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.

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

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