Evaluate ∮e^{2z}/((z-1)(z-2)) by CIF

Cauchy's integral formula (CIF): if f(z) is analytic inside and on a simple closed curve C and a lies inside C, then ∮_C f(z)/(z − a) dz = 2πi f(a).

Solution

The poles z = 1 and z = 2 both lie inside C: |z| = 3, since |1| < 3 and |2| < 3.

Partial fractions: 1/((z − 1)(z − 2)) = 1/(z − 2) − 1/(z − 1).

So I = ∮_C e^{2z}/(z − 2) dz − ∮_C e^{2z}/(z − 1) dz.

Take f(z) = e^{2z}, which is analytic everywhere. By CIF:

∮_C e^{2z}/(z − 2) dz = 2πi f(2) = 2πi e⁴
∮_C e^{2z}/(z − 1) dz = 2πi f(1) = 2πi e²

I = 2πi e⁴ − 2πi e² = 2πi (e⁴ − e²)

Answer: ∮_C e^{2z}/((z − 1)(z − 2)) dz = 2πi (e⁴ − e²) ≈ 296.62 i