complex analysis is life - 5

[logistic_guy]

Solve.

$\displaystyle \large \int_C \pi e^{\pi \overline{z}} \ dz$

where $\displaystyle C$ is the path that goes from $\displaystyle 0$ to $\displaystyle 1$, then from $\displaystyle 1$ to $\displaystyle 1 + i$, then from $\displaystyle 1 + i$ to $\displaystyle i$, then from $\displaystyle i$ to $\displaystyle 0$.

 

[logistic_guy]

This is just a combination of all the paths (contours) that we have solved in the previous exercises.

Then,

$\displaystyle \large \int_C \pi e^{\pi \overline{z}} \ dz = \int_{C_1} \pi e^{\pi \overline{z}} \ dz + \int_{C_2} \pi e^{\pi \overline{z}} \ dz + \int_{C_3} \pi e^{\pi \overline{z}} \ dz + \int_{C_4} \pi e^{\pi \overline{z}} \ dz$


$\displaystyle = \large (e^{\pi} - 1) + (2e^{\pi}) + (e^{\pi} - 1) + (-2)$


$\displaystyle = \large e^{\pi} - 1 + 2e^{\pi} + e^{\pi} - 1 - 2$


$\displaystyle = \large 4e^{\pi} - 4$


$\displaystyle = \large 4(e^{\pi} - 1)$