October 23, 2007
PHY 711 – Contour Integration
These notes summarize some basic properties of complex functions and their integrals.
An analytic function f(z) in a certain region of the complex plane zis one which takes a
single (non-infinite) value and is differentiable within that region. Cauchy’s theorm states
that a closed contour integral of the function within that region has the value
IC
f(z) = 0.(1)
As an example, functions composed of integer powers of z–
f(z) = zn,for n= 0,1,±2,±3.... (2)
fall in this catogory. Notice that non-integral powers are generally not analytic and that
n=−1 is also special. In fact, we can show that
IC
dz
z= 2πi. (3)
This result follows from the fact that we can deform the contour to a unit circle about
the origin so that z= eiθ. Then
IC
dz
z=Z2π
0
eiθ
eiθ idθ = 2πi. (4)
One result of this analysis is the Cauchy integral formula which states that for any analytic
function f(z) within a region C,
f(z) = 1
2πi IC
f(z0)
z0−zdz0.(5)
Another result of this analysis is the Residue Theorm which states that if the complex
function g(z) has poles at a finite number of points zpwithin a region Cbut is otherwise
analytic, the contour integral can be avaluated according to
IC
g(z)dz = 2πi X
p
Res(gp),(6)
where the residue is given by
Res(gp)≡lim
z→zp(1
(m−1)!
dm−1
dzm−1((z−zp)mg(z))),(7)
where mdenotes the order of the pole.
