∫∞0cosaxcoshxdx
a is a real number.
Hint: Consider a suitable rectangular contour
I know that 1coshz has simple poles when z=(2n−1)π2i.
What should I do next?
I would appreciate any help. Thank you.
Answer
First off, let us notice that the function cos(ax)/cosh(x) does not change under x↦−x. So,
∞∫0cosaxcoshxdx=12∞∫−∞cosaxcoshxdx.
Next, we should take the contour γM to be like that. Line (1) is a part of real line [−M,M], and the upper line (3) is the same line translated by πi.
The integral over line (3) could be rewritten as
∫(3)cosaxcoshxdx=∫(1)cosa(x+πi)cosh(x+πi)dx=(∗).
Notice, that cosh(x+πi)=−coshx and
cosa(x+πi)=cos(ax)cos(aπi)−sin(ax)sin(aπi).
Therefore,
(∗)=−cos(aπi)∫(1)cosaxcoshxdx+sin(aπi)∫(1)sinaxcoshxdx.
Last integral clearly equals zero, as it is the sum of symmetric and antisymmetric function over symmetric interval.
Integrals over lines (2) and (4) go to zero as M→∞. So, we have now, via residuals theorem
2πirescosaxcoshx=lim
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