Calculating the residues of $f(z)=\frac{e^{az}}{1+e^z}$
2 Answers
Seems like use l'Hospital is straightforward enough: $$ \lim_{z\to i\pi(1+2k)}\frac{(z-i\pi(1+2k)) \mathrm{e}^{az}}{1+\mathrm{e}^z} = \mathrm{e}^{a i\pi(1+2k)} \lim_{z\to 0}\frac{z \mathrm{e}^{az}}{1-\mathrm{e}^z} = \mathrm{e}^{a i\pi(1+2k)} \lim_{z\to 0}\frac{\mathrm{e}^{az}\left(1 + z a\right)}{-\mathrm{e}^z} = -\mathrm{e}^{a i\pi(1+2k)} $$
Added: Then the integral is the sum over poles: $$ \int_{-\infty}^\infty \frac{\mathrm{e}^{a z}}{1+\mathrm{e}^z} \mathrm{d} z = 2 \pi i \sum_{k=0}^\infty -\mathrm{e}^{a i\pi(1+2k)} = -2 \pi i \left(\frac{e^{i \pi a}}{-1+e^{2 i \pi a}} \right) = \frac{\pi}{\sin(\pi a)} $$
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0Thanks for giving a nice proof of the residues. Can you please tell about the circular path the this integral tends to zero? – 2018-12-21
When $g(z)$ has a zero of order one at $z = z_0$ (true here since all $g''(z_0) \neq 0$), the residue of $f(z)/g(z)$ at $z = z_0$ is $f(z_0)/g'(z_0)$.
So for the pole at $z = (1 + 2\pi)ki$, the residue is $e^{(1 + 2\pi)aki}/e^{(1 + 2\pi)ki} = - e^{(1 + 2\pi)aki}$.
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0Ah brilliant, I see what I did wrong – 2011-11-27