2-D inverse Fourier transform of Heaviside function

Question

Now I have a Heaviside function $H(K-\sqrt{k^2+l^2})$ in a 2D $\hat k$ space, where $k$ and $l$ are two variables in that space. In a paper, it is said that the inverse Fourier transform of this Heaviside function is: $$ \frac{K}{\sqrt{x^2+y^2}} J_1(K\sqrt{x^2+y^2}) $$ where $x$ and $y$ are variables in 2D real space and $J_1$ is the first order Bessel function.

How to derive this result?

Attempts:

I try to perform this transform by: $$ \int_{-\infty}^{+\infty} d k \int_{-\infty}^{+\infty} d l \ H(K-\sqrt{k^2+l^2}) e^{i2\pi(kx+ly)} $$ Using the properity of Heaviside function, the integral becomes: $$ \iint_{k^2+l^2 \leq K^2} dk \ dl \ e^{i2\pi(kx+ly)} $$ Switch to polar coordinates(set $k=r\cos\theta$ and $l=r\sin\theta$): $$ \int_0^{K} dr \int_0^{2\pi} d\theta \ r e^{i2\pi r(x\cos\theta+y\sin\theta)} $$ After integral over $r$, we have: $$ \int_0^{2\pi} d\theta \ \frac{-1+[1-i2\pi K(x\cos\theta+y\sin\theta)]e^{i2\pi K(x\cos\theta+y\sin\theta)}} {4\pi^2(x\cos\theta+y\sin\theta)^2} $$ I don't know how to deal with this integral and how to transform it into Bessel function.

Answer 1

Let's take as our starting point the integral $$ I:=\int_0^{K} dr \int_0^{2\pi} d\theta\, r e^{i2\pi r(x\cos\theta+y\sin\theta)}. \tag{1} $$ With $(x,y)=(\rho\cos\phi,\rho\sin\phi)$, we may compute the angular integral with the help of the Jacobi-Anger expansion: \begin{align} \int_0^{2\pi} d\theta\,e^{i2\pi r\rho\cos(\theta-\phi)}&= \int_{0}^{2\pi} d\theta\,\sum_{n=-\infty}^{\infty}i^nJ_n(2\pi r\rho)e^{in(\theta-\phi)} \\ &=2\pi J_0(2\pi r\rho). \tag{2} \end{align} Now, to compute the radial integral, we use the identity $\frac{d}{dx}(xJ_1(x))=xJ_0(x)$: \begin{align} \int_0^K2\pi J_0(2\pi r\rho)r\,dr&=\frac{1}{2\pi\rho^2}\int_0^{2\pi K\rho}\xi J_0(\xi)\,d\xi \\ &=\frac{K}{\rho}J_1(2\pi K\rho) \\ &=\frac{K}{\sqrt{x^2+y^2}} J_1\left(2\pi K\sqrt{x^2+y^2}\right). \tag{3} \end{align} (The factor of $2\pi$ in the argument of $J_1$ probably has to do with the convention used in the definition of the Fourier transform.)