 sin −1 +c
2 a

13
 ∞ 1
Z
dx = π cosec pπ for p < 1
0 (1 + x) x p
r
∞ ∞ 1 π
Z Z
2 2
cos( x ) dx = sin ( x ) dx =
0 0 2 2
Z ∞ √
exp(− x2 /2σ 2 ) dx = σ 2π
−∞
 √
Z ∞  1 × 3 × 5 × · · · (n − 1)σ n+1 2π for n ≥ 2 and even
n 2 2
x exp(− x /2σ ) dx =
−∞ 
0 for n ≥ 1 and odd
Z Z
sin x dx = − cos x + c sinh x dx = cosh x + c
Z Z
cos x dx = sin x + c cosh x dx = sinh x + c
Z Z
tan x dx = − ln(cos x) + c tanh x dx = ln(cosh x) + c
Z Z
cosec x dx = ln(cosec x − cot x) + c cosech x dx = ln [tanh( x/2)] + c
Z Z
sec x dx = ln(sec x + tan x) + c sech x dx = 2 tan−1 ( ex ) + c
Z Z
cot x dx = ln(sin x) + c coth x dx = ln(sinh x) + c

sin (m − n) x sin (m + n) x
Z
sin mx sin nx dx = − +c if m2 6= n2
2(m − n) 2(m + n)
sin (m − n) x sin (m + n) x
Z
cos mx cos nx dx = + +c if m2 6= n2
2(m − n) 2(m + n)

Standard substitutions
If the integrand is a function of: substitute:
p
( a2 − x2 ) or a2 − x2 x = a sin θ or x = a cos θ
p
( x2 + a2 ) or x2 + a2 x = a tan θ or x = a sinh θ
p
( x2 − a2 ) or x2 − a2 x = a sec θ or x = a cosh θ

If the integrand is a rational function of sin x or cos x or both, substitute t = tan( x/2) and use the results:
2t 1 − t2 2 dt
sin x = cos x = dx = .
1 + t2 1 + t2 1 + t2

If the integrand is of the form: substitute:

dx
Z
p px + q = u2
( ax + b) px + q

dx 1
Z
q ax + b = .
( ax + b) px2 + qx + r u

14
Integration by parts
b
b Z b
Z 
u dv = uv − v du
a a a

Differentiation of an integral
If f ( x, α ) is a function of x containing a parameter α and the limits of integration a and b are functions of α then
∂
Z b(α ) Z b(α )
d db da
f ( x, α ) dx = f (b, α ) − f ( a, α ) + f ( x, α ) dx.
dα a (α ) dα dα a (α ) ∂α
Special case,
d
Z x
f ( y) dy = f ( x).
dx a

Dirac δ-‘function’
1 ∞
Z
δ (t − τ ) = exp[iω(t − τ )] dω.
2π −∞
Z ∞
If f (t) is an arbitrary function of t then δ (t − τ ) f (t) dt = f (τ ).
−∞
Z ∞
δ (t) = 0 if t 6= 0, also δ (t) dt = 1
−∞

Reduction formulae

Factorials

n! = n(n − 1)(n − 2) . . . 1, 0! = 1.
Stirling’s formula for large n: ln(n!) ≈ n ln n − n.
Z ∞ Z ∞ √ √
For any p > −1, x p e− x dx = p x p−1 e− x dx = p!. (− 1/2)! = π, ( 1/2)! = π/ ,
2 etc.
0 0
Z 1 p!q!
For any p, q > −1, x p (1 − x)q dx = .
0 ( p + q + 1)!

Trigonometrical

If m, n are integers,
Z π/ 2
m − 1 π/ 2 n − 1 π/ 2
Z Z
sin m θ cos n θ dθ =sin m−2 θ cosn θ dθ = sin m θ cosn−2 θ dθ
0 m+n 0 m+n 0
and can therefore be reduced eventually to one of the following integrals
Z π/ 2 Z π/ 2 Z π/ 2 Z π/ 2
1 π
sin θ cos θ dθ = , sin θ dθ = 1, cos θ dθ = 1, dθ = .
0 2 0 0 0 2

Other

r
∞ (n − 1) 1 π 1
Z
If In = xn exp(−α x2 ) dx then In = In − 2 , I0 = , I1 = .
0 2α 2 α 2α

15
 11. Differential Equations

Diffusion (conduction) equation

∂ψ
= κ ∇2ψ
∂t

Wave equation
1 ∂2ψ
∇2ψ =
c2 ∂t2

Legendre’s equation
d2 y dy
(1 − x2 ) − 2x + l (l + 1) y = 0,
dx2 dx
 l
1 d
l
solutions of which are Legendre polynomials Pl ( x), where Pl ( x) = l x2 − 1 , Rodrigues’ formula so
2 l! dx
1 2
P0 ( x) = 1, P1 ( x) = x, P2 ( x) = (3x − 1) etc.
2

Recursion relation

1
Pl ( x) = [(2l − 1) xPl −1 ( x) − (l − 1) Pl −2( x)]
l

Orthogonality

Z 1 2
Pl ( x) Pl 0 ( x) dx = δll 0
−1 2l + 1

Bessel’s equation
d2 y dy
x2 +x + ( x2 − m2 ) y = 0,
dx2 dx
solutions of which are Bessel functions Jm ( x) of order m.

Series form of Bessel functions of the first kind

∞
(−1)k ( x/2)m+2k
Jm ( x ) = ∑ k!(m + k)!
(integer m).
k=0

The same general form holds for non-integer m > 0.

16
 13. Functions of Several Variables

∂φ
If φ = f ( x, y, z, . . .) then implies differentiation with respect to x keeping y, z, . . . constant.
∂x
∂φ ∂φ ∂φ ∂φ ∂φ ∂φ
dφ = dx + dy + dz + · · · and δφ ≈ δx + δy + δz + · · ·
∂x ∂y ∂z ∂x ∂y ∂z
  
∂φ ∂φ ∂φ 
where x, y, z, . . . are independent variables. is also written as or when the variables kept
∂x ∂x ∂x 
y,... y,...
constant need to be stated explicitly.
∂ 2φ ∂2φ
If φ is a well-behaved function then = etc.
∂x ∂y ∂y ∂x
If φ = f ( x, y),
       
∂φ 1 ∂φ ∂x ∂y
=   , = −1.
∂x y ∂x ∂x y ∂y φ ∂φ x
∂φ y

Taylor series for two variables

If φ( x, y) is well-behaved in the vicinity of x = a, y = b then it has a Taylor series
 2 2 
∂φ ∂φ 1 2∂ φ ∂2φ 2∂ φ
φ( x, y) = φ( a + u, b + v) = φ( a, b) + u +v + u + 2uv +v +···
∂x ∂y 2! ∂x2 ∂x ∂y ∂y2
where x = a + u, y = b + v and the differential coefficients are evaluated at x = a, y=b

Stationary points
∂φ ∂φ ∂2φ ∂2φ ∂2φ
A function φ = f ( x, y) has a stationary point when = = 0. Unless 2 = = = 0, the following
∂x ∂y ∂x ∂y 2 ∂x ∂y
conditions determine whether it is a minimum, a maximum or a saddle point.

∂2φ ∂2φ 
Minimum: > 0, or > 0, 

  2 2
∂x2 ∂y2 ∂2φ ∂2φ ∂φ
2 2 and >
∂φ ∂φ  2
∂x ∂y 2 ∂x ∂y
Maximum: 2
< 0, or 2
< 0, 


∂x ∂y
 2
∂2φ ∂2φ ∂2φ
Saddle point: <
∂x2 ∂y2 ∂x ∂y
∂2φ ∂2φ ∂2φ
If = = = 0 the character of the turning point is determined by the next higher derivative.
∂x2 ∂y2 ∂x ∂y

Changing variables: the chain rule

If φ = f ( x, y, . . .) and the variables x, y, . . . are functions of independent variables u, v, . . . then

∂φ ∂φ ∂x ∂φ ∂y
= + + ···
∂u ∂x ∂u ∂y ∂u
∂φ ∂φ ∂x ∂φ ∂y
= + + ···
∂v ∂x ∂v ∂y ∂v
etc.

18
Changing variables in surface and volume integrals – Jacobians
If an area A in the x, y plane maps into an area A 0 in the u, v plane then
 
 ∂x ∂x 
 
Z Z  ∂u ∂v 
f ( x, y) dx dy = f (u, v) J du dv where J =   

A A0  ∂y ∂y 
 
∂u ∂v
∂( x, y)
The Jacobian J is also written as . The corresponding formula for volume integrals is
∂(u, v)
 
 ∂x ∂x ∂x 
 
 ∂u ∂v ∂w 
 
Z Z  ∂y ∂y ∂y 
f ( x, y, z) dx dy dz = f (u, v, w) J du dv dw where now J =  

V V0  ∂u ∂v ∂w 
 
 ∂z ∂z ∂z 
 
∂u ∂v ∂w

14. Fourier Series and Transforms

Fourier series
If y( x) is a function defined in the range −π ≤ x ≤ π then
M M0
y( x) ≈ c0 + ∑ cm cos mx + ∑ sm sin mx
m=1 m=1

where the coefficients are

1
Z π
c0 = y( x) dx
2π −π
1
Z π
cm = y( x) cos mx dx (m = 1, . . . , M)
π −π
1
Z π
sm = y( x) sin mx dx (m = 1, . . . , M 0 )
π −π
with convergence to y( x) as M, M 0 → ∞ for all points where y( x) is continuous.

Fourier series for other ranges

Variable t, range 0 ≤ t ≤ T, (i.e., a periodic function of time with period T, frequency ω = 2π/ T).
y(t) ≈ c0 + ∑ cm cos mωt + ∑ sm sin mωt
where
ω T ω T ω T
Z Z Z
c0 = y(t) dt, cm = y(t) cos mωt dt, sm = y(t) sin mωt dt.
2π 0 π 0 π 0
Variable x, range 0 ≤ x ≤ L,
2mπx 2mπx
y( x) ≈ c0 + ∑ cm cos + ∑ sm sin
L L
where
1 L 2 L 2mπx 2 L 2mπx
Z Z Z
c0 = y( x) dx, cm = y( x) cos dx, sm = y( x) sin dx.
L 0 L 0 L L 0 L

19
 18. Statistics

Mean and Variance

A random variable X has a distribution over some subset x of the real numbers. When the distribution of X is
discrete, the probability that X = x i is Pi . When the distribution is continuous, the probability that X lies in an
interval δx is f ( x)δx, where f ( x) is the probability density function.
Z
Mean µ = E( X ) = ∑ Pi xi or x f ( x) dx.
Z
Variance σ 2 = V ( X ) = E[( X − µ )2 ] = ∑ Pi (xi − µ )2 or ( x − µ )2 f ( x) dx.

Probability distributions
2 x Z
2
Error function: erf( x) = √ e− y dy
π 0
 
n x n− x
Binomial: f ( x) = p q where q = (1 − p), µ = np, σ 2 = npq, p < 1.
x
µ x −µ
Poisson: f ( x) = e , and σ 2 = µ
x!
 
1 ( x − µ )2
Normal: f ( x) = √ exp −
σ 2π 2σ 2

Weighted sums of random variables

If W = aX + bY then E(W ) = aE( X ) + bE(Y ). If X and Y are independent then V (W ) = a 2 V ( X ) + b2 V (Y ).

Statistics of a data sample x 1 , . . . , xn

1
Sample mean x =
n ∑ xi
 
1 1
Sample variance s =
n
2
∑( x i − x ) 2
=
n∑ i
x2 − x2 = E( x2 ) − [E( x)]2

Regression (least squares fitting)

To fit a straight line by least squares to n pairs of points ( x i , yi ), model the observations by y i = α + β( xi − x) + i ,
where the i are independent samples of a random variable with zero mean and variance σ 2 .
1 1 1
Sample statistics: s 2x =
n ∑( x i − x ) 2 , s2y =
n ∑ ( y i − y) 2 , s2xy =
n ∑(xi − x)( yi − y).
s2xy n
Estimators: α b=
b = y, β ; E(Y at x) = α b ( x − x); σb 2 =
b+β (residual variance),
s2x n−2
1 s4
where residual variance = ∑{ yi − α b ( xi − x)}2 = s2 − xy .
b −β y
n s2x

b2
b are σ σb 2
b and β
Estimates for the variances of α and 2 .
n ns x

s2xy
Correlation coefficient: ρ
b=r= .
sx s y

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