Mathematical Tools for Physics

Complex Variables

In the calculus of functions of a complex variable there are three fundamental tools, the same fundamental tools
as for real variables. Differentiation, Integration, and Power Series. I’ll first introduce all three in the context of
complex variables, then show the relations between them. The applications of the subject will form the major
part of the chapter.

14.1 Differentiation
When you try to differentiate a continuous function is it always differentiable? If it’s differentiable once is it
differentiable again? The answer to both is no. Take the simple absolute value function of the real variablex.

f(x) =|x|=

{

x (x≥ 0 )

−x (x < 0 )

This has a derivative for allxexcept zero. In that case the limit

f(x+ ∆x)−f(x)

∆x

−→







1 (x > 0 )

− 1 (x < 0 )

? (x= 0)

(1)

has a different result depending on whether∆x→ 0 through positive or through negative values.
If you integrate this function,

∫x

0

|x′|dx′=

{

x^2 / 2 (x≥ 0 )

−x^2 / 2 (x < 0 )

the result has a derivative everywhere, including the origin, but you can’t differentiate it twice. A few more
integrations and you can produce a function that you can differentiate 42 times but not 43.

418