Hi everyone, I have a question concerning the derivation of the J_0(t). In my book, it states that the inverse laplace transform of (s^2+1)^-1/2 is this function. It gives me a contour to integrate around and derive it. The problem is this: I always get an extra I in the answer. This is the contour: Take an infinite semicircle starting from 1 all the way to negative infinity. Inside the semicircle there is a dogbone contour. Now, since there are no singularities inside the contour, it is zero. Thus, the Integral of the dogbone + the vertical line = 0, because the rest vanishes identically. Next, we divide by 2pi*i to get the old function back because by definition the function equals the vertical integral divided by 2pi*i. After this, we only need to find the limits between the branch points and we can write it as the poisson integral representation of the bessel function. I get an extra i because when I parametrize the dogbone, we have the integrand multiplied by i. What I got for the limits on each side of the contour is exp(-pi*i/2)integrand for the limit on the left and exp(-3pi*i/2) . Thus negative Integral divided by pi is the desired answer. I looked to confirm my answer on wikipedia under bessel functions, scrolled down, and I was a constant off. Where did I go wrong?(adsbygoogle = window.adsbygoogle || []).push({});

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# Derviation of bessel function of first kind via contour integration

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