Mathematical Tools for Physics

3—Complex Algebra 73

and other integerskjust keep repeating these three values.

e^6 πi/^5

e^4 πi/^5

e^8 πi/^5

e^2 πi/^5

1

5 throots

The roots are equally spaced around the unit circle. If you want thenthroot, you do the same sort of
calculation: the 1 /npower and the integersk= 0, 1 , 2 ,...,(n−1). These arenpoints equally spaced around
the circle.

3.4 Logarithms
The logarithm is the inverse function for the exponential. Ifew=zthenw= lnz. To determine what this is, let

w=u+iv and z=reiθ, then eu+iv=eueiv=reiθ

This implies thateu=r and sou= lnr, but it doesn’t implyv=θ. Remember the periodic nature of the
exponential function?eiθ=ei(θ+2nπ), so you can conclude instead thatv=θ+ 2nπ.

lnz= ln

(

reiθ

)

= lnr+i(θ+ 2nπ) (11)

has an infinite number of possible values. Is this bad? You’re already familiar with the square root function, and
that hastwopossible values,±. This just carries the idea farther. For exampleln(−1) =iπor 3 iπor− 7 iπetc.
As with the square root, the specific problem that you’re dealing with will tell you which choice to make.