Question

$\small\displaystyle \sf \red{Evaluate \int_{C} \frac{1}{2z + 3} dz \: where \: c \: is \: a \: circle \: |z| = 2}$​

Answer 1

$\displaystyle \tt\frac{1}{2z + 3} dz$

$\displaystyle \tt \int_{c} \frac{1}{2(z + \frac{3}{2})dz }$

$\displaystyle \frac{1}{2} \tt \int_{c} \frac{1}{z + \frac{3}{2} dz}$

$\displaystyle \frac{1}{2} \tt 2\pi if \bigg ( \frac{ - 3}{2} \bigg)$

$\displaystyle \tt\pi i \frac{1}{2z + 3} dz$

$\displaystyle \tt\pi i$

Answer 2

Answer:

Question-

$\small\displaystyle \sf \red{Evaluate \int_{C} \frac{1}{2z + 3} dz \: where \: c \: is \: a \: circle \: |z| = 2}$

Answer-

$= \int \limits_{c} \frac{1}{2(z + \frac{3}{2})dz }$

$= \frac{1}{2} \int \limits_{c} \: \frac{1}{z + \frac{3}{2}dz }$

$= \frac{1}{2} 2 \pi \: i \int( \frac{ - 3}{2} )$

$= \pi \: i \frac{1}{2z + 3} dz$

$= \pi \: i$