I have to calculate the following integral (1) by definition and (2) by using the primitive:
$\int_\alpha{\sinh(z)dz}$
where $\alpha$ is the line from $0$ to $\pi$ and from $\pi$ to $\pi + i\pi$.
I am kind of familiar with the Cauchy integral theorem, but not sure how/if I have to use it here.
I wanted to write $\alpha$ as: $\alpha: [a,b] \rightarrow \mathbb{C}$, and maybe splitting it up for the two lines? Then perhaps use:
$\int_\alpha{f(z)dz} = \int_a ^b f(\alpha(t))\alpha'(t)dt$.
Not sure if this is useful, so a little help is appreciated.
On the line from 0 to $\pi$, we can write z= x so that the integral becomes $\int_0^\pi \frac{e^x- e^{-x}}{2}dx$ which is easy to integrate. On the line from $\pi$ to $\pi+ i\pi$ we can write $z= \pi+ ix$ so that the integral becomes $\int_0^\pi e^{\pi+ ix}- e^{-\pi-ix}{2} i dx= e^\pi\frac{i}{2}\int_0^{\pi} e^{ix}dx+ \frac{i}{2}e^{-\pi}\int_0^\pi e^{-ix}dx$