Mathematical Tools for Physics

3—Complex Algebra 68

Complex Exponentials
A function that is central to the analysis of differential equations and to untold other mathematical ideas: the
exponential, the familiarex. What is this function for complex values of the exponent?

ez=ex+iy=exeiy (4)

This means that all I have to do is work out the value for the purely imaginary exponent and the general case is
then just a product. There are several ways to work this out and I’ll pick one, leaving another for you, problem 8.
Whatevereiyis, it has a real and an imaginary part,

eiy=f(y) +ig(y)

Now in order to figure out what the two functionf andgare, I’ll find a differential equation that they satisfy.
Differentiate this equation with respect toy.

d

dy

eiy=ieiy=f′(y) +ig′(y)

=i

[

f(y) +ig(y)

]

=if(y)−g(y)

Equate the real and imaginary parts.

f′(y) =−g(y) and g′(y) =f(y) (5)

You can solve simultaneous differential equations several ways, and here the simplest is just to eliminate one of
the unknown functions between them. Differentiate the first equation and eliminateg.

f′′=−g′, then f′′=−f

This is the standard harmonic oscillator equation, so the solution is a combination of sines and cosines.

f(y) =Acosy+Bsiny

You find the unknown constantsAandBby using initial conditions onf, and those values come from the value
ofeiyat zero.
ei^0 = 1 =f(0) +ig(0), so f(0) = 1 and g(0) = 0