1550251515-Classical_Complex_Analysis__Gonzalez_

296 Chapter^5

ID IB

I

GJ

I

I I

(^00) I I I ..
I I^2 3 x
I I
le zo IA
Fig. 5.41
0 and -oo, z goes into G 1 and returns to the position z 0 with the value
w 1 (z 0 ). This is in accordance with (5.22-1).

It should be noted that on top of z = 0 there are two branch points;

at one of them w 0 (0) = w 1 (0) = w 2 (0) = ,,/2,i, while at the other w3(0) =

w 4 (0) == ws(O) = -,,/2,i. On the other hand, on top of z = 2 there are three

branch points, and we have w 0 (2) = w 3 (2) = ~' w 1 (2) = w 4 (2) = 'T/~,
w 2 (2) = w 5 (2) = 'T/^2 ~. However, above z = oo there is just one branch
point, and w 0 (oo) = w 1 (oo) = w 2 (oo) = .. · = w 5 (oo) = oo: i.e., all
branches assume the value oo at oo. Such a branch point is called an
· algebraic pole.

5.23 The Logarithmic Function

In Section 1.15 we have seen that given any complex number z -/:- 0 there

are infinitely many values of w such that

(5.23-1)

which are given by the formula

w = ln Jz J + i arg z = ln r + i( (} + 2k7r) (5.23-2)

where r = JzJ, (} = Argz, and k is any integer. Thus the inverse of the

exponential function (to the base e) is a multiple-valued function defined in

C - { 0}. This function is called the natural logarithmic function, or briefly,

the logarithmic function, t denoted

w = logz (5.23-3)

tThe base e being always understood, since logarithms to any other base are

rarely used in complex analysis. If in some case a logarithm to a base other than

. e were to be used, that base will be clearly specified.