I'm trying to determine the expression for the Fourier transform of a function defining half an ellipse. It's been awhile since I've done Fourier transforms by hand. Obviously I can plug the expression into Wolfram or Mathematica, but I want to be able to check the solution they provide and understand how these tools operate. My function is defined as
$ f(x) = \frac{b}{a} \sqrt{a^2 -x^2}$
where $a$ and $b$ are constants defining the "semi major" and "semi minor" axis, respectively.
Plugging the equation into Wolfram generates a transform that looks rather complex and involves a "modified Bessel function of the second kind" which quite honestly has crippled my confidence in solving this on my own. What leads to a "modified Bessel function of the second kind" and how would I identify it in the process of applying the transform? I can honestly say, I would never identify it myself and probably give up in a mess of expressions.
Can someone help walk me through the process of solving the transform by hand and comment on my questions concerning the "Bessel function"?
I should note that I would like to use the convention for the Fourier transform and its inverse as follows
$F(k) = \frac{1}{2\pi} \int_{-\infty}^{\infty} f(x)e^{-ikx}dx$
$f(x) = \int_{-\infty}^{\infty} F(k)e^{ikx}dk$