solved Evaluate the line integral of 1/z

Question : Evaluate the line integral of 1/z along unit circle in counter clockwise direction starting from z = 1

Solution:

Since a complex line integral is evaluated as \[\int_{C}{f(z)}dz=\int_{t=a}^{b}f\left(z\left(t\right)\right)\frac{dz(t)}{dt}dt\] Given function \[f\left(z\right)=\frac{1}{z}\] and the parametrization of unit circle C is \[z\left(t\right)=\cos{t}\ +\ isin\ t\ =\ e^{it}\ ,\ 0\ \le\ t\ \le\ 2\pi\] therefore \[f\left(z\left(t\right)\right)=\frac{1}{e^{it}}=e^{-it}\] And \[\frac{dz(t)}{dt}\ =\ -\ sin\ t\ +\ i\ cost\ =\ ie^{it}\] Substituting in to the formula Now \[\int_{C}\frac{1}{z}dz=\int_{t=0}^{2\pi}e^{-it}\ ie^{it}dt=\int_{t=0}^{2\pi}i\ dt=i\left[t\right]_0^{2\pi}=2\pi i\] \[\int_{C}\frac{1}{z}dz=2\pi i\] is the required value of the line integral of along unit circle in counter clockwise direction starting from z = 1