Why this $$ \Im \frac{-2}{1+e^{-s + i a }} $$
equals to this expression:
$$\\\ \frac{\sin(a)}{\cos(a)+\cosh(s)} $$
I was trying to evaluate the Fourier transform of a hyperbolic function and my textbook and the other sources say this equality holds on. I only got to:
$$ \Im \frac{e^{-ia}}{e^{-ia}+e^{-s}} $$
Well, generally the imaginary part of $e^{-ia}$ is $\sin(a)$, but $e^{-s}$ is real. So I don't understand the $\cosh (s)$ part. I am just so confused.
The fourier tranform has been done on this function:
$$f(x) = \frac{\sinh(ax)}{\sinh(\pi x)}$$
I get a slightly different result. Consider the following calculation. \begin{align*} \Im \frac{1}{1+e^{-s+ia}} &= \Im \frac{1}{1+e^{-s}(\cos(a)+i \sin(a))} \\ &=\Im \frac{1}{1+e^{-s}\cos(a) + ie^{-s}\sin(a)} \cdot \frac{1+e^{-s}\cos(a) - ie^{-s}\sin(a)}{1+e^{-s}\cos(a) - ie^{-s}\sin(a)} \\ &= \Im \frac{1+e^{-s}\cos(a) - ie^{-s}\sin(a)}{(1+e^{-s}\cos(a))^2 + (e^{-s}\sin(a))^2} \\ &= \frac{-e^{-s}\sin(a)}{1+2e^{-s}\cos(a)+e^{-2s}\cos(a)^2 + e^{-2s}\sin(a)^2} \\ &= \frac{ -e^{-s}\sin(a)}{1+2e^{-s}\cos(a)+e^{-2s}}\\ &= \frac{ -\sin(a)}{e^{s} +2\cos(a)+e^{-s}}\\ &=\frac{-\sin(a)}{2\left( \frac{e^s+e^{-s}}{2}+\cos(a) \right)} = -\frac{1}{2}\frac{\sin(a)}{\cosh(s)+\cos(a)} \end{align*}
I get a factor of $-1/2$ in front of your suggested result.
Your confusion about the expression $$ \Im \frac{e^{-ia}}{e^{-ia}+e^{-s}} $$ might arise because you can not see the real and the imaginary parts from this expression, as there is also an $i$ in the denominator of the fraction.