# Solving Integral using Residues

1. Feb 12, 2009

### nassboy

I've already posted this question, but I think I need to clarify my approach to the problem.

I'm trying to solve this integral using the method of residues.

$$\int_{\text{yo}}^{\text{ys}} \frac{1}{\sqrt{c+e^{-y}+y}} \, dy$$

First I've changed variables to so that

$$w=\text{ArcTanh}\left[\frac{-2 y+\text{yo}+\text{ys}}{\text{yo}-\text{ys}}\right]$$

and now the integral reads

$$\int_{-\infty }^{\infty } \frac{\text{ys} \text{sech}^2(w)-\text{yo} \text{sech}^2(w)}{2 \sqrt{c+e^{\frac{1}{2} (\text{yo} \tanh (w)-\text{ys} \tanh (w)-\text{yo}-\text{ys})}+\frac{1}{2} (\text{yo} (-\tanh (w))+\text{ys} \tanh (w)+\text{yo}+\text{ys})}} \, dw$$

I not really sure how to find the poles or how to proceed from here....

I think there is a pole at $$\left\{\left\{w = \text{ArcTanh}\left[\frac{2 c+\text{yo}+\text{ys}-2 \text{ProductLog}\left[-e^c\right]}{\text{yo}-\text{ys}}\right]\right\}\right\}$$

Any help would be appreciated

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