Answer to Question #216748 in Complex Analysis for subodh gupta

Question

Find the value of ∫c 1/𝑍𝑑𝑧, where C is circle 𝑧=𝑒^𝑖∅, 0≤∅≤𝜋

Answer 1

$I=\int _C \dfrac{1}{z}dz\ \ \ where, z=e^{i\theta},\ \ 0\leq\theta \leq \pi$

We know that

$\int_C f(z)dz=\int _a^b f(w(\theta))\cdot w'(\theta)d\theta\ \ \ \ \ , a\leq \theta \leq b$

$f(z)=\dfrac{1}{z}$ , $w(\theta )= e^{i\theta},\ \ w'(\theta)=e^{i\theta}\cdot i=ie^{i\theta}$

$I=\int _C \dfrac{1}{z}dz=\int_0^{\pi}f(e^{i\theta})\cdot ie^{i\theta}d\theta=\int_0^\pi \dfrac{1}{e^{i\theta}}\cdot ie^{i\theta}d\theta\\\ \\\implies I=\int_0^\pi id\theta=i\theta]_0^{\pi}=\pi i$

Hence,

$I=\int _C \dfrac{1}{z}dz=\pi i$