Mathematical Methods for Physics and Engineering : A Comprehensive Guide

(Darren Dugan) #1

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

)


.

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