Evaluate ∮e^z/((z-1)(z-3)) on |z|=2

Statement of Cauchy's integral formula

If f(z) is analytic inside and on a simple closed contour C and a is any point inside C, then

f(a) = (1/2πi) ∮_C f(z)/(z − a) dz, that is, ∮_C f(z)/(z − a) dz = 2πi f(a).

Solution

C: |z| = 2. The poles are z = 1 (|1| < 2, inside) and z = 3 (|3| > 2, outside).

Write the integrand as e^z/(z − 3) · 1/(z − 1). Let f(z) = e^z/(z − 3). It is analytic inside and on C, because its only singularity is z = 3, which is outside C.

By CIF with a = 1:

∮_C e^z/((z − 1)(z − 3)) dz = ∮_C f(z)/(z − 1) dz = 2πi f(1) = 2πi · e/(1 − 3) = −πi e

Answer: −πie ≈ −8.54 i