Find the contour integral. Let $f(z)=π \exp(π\cdot\overline{z})$. Let $C$ be the square whose vertices are $0,1,(1+i)$, and $i$. 

How can I evaluate the contour integral of $f(z)$ over $C$?


We have
$$
\int_C\pi e^{\pi\bar{z}}\,\mathrm{d}\bar{z}=0
$$
On $[0,1]$ and $[1+i,i]$, $\mathrm{d}\bar{z}=\mathrm{d}z$, and on $[1,1+i]$ and $[i,0]$, $\mathrm{d}\bar{z}=-\mathrm{d}z$. Thus,
$$
\begin{align}
\int_C\pi e^{\pi\bar{z}}\,\mathrm{d}z
&=\int_C\pi e^{\pi\bar{z}}\,\mathrm{d}\bar{z}+2\int_{[1,1+i]\cup[i,0]}\pi e^{\pi\bar{z}}\,\mathrm{d}z\\
&=2\pi i\int_0^1e^{\pi(1-it)}\,\mathrm{d}t-2\pi i\int_0^1e^{-\pi it}\,\mathrm{d}t\\
&=2\pi i(e^\pi-1)\int_0^1e^{-\pi it}\,\mathrm{d}t\\[5pt]
&=-2(e^\pi-1)(e^{-\pi i}-1)\\[12pt]
&=4(e^\pi-1)
\end{align}
$$
