So I'm integrating
$$I = \int_C e^z dz$$ where C is defined as the curve along $Re(z)=1$ and $Im(z)=0$ to $\pi$.
I know that $e^z$ is analytic, but I don't think any of the integral theorems I know work on this since I don't think it's a closed contour (it's basically a line, right?)
Anyway, I tried parametrizing it as $z=e^{\iota\theta}$, and we get $dz = \iota e^{\iota\theta}d\theta$, so that we're integrating $e^{e^{\iota\theta}}$.
I have two questions essentially;
- Am I doing this right?
- What limits do I use? $0$ to $\pi$? Since the $1$ is constant? Or something else? I think this contour goes from $z=1$ to $z = \iota\pi$, so I'm guessing that the limits should be what I mentioned, but I'm not completely sure.
Can someone help explain please?
Since $\exp'=\exp$ and your path goes form $1$ to $1+\pi i$,$$\int_Ce^z\,\mathrm dz=e^{1+\pi i}-e^1=-e-e=-2e.$$