Question

Calculate using Cauchys integral formula

Answer 1

9πi/2

Explanation:

`\oint\limits_C {(5z^(2)+4)/((z-1)(z+3)) } \, dz,\ C: |z|=2`

Let f(z) = (5z² + 4) / (z + 3).

`\oint\limits_C {(f(z))/(z-1) } \, dz,\ C: |z|=2`

Using Cauchy's integral formula:

`2\pi i\ f(1)\\2\pi i\ ((5(1)^(2)+4)/(1+3)) \\2\pi i\ ((9)/(4)) \\(9\pi i)/(2)`