Evaluating the Integral of $\pi e^{\pi \overline z}$ with respect to $z$

Question

I've been given a question as part of the homework on Complex Integration. I cannot seem to think how to integrate it especially with the presence of the conjugate of $z$ in the expression

$$\displaystyle\int_\gamma \pi e ^{\pi \overline z} dz$$ where $\gamma$ is the line segment from $i$ to $0$. What method can I use to integrate the expression having $\overline z$ with respect to $z$?

Note: Here $\overline z$ is the conjugate of $z$

Answer 1

Parametrize the path $\gamma$:

$$ \gamma = \{z = x + i y ~|~ x = 0 ~\mbox{ and } y = 1 - t, ~ 0\leq 0 \leq 1\} $$

Along this path

$$ {\rm d}z = -i{\rm d}t $$

So that

$$ \int_\gamma \pi e^{\pi \overline{z}}~{\rm d}z = -i\pi \int_0^1 e^{-i\pi (1 - t)}{\rm d}t = (\cdots) $$