Mathematical Methods for Physics and Engineering : A Comprehensive Guide

COMPLEX NUMBERS AND HYPERBOLIC FUNCTIONS

We may use (3.34) to investigate further the properties of Lnz. We have already

noted that the argument of a complex number is multivalued, i.e. argz=θ+2nπ,

wherenis any integer. Thus, in polar form, the complex numberzshould strictly

be written as

z=rei(θ+2nπ).

Taking the logarithm of both sides, and using (3.34), we find

Lnz=lnr+i(θ+2nπ), (3.35)

where lnris the natural logarithm of the real positive quantityrandsois

written normally. Thus from (3.35) we see that Lnzis itself multivalued. To avoid

this multivalued behaviour it is conventional to define another function lnz,the

principal valueof Lnz, which is obtained from Lnzby restricting the argument

ofzto lie in the range−π<θ≤π.

EvaluateLn (−i).

By rewriting−ias a complex exponential, we find

Ln (−i)=Ln

[

ei(−π/2+2nπ)

]

=i(−π/2+2nπ),

wherenis any integer. Hence Ln (−i)=−iπ/ 2 , 3 iπ/ 2 ,.... We note that ln(−i), the
principal value of Ln (−i), is given by ln(−i)=−iπ/2.

Ifzandtare both complex numbers then thezth power oftis defined by

tz=ezLnt.

Since Lntis multivalued, so too is this definition.

Simplify the expressionz=i−^2 i.

Firstly we take the logarithm of both sides of the equation to give

Lnz=− 2 iLni.

Now inverting the process we find

eLnz=z=e−^2 iLni.

We can writei=ei(π/2+2nπ),wherenis any integer, and hence

Lni=Ln

[

ei(π/2+2nπ)

]

=i

(

π/2+2nπ

)

.

We can now simplifyzto give

i−^2 i=e−^2 i×i(π/2+2nπ)

=e(π+4nπ),

which, perhaps surprisingly, is a real quantity rather than a complex one.

Complex powers and the logarithms of complex numbers are discussed further

in chapter 24.