Understanding the Cauchy Integral Formula and Evaluating Complex Integrals

[NewtonianAlch]

Homework Statement

Evaluating using CIF.

|z| = 4

Integral ${\frac {{{\rm e}^{2\,iz}}{\it dz}}{ \left( 3\,z-1 \right) ^{2}}}$

The Attempt at a Solution

So the singularity here is z = 1/3 which is inside the circle.

Therefore using the formula $2\,i\pi \,f$ and substituting in the z = 1/3

We get f(1/3) = exp(2/3i)

So I get the answer of 2*Pi*i(exp(2/3i))

However the answer given is (-4Pi/9)(exp(2/3i))

I'm thinking it has something to do with the repeated root, but I'm not sure.

 

[NewtonianAlch]

Found out that when dealing with powers, a generalised CIF has to be used. Now however I get -4Pi(exp(2/3i))

Where did that 1/9 come from?