complex analysis is life - 3

[logistic_guy]

Solve.

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

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

 

[logistic_guy]

We start with:

$\displaystyle z = x + iy$
$\displaystyle \overline{z} = x - iy$

By looking at the path, we find that $\displaystyle y = 1$, then we have:

$\displaystyle z = x + i$
$\displaystyle \overline{z} = x - i$

This gives us the integral:

$\displaystyle \int_{1}^{0} \pi e^{\pi(x - i)} \ dx$

 

[logistic_guy]

$\displaystyle \int_{1}^{0} \pi e^{\pi(x - i)} \ dx$

Let's solve it.

$\displaystyle \int_{1}^{0} \pi e^{\pi(x - i)} \ dx = \pi e^{-i\pi}\int_{1}^{0}e^{\pi x} \ dx = e^{-i\pi}e^{\pi x}\bigg |_{1}^{0} = e^{-i\pi}(1 - e^{\pi})$

$\displaystyle = (\cos \pi - i\sin \pi)(1 - e^{\pi}) = (-1)(1 - e^{\pi}) = e^{\pi} - 1$