The complex logarithm, exponential and power functions

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 differ 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

defined such that:

?¦Ð < Arg z ¡Ü ¦Ð .

(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 + 2¦Ðn ,

n = 0 , ¡À1 , ¡À2 , ¡À3 , . . . ,

?¦Ð < Arg z ¡Ü ¦Ð .

One can also express Arg z in terms of arg z as follows:





1 arg z

,

?

Arg z = arg z + 2¦Ð

2

2¦Ð

(2)

(3)

where [ ] denotes the greatest integer function. That is, [x] is defined to be the

largest integer less than or equal to the real number x. Consequently, [x] is the

unique integer that satisfies the inequality

x ? 1 < [x] ¡Ü 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).

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For example, [1.5] = [1] = 1 and [?0.5] = ?1. One can check that Arg z as

defined in eq. (3) does fall inside the principal interval specified by eq. (1).

The multi-valued function arg z satisfies the following properties,

arg(z1 z2 ) = arg z1 + arg z2 ,

 

z1

= arg z1 ? arg z2 .

arg

z2

 

1

= arg z = ? arg z .

arg

z

(5)

(6)

(7)

Eqs. (5)¨C(7) should be viewed as set equalities, i.e. the elements of the sets indicated 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 6= 2 arg z ,

arg z ? arg z 6= 0 ,

(8)

arg(1) = arg z ? arg z 6= 0 .

(9)

which, by virtue of eqs. (5) and (6), yield:

arg z 2 = arg z + arg z 6= 2 arg z ,

For example, arg(1) = 2¦Ðn, for n = 0 ¡À 1, ¡À2, . . .. More generally,

arg z n = arg z + arg z + ¡¤ ¡¤ ¡¤ arg z 6= n arg z .

{z

}

|

(10)

Arg (z1 z2 ) = Arg z1 + Arg z2 + 2¦ÐN+ ,

(11)

Arg (z1 /z2 ) = Arg z1 ? Arg z2 + 2¦ÐN? ,

(12)

n

We also note some properties of the the principal value of the argument.

where the integers N¡À are determined as follows:

?

?

if Arg z1 ¡À Arg z2 > ¦Ð ,

??1 ,

N¡À =

0,

if ?¦Ð < Arg z1 ¡À Arg z2 ¡Ü ¦Ð ,

?

?

1,

if Arg z1 ¡À Arg z2 ¡Ü ?¦Ð .

If we set z1 = 1 in eq. (12), we find that

(

Arg z ,

Arg(1/z) = Arg z =

?Arg z ,

if Im z = 0 and z 6= 0 ,

if Im z 6= 0 .

(13)

(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 ) = n Arg z + 2¦ÐNn ,

2

(15)

where the integer Nn is given by:





1

n

Nn =

?

Arg z ,

2 2¦Ð

(16)

and [ ] is the greatest integer bracket function introduced in eq. (4).

2. Properties of the real-valued logarithm, exponential and power functions

Consider the logarithm of a positive real number. This function satisfies a

number of properties:

eln x = x ,

(17)

ln(ea ) = a ,

(18)

ln(xy) = ln(x) + ln(y) ,

 

x

ln

= ln(x) ? ln(y) ,

y

 

1

= ? ln(x) ,

ln

x

ln xp = p ln x ,

(19)

(20)

(21)

(22)

for positive real numbers x and y and arbitrary real numbers a and p. Likewise,

the power function defined over the real numbers satisfies:

xa = ea ln x ,

(23)

xa xb = xa+b ,

(24)

xa

= xa?b ,

xb

1

= x?a ,

xa

(xa )b = xab ,

(xy)a = xa y a ,

 a

x

= xa y ?a ,

y

(25)

(26)

(27)

(28)

(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 defined over

3

the real numbers. This function satisfies:

ax = ex ln a ,

(30)

ax ay = ax+y ,

(31)

ax

= ax?y ,

ay

1

= a?x ,

ax

(ax )y = axy ,

(32)

(33)

(34)

(ab)x = ax bx ,

(35)

 a x

= ax b?x .

(36)

b

for positive real numbers a and b and arbitrary real numbers x and y.

We would like to know which of these relations are satisfied when these functions are extended to the complex plane. It is dangerous to assume that all of

the above relations are valid in the complex plane without modification, as this

assumption can lead to seemingly paradoxical conclusions. Here are three examples:

1. Since 1/(?1) = (?1)/1 = ?1,

r

r

1

1

?1

i

= =

= .

?1

i

1

1

(37)

Hence, 1/i = i or i2 = 1. But i2 = ?1, so we have proven that 1 = ?1.

2. Since 1 = (?1)(?1),

p

¡Ì

¡Ì

¡Ì

1 = 1 = (?1)(?1) = ( ?1)( ?1) = i ¡¤ i = ?1 .

(38)

3. To prove that ln(?z) = ln(z) for all z 6= 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 first 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)¨C(36) and avoid pitfalls that can lead to false results.

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3. Definition of the complex exponential function

We begin with the complex exponential function, which is defined via its power

series:

¡Þ

X

zn

ez =

,

n!

n=0

where z is any complex number. Using this power series definition, one can verify

that:

ez1 +z2 = ez1 ez2 ,

for all complex z1 and z2 .

(39)

In particular, if z = x + iy where x and y are real, then it follows that

ez = ex+iy = ex eiy = ex (cos y + i sin y) .

One can quickly verify that eqs. (30)¨C(33) are satisfied by the complex exponential

function. In addition, eq. (34) clearly holds when the outer exponent is an integer:

(ez )n = enz ,

n = 0 , ¡À1 , ¡À2 , . . . .

(40)

If the outer exponent is a non-integer, then the resulting expression is a multivalued 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:

e2¦Ðin = 1 ,

n = 0 , ¡À1 , ¡À2 , ¡À3 , . . . .

(41)

4. Definition of the complex logarithm

In order to define the complex logarithm, one must solve the complex equation:

z = ew ,

(42)

for w, where z is any non-zero complex number. If we write w = u + iv, then

eq. (42) can be written as

eu eiv = |z|ei arg z .

(43)

Eq. (43) implies that:

|z| = eu ,

v = arg z .

The equation |z| = eu 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 + 2¦Ðn) , n = 0 , ¡À1 , ¡À2 , ¡À3 , . . .

(44)

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