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Math Help - [SOLVED] Tricks to evaluate this integral?

  1. #1
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    [SOLVED] Tricks to evaluate this integral?

    Let J_0 (x) = 2/\pi \int_{0}^{\pi/2} cos (x cos \theta) d\theta. Show that \int_{0}^{\infty} J_0(x) \exp(-ax) dx = 1/\sqrt{(1+a^2)} if a>0.

    I'm guessing that you have to reverse the order of integration in order to make the integrals easier to evaluate, but it still seems quite tricky. Any ideas?
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  2. #2
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    Yes, just reverse integration order.

    We need to deal with \int_{0}^{\infty }{e^{-ax}\cos (x\cos \theta )\,dx}. The best way on solving it, it's via complex numbers, but to do that, first put \beta=\cos\theta. (It's just a constant, it's just to make computations easier.)

    Thus the integral equals \int_{0}^{\infty }{e^{-ax}\cos (\beta x)\,dx}=\text{Re}\int_{0}^{\infty }{e^{-(a+\beta i)x}\,dx}=\text\,\frac{1}{a+\beta i}=\frac{a^{2}}{a^{2}+\beta ^{2}}, thus the remaining challenge is to compute \frac{2}{\pi }\int_{0}^{\frac{\pi }{2}}{\frac{a}{a^{2}+\cos ^{2}\theta }\,d\theta }.
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