Contour Integral question

Question

Evaluate the contour integral (e^z)/z(1-z)^3 dz with the circle |z| = 2 using residue theorem.

What will be the exact answer?

Answer 1

First of all, the function has a single pole at z=0 and a pole of order 3 at z=1. For completeness the function has an essential singularity at z=∞.

Both poles are within the circle |z|=2 therefore, for the Theorem of Residues, the integral is equal to the sum of both residues.

Residue at z=0

In this case, the residue is given by the limit as z to 1 of your function multiplied by the term z (which will remove the singularity at the denominator. Hence, you have the function e^z/(z-1)^3 and if you consider the limit as z approaches to 0 you will get -1. Thus the residue of the function at z=0 is equal to -1.

Residue at z=1

By applying the definition of residue of a function which has a pole of order 3 at z=1 you have to evaluate the limit as z to 1 of the second derivative (the order of the pole minus one tells you how many derivatives you need to evaluate) of your function multiplied by the term (z-1)^3 (which will remove the singularity at the denominator). Hence, you need to evaluate the second derivative of the function e^z/z which is given by

(z*e^z)/z^2 - (z^2e^z-2ze^z)/z^4

if you now evaluate the limit as z to 1 you get the value e (simply substitute z=1 in the above function)

Since the pole is of order 3, you need to multiply this limit by the factor 1/(3-1)!=1/2 (in general for a pole of order m you need to multiply the limit by 1/(m-1)! )

To conclude, the integral I is equal to the sum of both residues ---> I=-1+e/2

Best,

Davide