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I want to evaulate $\displaystyle{ \int_{-\infty}^{\infty} \frac{e^{kx}}{1+e^x} dx, \ k \in \mathbb{R} }$ via the Residue Theorem over the contour $[x=\pm a, y=0, y=2\pi]$ oriented counterclockwise.

  1. Lets define the complex function, $\displaystyle{ f(z) = \frac{e^{kz}}{1+e^z} }$.

  2. The function has singularities at $z_n = i\pi(2n+1), \ n \in \mathbb{Z}$ and thus we have a single singularity within the desired domain at $n=0$, $z_0 = i\pi$.

Any tips on how to proceed?

  • 1
    [This](http://math.stackexchange.com/questions/110457/closed-form-of-integral) and [this](http://math.stackexchange.com/questions/111435/infinite-series-representation-limited-or-not) are very related.2012-07-06
  • 0
    Note the integral converges $\iff a \in (0,1)$2012-07-06

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