complex analysis is life

[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 $\displaystyle 1$.

 

[logistic_guy]

We know that $\displaystyle z$ is a complex variable, so it has the form:

$\displaystyle z = x + iy$

$\displaystyle \overline{z}$ is the conjugate of $\displaystyle z$ and it has the form:

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

Since the path lies on the $\displaystyle x$-axis, $\displaystyle y = 0$ during the whole path.

Therefore,

$\displaystyle \overline{z} = x - i(0) = x$

Then, our integral becomes:

$\displaystyle \large \int_{0}^{1} \pi e^{\pi x} \ dx$

 

[logistic_guy]

$\displaystyle \large \int_{0}^{1} \pi e^{\pi x} \ dx$

It's straightforward to solve this integral.

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