Question

Use Cauchy’s Integral formulas to evaluate the following integral along the

indicated closed contours.

∮

z

2+3z+2i

z

2+3z−4

dz;

C

(a) |z| = 2 (b) |z + 5| =

3

2

Answer 1

Cauchy’s integral formula:

$f(z_0)=\frac{1}{2\pi i}\int_C \frac{f(z)}{z-z_0}dz$

 for any z₀ inside C

we have:

$\int_C \frac{z^2+3z+2i}{z^2+3z-4}dz$

$z^2+3z-4=0$

$z=\frac{-3\pm \sqrt{9+16}}{2}$

so,

$z_0=-4,1$

$\frac{z^2+3z+2i}{z^2+3z-4}=\frac{z^2+3z+2i}{(z+4)(z-1)}$

a)

for  |z| = 2:

$z_0=1$ is inside C

then:

$f(z)=\frac{z^2+3z+2i}{z+4}$

$\int_C \frac{z^2+3z+2i}{z^2+3z-4}dz=2\pi if(1)=2\pi i(4+2i)/5=\pi(-4 +8i)/5$

b)

for  |z + 5| = 3/2:

$z_0=-4$ is inside C

then:

$f(z)=\frac{z^2+3z+2i}{z-1}$

$\int_C \frac{z^2+3z+2i}{z^2+3z-4}dz=2\pi if(-4)=-2\pi i(4+2i)/5=\pi(4 -8i)/5$