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In a paper I came across (page 10, section 7), the authors state that $\int_{-\infty}^{\infty} \frac{dx}{(b^{2}+x^{2})\cosh ax} $ can be evaluated by "closing the real axis with a semi-circle centered at the origin located in the upper half-plane. An elementary estimate shows that the integral over the circular boundary vanishes as the radius goes to infinity."

But I don't think that the integral over the circular boundary is going to vanish if one simply lets the radius go to infinity in a continuous manner. Furthermore, wouldn't it be easier to use a rectangular contour?

Random Variable
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  • The top half of the circle tends to zero. The real part does not. A semicircle is the simplest contour to use for this integral. – Argon Apr 06 '12 at 19:54

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You're right, the argument in the paper isn't rigorous because one would have to consider what happens when $|\cosh ax|\lt1$. Continously taking either a semicircle or a rectangle to infinity would cross the poles. You're also right that showing how to avoid this complication is easier for a rectangle than for a semicircle. Since $\cosh (x+\mathrm iy)=\cosh x\cos y+\mathrm i\sinh x\sin y$, rectangles at arbitrary distances from the origin can be chosen such that $|\cosh ax|\ge1$, and then the factor $b^2+x^2$ in the denominator ensures convergence.

Of course, since the integrals along the rectangular contours converge, the integrals along the semicircular contours also converge, since they're equal if they enclose the same poles; but this would be more difficult to show.

joriki
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  • Say $b=1$ and $a = \pi $. My idea would be to use a rectangle with vertices at $N, N+iN,-N+iN,$ and $-N$ where $N$ is an integer greater than $1$ so that I'm not going through a pole. Then showing the integral tends to zero around the three other sides is an easy application of the ML inequality. But if that works, then I guess I could instead use a circle with a radius of $N$. – Random Variable Apr 06 '12 at 20:48
  • @RandomVariable: Yes, but if you did it directly using a semi-circular contour, you'd need a slightly more complicated argument. – joriki Apr 06 '12 at 21:14
  • Are you saying that I need additional justification that I won't be integrating through a pole as the size of the semi-circle increases? I was thinking that I could just increase the size discretely and there wouldn't be a problem. – Random Variable Apr 06 '12 at 21:35
  • @RandomVariable: No, integrating through a pole isn't a problem, you can avoid that by increasing the size discretely; but it's not obvious that $|\cosh ax|\ge1$ on the semi-circle. It's intuitively clear that even if that's not the case, as the curvature of the semi-circle decreases, any contributions with $|\cosh ax|\lt1$ wouldn't matter compared to the $x^{-2}$ decay, but it would be a bit of work to prove that rigorously. – joriki Apr 06 '12 at 21:53
  • But if we know that the integral evaluates to zero along the three sides of the rectangle, then it must also evaluate to zero along the semi-circle, right? And could you argue that along the vertical sides of the rectangle the integral must tend to zero since the real part of $z$ becomes increasing large there and thus the modulus of $\cosh az$ becomes increasingly large? And then we could just use the ML inequality to estimate the integral along the top of the rectangle. – Random Variable Apr 06 '12 at 22:18
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    @RandomVariable: I think that was a misunderstanding. I had argued from the beginning that the integrals along the semi-circles must converge because the integrals along the rectangles do. I was merely saying that the latter is easier to show directly, whereas the convergence of the integrals along the semi-circles is easy to infer from the convergence of the integrals along the rectangles, but is more difficult to rigorously prove directly. – joriki Apr 06 '12 at 22:22
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    @RandomVariable: Yes, the modulus of $\cosh az$ increases exponentially with the real part of $z$. And yes, the ML inequality is the basis for all this; but to easily apply it we need $|\cosh ax|\ge1$, which we can ensure easily at the top of the rectangle but not as easily along the semi-circle. – joriki Apr 06 '12 at 22:24