Mathematical Methods for Physics and Engineering : A Comprehensive Guide

COMPLEX VARIABLES

derived from them (e.g. tan and tanh), the identities they satisfy and their

derivative properties are also just as for real variables. In view of this we will not

give them further attention here.

The inverse function of expzis given byw, the solution of

expw=z. (24.19)

This inverse function was discussed in chapter 3, but we mention it again here

for completeness. By virtue of the discussion following (24.18),wis not uniquely

defined and is indeterminate to the extent of any integer multiple of 2πi.Ifwe

expresszas

z=rexpiθ,

whereris the (real) modulus ofzandθis its argument (−π<θ≤π), then

multiplyingzby exp(2ikπ), wherekis an integer, will result in the same complex

numberz. Thus we may write

z=rexp[i(θ+2kπ)],

wherekis an integer. If we denotewin (24.19) by

w=Lnz=lnr+i(θ+2kπ), (24.20)

where lnris the natural logarithm (to basee) of the real positive quantityr,then

Lnzis an infinitely multivalued function ofz.Itsprincipal value, denoted by lnz,

is obtained by takingk= 0 so that its argument lies in the range−πtoπ. Thus

lnz=lnr+iθ, with−π<θ≤π. (24.21)

Now that the logarithm of a complex variable has been defined, definition (24.16)

of a general power can be extended to cases other than those in whichais real

and positive. Ift(=0)andzare both complex, then thezth power oftis defined

by

tz= exp(zLnt). (24.22)

Since Lntis multivalued, so is this definition. Its principal value is obtained by

giving Lntits principal value, lnt.

Ift(= 0) is complex butzis real and equal to 1/n, then (24.22) provides a

definition of thenth root oft. Because of the multivaluedness of Lnt, there will

be more than onenth root of any givent.

Show that there are exactlyndistinctnth roots oft.

From (24.22) thenth roots oftare given by

t^1 /n=exp

(

1

n

Lnt

)

.