complex analysis is life - 4

[logistic_guy]

Solve.

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

where $\displaystyle C$ is the path that goes from $\displaystyle i$ to $\displaystyle 0$.

 

[logistic_guy]

We have:

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

By observing the path, we see that $\displaystyle x = 0$, then we have:

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

This gives us the integral:

$\displaystyle \large \int_{1}^{0} \pi e^{-i\pi y} \ i dy$

 

[logistic_guy]

$\displaystyle \large \int_{1}^{0} \pi e^{-i\pi y} \ i dy$

Let us solve this bastard integral.

$\displaystyle \large i\pi\int_{1}^{0} e^{-i\pi y} \ dy = -e^{-i\pi y}\bigg |_{1}^{0} = -(1 - e^{-i\pi}) = e^{-i\pi} - 1 = \cos \pi - i\sin \pi - 1 = -1 - 1 = -2$

 

[khansaheb]

... bastard.....

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