Let $C$ denote the unit circle in the complex plane. Find the line integral of $\exp(2\overline z)$ over $C$. I tried to find this integral by using direct definition of contour integral but couldn't work out the whole thing. My attempt:
We denote $C$ in parametric form:$\exp(it)$ $0\leq t\leq 2\pi$
By the definition of contour integral
$\oint _C$ $\exp(2\overline z) dz$ $=$ $\int_{t=0}^{2\pi} \exp(2\exp(-it)) i\exp(it) dt$ =$\int_{t=0}^{2\pi} \exp(2(\cos t -i \sin t) i\exp(it) dt$ =$\int_{t=0}^{2\pi} i \exp(2(\cos t)) \exp i(t-2\sin t)dt$ =$\int_{t=0}^{2\pi} \exp(2(\cos t))(\cos (t-2\sin t)+i\sin (t-2\sin t) )dt$
How to proceed further? (I am not pretty good at using latex . Apologies for messy writing)
$$e^{2\overline{z}}= \sum_{n=0}^\infty \frac{(2\overline{z})^n}{n!}$$
$$J=\oint_C e^{2\overline{z}}\,dz = \oint_C \sum_{n=0}^\infty \frac{(2\overline{z})^n}{n!} \, dz = \sum_{n=0}^\infty \frac{2^n}{n!} \oint_C{\overline{z}^n} \, dz$$
$$\oint_C \overline{z}^n \, dz = \int_0^{2\pi}e^{-i n \theta}\, i\,e^{i\theta}\,d\theta = i \int_0^{2\pi} e^{i(1-n)\theta} \, d\theta= \left\{ \begin{aligned}2\pi i, & \quad n=1 \\0, & \quad \text{otherwise} \end{aligned} \right.$$
$$J= {2\cdot 2\pi i}=4\pi i.$$