Mathematical Tools for Physics

14—Complex Variables 421

The common formulas for differentiation are exactly the same for complex variables as they are
for real variables, and their proofs are exactly the same. For example, the product formula:

f(z+ ∆z)g(z+ ∆z)−f(z)g(z)

∆z

=

f(z+ ∆z)g(z+ ∆z)−f(z)g(z+ ∆z) +f(z)g(z+ ∆z)−f(z)g(z)

∆z

=

f(z+ ∆z)−f(z)

∆z

g(z+ ∆z) +f(z)

g(z+ ∆z)−g(z)

∆z

As∆z→ 0 , this becomes the familiarf′g+fg′. That the numbers are complex made no difference.
For integer powers you can use induction, just as in the real case:dz/dz= 1and

If

dzn

dz

=nzn−^1 , then use the product rule

dzn+1

dz

=

d(zn.z)

dz

=nzn−^1 .z+zn.1 = (n+ 1)zn

The other differentiation techniques are in the same spirit. They follow very closely from the definition. For
example, how do you handle negative powers? Simply note thatznz−n= 1and use the product formula. The
chain rule, the derivative of the inverse of a function, all the rest, are close to the surface.

14.2 Integration
The standard Riemann integral of section1.6is
∫b

a

f(x)dx= lim

∆xk→ 0

∑N

k=1

f(ξk)∆xk

The extension of this to complex functions is direct. Instead of partitioning the intervala < x < bintoNpieces,
you have to specify a curve in the complex plane and partitionitintoN pieces. The interval is the complex
number∆zk=zk−zk− 1.

∫

C

f(z)dz= lim

∆zk→ 0

∑N

k=1

f(ζk)∆zk

z 0 z^1

z 2 z 3 z (^4) z
5
z 6
ζ 1
ζ 2 ζ^3 ζ 4 ζ 5 ζ^6