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As one step in Ahlfors derivation of Stirling's formula (2nd edition pages 199-200), he uses residues to evaluate the integral $$\int\limits_{\xi = n+1/2} \frac{d\zeta}{\left|\zeta +z\right|^2}$$

over a vertical line $\xi = n + \frac{1}{2}$ where $\zeta = \xi + i\eta$, $z$ is a fixed point with positive real part, and $n$ is a positive integer. Could someone explain how residues could be used in this case,since the contour is not closed? To set the context, Ahlfors has already applied the residue theorem over a rectangle, of which the vertical line $\xi = n + \frac{1}{2}$ makes up the right side.

Siminore
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Russell
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  • This sort of thing is a standard technique in complex analysis. Have you seen, for example, computations of various integrals over the real line (for example, $1/p(z)$ with $p$ a polynomial) via contour integrals? – anomaly Aug 11 '15 at 17:06
  • Yes, now I see that I can create a closed contour with a semicircle and pass to the limit. I was needlessly concerned about working with the original contour. Thanks. – Russell Aug 11 '15 at 22:05

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