Physics 116A Winter 2011

The complex logarithm, exponential and power functions

In these notes, we examine the logarithm, exponential and power functions, where
the arguments

of these functions can be complex numbers. In particular, we are

interested in how their properties dier from the properties of the corresponding
real-valued functions.

1. Review of the properties of the argument of a complex number

Before we begin, I shall review the properties of the argument of a non-zero
complex number z, denoted by arg z (which is a multi-valued function), and the
principal value of the argument, Arg z, which is single-valued and conventionally
dened such that:
< Arg z . (1)
Details can be found in the class handout entitled, The argument of a complex
number. Here, we recall a number of results from that handout. One can regard
arg z as a set consisting of the following elements,
arg z = Arg z + 2n, n = 0 , 1 , 2 , 3 , . . . , < Arg z . (2)
One can also express Arg z in terms of arg z as follows:
Arg z = arg z + 2
_
1
2

arg z
2
_
, (3)
where [ ] denotes the greatest integer function. That is, [x] is dened to be the
largest integer less than or equal to the real number x. Consequently, [x] is the
unique integer that satises the inequality
x 1 < [x] x, for real x and integer [x] . (4)

Note that the word argument has two distinct meanings. In this context, given a function
w = f(z), we say that z is the argument of the function f. This should not be confused with
the argument of a complex number, arg z.

The following three books were particularly useful in the preparation of these notes:
1. Complex Variables and Applications, by James Ward Brown and Ruel V. Churchill (McGraw
Hill, New York, 2004).
2. Elements of Complex Variables, by Louis L. Pennisi, with the collaboration of Louis I. Gordon
and Sim Lasher (Holt, Rinehart and Winston, New York, 1963).
3. The Theory of Analytic Functions: A Brief Course, by A.I. Markushevich (Mir Publishers,
Moscow, 1983).
1
For example, [1.5] = [1] = 1 and [0.5] = 1. One can check that Arg z as
dened in eq. (3) does fall inside the principal interval specied by eq. (1).
The multi-valued function arg z satises the following properties,
arg(z
1
z
2
) = arg z
1
+ arg z
2
, (5)
arg
_
z
1
z
2
_
= arg z
1
arg z
2
. (6)
arg
_
1
z
_
= arg z = arg z . (7)
Eqs. (5)(7) should be viewed as set equalities, i.e. the elements of the sets indi-
cated by the left-hand side and right-hand side of the above identities coincide.
However, the following results are not set equalities:
arg z + arg z = 2 arg z , arg z arg z = 0 , (8)
which, by virtue of eqs. (5) and (6), yield:
arg z
2
= arg z + arg z = 2 arg z , arg(1) = arg z arg z = 0 . (9)
For example, arg(1) = 2n, for n = 0 1, 2, . . .. More generally,
arg z
n
= arg z + arg z + arg z
. .
n
= n arg z . (10)
We also note some properties of the the principal value of the argument.
Arg (z
1
z
2
) = Arg z
1
+ Arg z
2
+ 2N
+
, (11)
Arg (z
1
/z
2
) = Arg z
1
Arg z
2
+ 2N

, (12)
where the integers N

are determined as follows:

N

=
_

_
1 , if Arg z
1
Arg z
2
> ,
0 , if < Arg z
1
Arg z
2
,
1 , if Arg z
1
Arg z
2
.
(13)
If we set z
1
= 1 in eq. (12), we nd that
Arg(1/z) = Arg z =
_
Arg z , if Im z = 0 and z = 0 ,
Arg z , if Im z = 0 .
(14)
Note that for z real, both 1/z and z are also real so that in this case z = z and
Arg(1/z) = Arg z = Arg z. In addition, in contrast to eq. (10), we have
Arg(z
n
) = nArg z + 2N
n
, (15)
2
where the integer N
n
is given by:
N
n
=
_
1
2

n
2
Arg z
_
, (16)
and [ ] is the greatest integer bracket function introduced in eq. (4).
2. Properties of the real-valued logarithm, exponential and power func-
tions
Consider the logarithm of a positive real number. This function satises a
number of properties:
e
lnx
= x, (17)
ln(e
a
) = a , (18)
ln(xy) = ln(x) + ln(y) , (19)
ln
_
x
y
_
= ln(x) ln(y) , (20)
ln
_
1
x
_
= ln(x) , (21)
ln x
p
= p lnx, (22)
for positive real numbers x and y and arbitrary real numbers a and p. Likewise,
the power function dened over the real numbers satises:
x
a
= e
a lnx
, (23)
x
a
x
b
= x
a+b
, (24)
x
a
x
b
= x
ab
, (25)
1
x
a
= x
a
, (26)
(x
a
)
b
= x
ab
, (27)
(xy)
a
= x
a
y
a
, (28)
_
x
y
_
a
= x
a
y
a
, (29)
for positive real numbers x and y and arbitrary real numbers a and b. Closely
related to the power function is the generalized exponential function dened over
3
the real numbers. This function satises:
a
x
= e
xln a
, (30)
a
x
a
y
= a
x+y
, (31)
a
x
a
y
= a
xy
, (32)
1
a
x
= a
x
, (33)
(a
x
)
y
= a
xy
, (34)
(ab)
x
= a
x
b
x
, (35)
_
a
b
_
x
= a
x
b
x
. (36)
for positive real numbers a and b and arbitrary real numbers x and y.
We would like to know which of these relations are satised when these func-
tions are extended to the complex plane. It is dangerous to assume that all of
the above relations are valid in the complex plane without modication, as this
assumption can lead to seemingly paradoxical conclusions. Here are three exam-
ples:
1. Since 1/(1) = (1)/1 = 1,
_
1
1
=
1
i
=
_
1
1
=
i
1
.
Hence, 1/i = i or i
2
= 1. But i
2
= 1, so we have proven that 1 = 1.
2. Since 1 = (1)(1),
1 =

1 =
_
(1)(1) = (

1)(

1) = i i = 1 .
3. To prove that ln(z) = ln(z) for all z = 0, we proceed as follows:
ln(z
2
) = ln[(z)
2
] ,
ln(z) + ln(z) = ln(z) + ln(z) ,
2 ln(z) = 2 ln(z) ,
ln(z) = ln(z) .
Of course, all these proofs are faulty. The fallacy in the rst two proofs can
be traced back to eqs. (28) and (29), which are true for real-valued functions but
not true in general for complex-valued functions. The fallacy in the third proof
is more subtle, and will be addressed later in these notes. A careful study of the
complex logarithm, power and exponential functions will reveal how to correctly
modify eqs. (17)(36) and avoid pitfalls that can lead to false results.
4
3. Denition of the complex exponential function
We begin with the complex exponential function, which is dened via its power
series:
e
z
=

n=0
z
n
n!
,
where z is any complex number. Using this power series denition, one can verify
that:
e
z
1
+z
2
= e
z
1
e
z
2
, for all complex z
1
and z
2
. (37)
In particular, if z = x + iy where x and y are real, then it follows that
e
z
= e
x+iy
= e
x
e
iy
= e
x
(cos y + i sin y) .
One can quickly verify that eqs. (30)(33) are satised by the complex exponential
function. In addition, eq. (34) clearly holds when the outer exponent is an integer:
(e
z
)
n
= e
nz
, n = 0 , 1 , 2 , . . . . (38)
If the outer exponent is a non-integer, then the resulting expression is a multi-
valued power function. We will discuss this case in more detail in section 8.
Before moving on, we record one key property of the complex exponential:
e
2in
= 1 , n = 0 , 1 , 2 , 3 , . . . . (39)
4. Denition of the complex logarithm
In order to dene the complex logarithm, one must solve the complex equation:
z = e
w
, (40)
for w, where z is any non-zero complex number. If we write w = u + iv, then
eq. (40) can be written as
e
u
e
iv
= |z|e
i arg z
. (41)
Eq. (41) implies that:
|z| = e
u
, v = arg z .
The equation |z| = e
u
is a real equation, so we can write u = ln |z|, where ln |z| is
the ordinary logarithm evaluated with positive real number arguments. Thus,
w = u + iv = ln |z| + i arg z = ln |z| + i(Arg z + 2n) , n = 0 , 1 , 2 , 3 , . . .
(42)
5
We call w the complex logarithm and write w = ln z. This is a somewhat awkward
notation since in eq. (42) we have already used the symbol ln for the real logarithm.
We shall nesse this notational quandary by denoting the real logarithm in eq. (42)
by the symbol Ln. That is, Ln|z| shall denote the ordinary real logarithm of |z|.
With this notational convention, we rewrite eq. (42) as:
ln z = Ln|z| + i arg z = Ln|z| + i(Arg z + 2n) , n = 0 , 1 , 2 , 3 , . . . (43)
for any non-zero complex number z.
Clearly, ln z is a multi-valued function (as its value depends on the integer n).
It is useful to dene a single-valued function complex function, Ln z, called the
principal value of lnz as follows:
Lnz = Ln|z| + iArg z , < Arg z , (44)
which extends the denition of Ln z to the entire complex plane (excluding the
origin, z = 0, where the logarithmic function is singular). In particular, eq. (44)
implies that Ln(1) = i. Note that for real positive z, we have Arg z = 0, so
that eq. (44) simply reduces to the usual real logarithmic function in this limit.
The relation between ln z and its principal value is simple:
lnz = Ln z + 2in, n = 0 , 1 , 2 , 3 , . . . .
5. Properties of the complex logarithm
We now consider which of the properties given in eqs. (17)(22) apply to the
complex logarithm. Since we have dened the multi-value function ln z and the
single-valued function Ln z, we should examine the properties of both these func-
tions. We begin with the multi-valued function ln z. First, we examine eq. (17).
Using eq. (43), it follows that:
e
lnz
= e
Ln|z|
e
iArg z
e
2in
= |z|e
iArg z
= z . (45)
Thus, eq. (17) is satised. Next, we examine eq. (18) for z = x + iy:
ln(e
z
) = Ln|e
z
| + i(arg e
z
) = Ln(e
x
) + i(y + 2k) = x + iy + 2ik = z + 2ik ,
where k is an arbitrary integer. In deriving this result, we used the fact that
e
z
= e
x
e
iy
, which implies that arg(e
z
) = y + 2k.

Thus,
ln(e
z
) = z + 2ik = z , unless k = 0 . (46)

Note that Arg e

z
= y + 2N, where N is chosen such that < y + 2N . Moreover,
eq. (2) implies that arg e
z
= Arg e
z
+ 2n, where n = 0, 1, 2, . . .. Hence, arg(e
z
) = y + 2k,
where k = n + N is still some integer.
6
This is not surprising, since ln(e
z
) is a multi-valued function, which cannot be
equal to the single-valued function z. Indeed eq. (18) is false for the multi-valued
complex logarithm.
As a check, let us compute ln(e
ln z
) in two dierent ways. First, using eq. (45),
it follows that ln(e
ln z
) = ln z. Second, using eq. (46), ln(e
ln z
) = ln z + 2ik. This
seems to imply that ln z = ln z + 2ik. In fact, the latter is completely valid as a
set equality in light of eq. (43).
We now consider the properties exhibited in eqs. (19)(22). Using the deni-
tion of the multi-valued complex logarithms and the properties of arg z given in
eqs. (5)(7), it follows that eqs. (19)(21) are satised as set equalities:
ln(z
1
z
2
) = ln z
1
+ ln z
2
, (47)
ln
_
z
1
z
2
_
= ln z
1
ln z
2
. (48)
ln
_
1
z
_
= ln z . (49)
However, one must be careful in employing these results. One should not make
the mistake of writing, for example, ln z + ln z
?
= 2 lnz or ln z ln z
?
= 0. Both
these latter statements are false for the same reasons that eqs. (8) and (9) are not
identities under set equality. In particular, the multi-valued complex logarithm
does not satisfy eq. (22) when p is an integer n:
ln z
n
= ln z + ln z + + ln z
. .
n
= nln z , (50)
which follows from eq. (10). If p is not an integer, then z
p
is a complex multi-
valued function, and one needs further analysis to determine whether eq. (22) is
valid. In section 6, we will prove [see eq. (60)] that eq. (19) is satised by the
complex logarithm only if p = 1/n where n is an integer. In this case,
ln(z
1/n
) =
1
n
ln z , n = 1 , 2 , 3 , . . . . (51)
We next examine the properties of the single-valued function Ln z. Again,
we examine the six properties given by eqs. (17)(22). First, eq. (17) is trivially
satised since
e
Ln z
= e
Ln|z|
e
iArg z
= |z|e
iArg z
= z . (52)
However, eq. (18) is generally false. In particular, for z = x + iy
Ln(e
z
) = Ln |e
z
| + i(Arg e
z
) = Ln(e
x
) + i(Arg e
iy
) = x + iArg (e
iy
)
= x + i arg(e
iy
) + 2i
_
1
2

arg(e
iy
)
2
_
= x + iy + 2i
_
1
2

y
2
_
= z + 2i
_
1
2

Im z
2
_
, (53)
7
after using eq. (3), where [ ] is the greatest integer bracket function dened in
eq. (4). Thus, eq. (18) is satised only when < y . For values of y outside
the principal interval, eq. (18) contains an additive correction term as shown in
eq. (53).
As a check, let us compute Ln(e
Lnz
) in two dierent ways. First, using eq. (52),
it follows that Ln(e
Ln z
) = Ln z. Second, using eq. (53),
Ln(e
Ln z
) = ln z + 2i
_
1
2

Im Ln z
2
_
= Ln z + 2i
_
1
2

Arg z
2
_
= Ln z ,
where we have used Im Ln z = Arg z [see eq. (44)]. In the last step, we noted
that
0
1
2

Arg(z)
2
< 1 ,
due to eq. (1), which implies that the integer part of
1
2
Arg z/(2) is zero. Thus,
the two computations agree.
We now consider the properties exhibited in eqs. (19)(22). Ln z may not
satisfy any of these properties due to the fact that the principal value of the
complex logarithm must lie in the interval < Im Ln z . Using the results
of eqs. (11)(16), it follows that
Ln (z
1
z
2
) = Ln z
1
+ Ln z
2
+ 2iN
+
, (54)
Ln (z
1
/z
2
) = Ln z
1
Ln z
2
+ 2iN

, (55)
Ln(z
n
) = nLn z + 2iN
n
(integer n) , (56)
where the integers N

= 1, 0 or +1 and N
n
are determined by eqs. (13) and
(16), respectively, and
Ln(1/z) =
_
Ln(z) + 2i , if z is real and negative ,
Ln(z) , otherwise .
(57)
Note that eq. (19) is satised if Re z
1
> 0 and Re z
2
> 0 (in which case N
+
= 0).
In other cases, N
+
= 0 and eq. (19) fails. Similar considerations also apply to
eqs. (20)(22). For example, eq. (21) is satised by Ln z unless Arg z =
(equivalently for negative real values of z), as indicated by eq. (57). In particular,
one may use eq. (56) to verify that:
Ln[(1)
1
] = Ln(1) + 2i = i + 2i = i = Ln(1) ,
as expected, since (1)
1
= 1.
We cannot yet check whether eq. (22) is satised if p is a non-integer, since in
this case z
p
is a multi-valued function. Thus, we now turn our attention to the
complex power functions (and the related generalized exponential functions).
8
6. Denition of the generalized power and exponential functions
The generalized complex power function is dened via the following equation:
w = z
c
= e
c lnz
, z = 0 . (58)
To motivate this denition, we rst note that if c = k is an integer, then for
z = |z|e
i arg z
,
z
k
= |z|
k
e
ki arg z
= e
k Ln|z|
e
ki arg z
= e
k(Ln|z|+i arg z)
= e
k ln z
.
In this case, w = z
k
is a single-valued function, since
z
k
= e
k lnz
= e
k(Ln z +2in)
= e
k Ln z
.
If c = 1/k (where k is an integer), then we have:
z
1/k
= |z|
1/k
e
i arg(z)/k
= e
Ln|z|/k
e
i arg(z)/k
= e
(Ln|z|+i arg z)/k
= e
ln(z)/k
,
where |z|
1/k
refers to the positive real kth root of |z|. Combining the two results
just obtained, we can easily prove that eq. (58) holds for any rational real number
c. Since any irrational real number can be approximated (to any desired accuracy)
by a rational number, it follows by continuity that eq. (58) must hold for any real
number c. These arguments provide the motivation for dening the generalized
complex power function as in eq. (58) for an arbitrary complex power c.
Note that due to the multi-valued nature of ln z, it follows that w = z
c
= e
c lnz
is also multi-valued for any non-integer value of c, with a branch point at z = 0:
w = z
c
= e
c ln z
= e
c Ln z
e
2inc
, n = 0 , 1 , 2 , 3 , . (59)
If c is a rational number of the form c = m/k, where m and k are integers with
no common divisor, then we may take n = 0, 1, 2, . . . , k 1 in eq. (59), since
other values of n will not produce any new values of z
m/k
. It follows that the
multi-valued function w = z
m/k
has precisely k distinct branches. If c is irrational
or complex, then the number of branches is innite (with one branch for each
possible choice of integer n).
Having dened the multi-valued complex power function, we are now able to
compute ln(z
c
):
ln(z
c
) = ln(e
c ln z
) = ln(e
c (Ln z+2im)
) = ln(e
c Ln z
e
2imc
)
= ln(e
c Ln z
) + ln(e
2imc
) = c (Lnz + 2im) + 2ik
= c ln z + 2ik = c
_
lnz +
2ik
c
_
, (60)
where k and m are arbitrary integers. Thus, ln(z
c
) = c ln z in the sense of set
equality (in which case the sets corresponding to ln z and lnz + 2ik/c coincide)
9
if and only if k/c is an integer for all values of k. The only possible way to satisfy
this latter requirement is to take c = 1/n, where n is an integer. Thus, eq. (51) is
now veried.
We can dene a single-valued power function by selecting the principal value
of ln z in eq. (58). Consequently, the principal value of z
c
is dened by
Z
c
= e
c Ln z
, z = 0 .
For a lack of a better notation, I will indicate the principal value by capitalizing
the variable Z as above. The principal value denition of z
c
can lead to some
unexpected results. For example, consider the principal value of the cube root
function w = Z
1/3
= e
Ln(z)/3
. Then, for z = 1, the principal value of
3

1 = e
Ln(1)/3
= e
i/3
=
1
2
_
1 + i

3
_
.
This may have surprised you, if you were expecting that
3

1 = 1. To obtain
the latter result would require a dierent choice of the principal interval in the
denition of the principal value of z
1/3
.
We are now in the position to check eq. (22) in the case that both the complex
logarithm and complex power function are dened by their principal values. That
is, we compute:
Ln(Z
c
) = Ln(e
c Ln z
) = c Lnz + 2iN
c
, (61)
after using eq. (53), where N
c
is an integer determined by
N
c

_
1
2

Im (c Lnz)
2
_
, (62)
and [ ] is the greatest integer bracket function dened in eq. (4). N
c
can be
evaluated by noting that:
Im (c Lnz) = Im {c (Ln|z| + iArg z)} = Arg z Re c + Ln|z| Im c .
Note that if c = n where n is an integer, then eq. (61) simply reduces to eq. (56),
as expected. We conclude that eq. (22) is generally false both for the multi-valued
complex logarithm and its principal value.
A function that in some respects is similar to the complex power function
is the generalized exponential function. A possible denition of the generalized
exponential function for c = 0 is:
w = c
z
= e
z lnc
= e
z(Ln c+2in)
, n = 0 , 1 , 2 , 3 , . (63)
However, the multi-valued nature of this function diers somewhat from the multi-
valued power function. In contrast to the latter, the generalized exponential
function possesses no branch point (or any other type of singularity) in the nite
10
complex z-plane. Thus, one can regard eq. (63) as dening a set of independent
single-valued functions for each value of n. Typically, the n = 0 case is the most
useful, in which case, we would simply dene:
w = c
z
= e
z Ln c
, c = 0 . (64)
This conforms with our denition of the exponential function in section 3 (where
c = e). Henceforth, we shall employ eq. (64) as the denition of the single-valued
generalized exponential function.

Some results for the principal value of the complex power function can be
immediately adapted to the generalized exponential function. For example, by an
almost identical computation as in eqs. (61) and (62), we nd that:
Ln(c
z
) = Ln(e
z Ln c
) = z Lnc + 2iN

c
, (65)
where N

c
is an integer determined by:
N

c

_
1
2

Im (z Ln c)
2
_
. (66)
7. Properties of the generalized power function
Let us examine the properties listed in eqs. (23)(29). Eq. (23) denes the
complex power function. It is tempting to write:
z
a
z
b
= e
a ln z
e
b lnz
= e
a lnz+b lnz
?
= e
(a+b) lnz
= z
a+b
. (67)
However, consider the case of non-integer a and b where a + b is an integer. In
this case, eq. (67) cannot be correct since it would equate a multi-valued function
z
a
z
b
with a single-valued function z
a+b
. In fact, the questionable step in eq. (67)
is false:
a ln z + b lnz
?
= (a + b) ln z [FALSE!!]. (68)
We previously noted that eq. (68) is false in the case of a = b = 1 [cf. eq. (8)]. A
more careful computation yields:
z
a
z
b
= e
a ln z
e
b ln z
= e
a(Ln z+2in)
e
b(Ln z+2ik)
= e
(a+b)Ln z
e
2i(na+kb)
,
z
a+b
= e
(a+b) ln z
= e
(a+b)(Ln z+2ik)
= e
(a+b)Ln z
e
2ik(a+b)
, (69)
where k and n are arbitrary integers. Hence, z
a+b
is a subset of z
a
z
b
. Whether
the set of values for z
a
z
b
and z
a+b
does or does not coincide depends on a and b.
However, in general, eq. (24) does not hold.

In practice, many textbooks treat the generalized exponential function as a single-valued

function, c
z
= e
z Ln c
, only when c is a positive real number. For any other value of c, the
multi-valued function c
z
= e
z ln c
is preferred. We shall not pursue this approach in these notes.
11
Similarly,
z
a
z
b
=
e
a ln z
e
b ln z
=
e
a(Ln z+2in)
e
b(Ln z+2ik)
= e
(ab)Ln z
e
2i(nakb)
,
z
ab
= e
(ab) ln z
= e
(ab)(Ln z+2ik)
= e
(ab)Ln z
e
2ik(ab)
, (70)
where k and n are arbitrary integers. Hence, z
ab
is a subset of z
a
/z
b
. Whether
the set of values z
a
/z
b
and z
ab
does or does not coincide depends on a and b.
However, in general, eq. (25) does not hold. Setting a = b in eq. (70) yields the
expected result:
z
0
= 1 , z = 0
for any non-zero complex number z. Setting a = 0 in eq. (70) yields the set
equality:
z
b
=
1
z
b
, (71)
i.e., the set of values for z
b
and 1/z
b
coincide. Thus, eq. (26) is satised. Note,
however, that
z
a
z
a
= e
a ln z
e
a ln z
= e
a(ln zln z)
= e
a ln 1
= e
2ika
,
where k is an arbitrary integer. Hence, if a is a non-integer, then z
a
z
a
= 1 for
k = 0. This is not in conict with the set equality given in eq. (71) since there
always exists at least one value of k (namely k = 0) for which z
a
z
a
= 1.
To show that eq. (27) can fail, we use eq. (46) in concluding that
(z
a
)
b
= (e
a lnz
)
b
= e
b ln(e
a ln z
)
= e
b(a lnz+2ik)
= e
ba lnz
e
2ibk
= z
ab
e
2ibk
,
where k is an arbitrary integer. Thus, z
ab
is a subset of (z
a
)
b
. The elements of z
ab
and (z
a
)
b
coincide if and only if b is an integer. For example, if z = a = b = i, we
nd that:
(i
i
)
i
= i
ii
e
2k
= i
1
e
2k
= ie
2k
, k = 0 , 1 , 2 , . (72)
On the other hand, eqs. (28) and (29) are satised by the multi-valued power
function, since
(z
1
z
2
)
a
= e
a ln(z
1
z
2
)
= e
a(ln z
1
+ln z
2
)
= e
a ln z
1
e
a lnz
2
= z
a
1
z
a
2
,
_
z
1
z
2
_
a
= e
a ln(z
1
/z
2
)
= e
a(ln z
1
ln z
2
)
= e
a lnz
1
e
a ln z
2
= z
a
1
z
a
2
.
We now repeat the above analysis for the principal value of the power function,
Z
c
= e
c Ln z
. In this case, the results are somewhat reversed from the case of the
12
multi-valued power function. In particular, eqs. (24)(26) are satised, whereas
eqs. (27)(29) may be violated. For example, for the single-valued power function,
Z
a
Z
b
= e
aLn z
e
bLn z
= e
(a+b)Ln z
= Z
a+b
, (73)
Z
a
Z
b
=
e
aLn z
e
bLn z
= e
(ab)Ln z
= Z
ab
, (74)
Z
a
Z
a
= e
aLn z
e
aLn z
= 1 . (75)
Setting a = b in eq. (74) yields Z
0
= 1 (for z = 0) as expected.
Eq. (27) may be violated since eq. (53) implies that
(Z
c
)
b
= (e
cLn z
)
b
= e
b Ln(e
cLn z
)
= e
b(cLn z+2iNc)
= e
bc Ln z
e
2ibNc
= Z
cb
e
2ibNc
,
where N
c
is an integer determined by eq. (62). As an example, if z = b = c = i,
eq. (62) gives N
c
= 0, which yields the principal value of (i
i
)
i
= i
ii
= i
1
= i.
However, in general N
c
= 0 is possible in which case eq. (27) does not hold.
Eqs. (28) and (29) may also be violated since eqs. (54) and (55) imply that
(Z
1
Z
2
)
a
= e
aLn(z
1
z
2
)
= e
a(Ln z
1
+Ln z
2
+2iN
+
)
= Z
a
1
Z
a
2
e
2iaN
+
, (76)
_
Z
1
Z
2
_
a
= e
aLn(z
1
/z
2
)
= e
a(Ln z
1
Ln z
2
+2iN

)
=
Z
a
1
Z
a
2
e
2iaN

, (77)
where the integers N

are determined from eq. (13).

8. Properties of the generalized exponential function
The generalized exponential function, w = c
z
(c = 0), is a single-valued func-
tion dened by eq. (64). Using this denition and the properties of the complex
exponential function e
z
, one can quickly check whether eqs. (31)(36) hold in the
complex plane. The proof of eqs. (31)(33) is nearly identical to the one given in
eqs. (73)(75):
c
z
1
c
z
2
= e
z
1
Ln c
e
z
2
Ln c
= e
(z
1
+z
2
)Ln c
= c
z
1
+z
2
, (78)
c
z
1
c
z
2
=
e
z
1
Ln c
e
z
2
Ln c
= e
(z
1
z
2
)Ln c
= c
z
1
z
2
, (79)
c
z
c
z
= e
zLn c
e
zLn c
= 1 . (80)
However, eq. (34) does not generally hold. Using eq. (65),
(c
z
1
)
z
2
= e
z
2
Ln(c
z
1)
= e
z
2
(z
1
Ln c+2iN

c
)
= e
z
2
z
1
Ln c
e
2iz
2
N

c
= c
z
1
z
2
e
2iz
2
N

c
, (81)
where N

c
is determined by eq. (66) [with z replaced by z
1
].
13
The case of c = e is noteworthy. Eq. (81) reduces to:
(e
z
1
)
z
2
= e
z
1
z
2
e
2iz
2
N

e
, N

e

_
1
2

Im z
1
2
_
. (82)
If z
2
= n where n is any integer, then e
2inN

e
= 1 (since N

e
is an integer by
denition of the bracket notation). Thus, we recover eq. (38).
Let us test eq. (82) by substituting z
1
= i. Then, N

e
= 1 and hence
(e
i
)
z
= e
iz
.
This result may seem strange, but it is a consequence of our denition of the
generalized exponential function, c
z
= e
z Ln c
, which employs the principal value
of the logarithm. Indeed
(e
i
)
z
= (1)
z
= e
z Ln(1)
= e
iz
,
since Ln(1) = i. We conclude that eq. (34) can be violated, even for the
ordinary exponential function.
Ultimately, the real diculty with (c
z
1
)
z
2
is that it is simultaneously a gen-
eralized exponential function and a generalized power function. Thus, if z
2
is a
non-integer, it may be more convenient to treat (c
z
1
)
z
2
as a multi-valued function.
That is, in this latter convention, we treat the generalized exponential function
c
z
1
= e
z
1
Ln c
as a single-valued function (using the principal value denition of
the logarithm in the exponent), whereas we treat the generalized power function
(c
z
1
)
z
2
= e
z
2
ln(c
z
1)
as a possible multi-valued function:
(c
z
1
)
z
2
= e
z
2
ln(c
z
1)
= e
z
2
ln(e
z
1
Ln c
)
= e
z
2
(z
1
Ln c+2ik)
= e
z
2
z
1
Ln c
e
2iz
2
k
= c
z
1
z
2
e
2ikz
2
,
where k is an arbitrary integer [see eq. (46)]. In particular, for z
2
a non-integer,
(c
z
1
)
z
2
is a multi-valued function, with branches corresponding to dierent choices
of k. For example, for c = z
1
= z
2
= i, we recover eq. (72). One might be
tempted to call the k = 0 branch the principal value of (c
z
1
)
z
2
, in which case
eq. (34) would be valid. Clearly, we must dene our conventions carefully if we
wish to manipulate expressions involving exponentials of exponentials.
Finally, eqs. (35) and (36) may be violated. The calculation is nearly identical
to the one given in eqs. (76) and (77):
(ab)
z
= e
zLn(ab)
= e
z(Ln a+Ln b+2iN
+
)
= a
z
b
z
e
2izN
+
,
_
a
b
_
z
= e
zLn(a/b)
= e
z(Ln aLn b+2iN

)
=
a
z
b
z
e
2izN

,
where the integers N

are determined from eq. (13) [with z

1
and z
2
replaced by
a and b, respectively]. If Re a > 0 and Re b > 0, then N

= 0, and eqs. (35) and

(36) are satised.
14