I had a question about the following. Consider the function $\frac{i}{4}H_{0}^{(1)}(\tilde{k}\rho)$ where $H_{n}^{(1)}$ is the Hankel function of the first kind of order $n.$ For context, it is a Green's function for a line source with a polarization along that same direction from electromagnetism. For electromagnetism, you can get Fourier decomposition of the Green's function. Therefore,
\begin{equation} \frac{i}{4}H_{0}^{(1)}(\tilde{k}\rho) = \iint \frac{d^2\mathbf{k}}{(2\pi)^2} e^{i\mathbf{k}\cdot\boldsymbol{\rho}} \frac{1}{k^2-\tilde{k}^2} \end{equation} where $\boldsymbol{\rho} = (x, y)$ is $2D$, $\rho = \sqrt{x^2 + y^2},$ and $\tilde{k}$ is complex. When I evaluate the integral on the right numerically, it seems to converge quite well with the left-hand side of the equation. Why is this? Isn't there a blow-up as $\rho\to 0$? That's one question. Another question is how do I get good Fourier representations of various derivatives $\left(\frac{d}{dx}, \frac{d}{dy}\right)$ of the left-hand side?
I am personally interested in a TE line source, and according to a textbook by Felsen it is related to derivatives of the left-hand side. However, even when I take one derivative \begin{equation} \frac{d}{dx}\frac{i}{4}H_{0}^{(1)}(\tilde{k}\rho) = -\frac{i\tilde{k}}{4}H_{1}^{(1)}(\tilde{k}\rho)\frac{x}{\sqrt{x^2+y^2}} \stackrel{?}{=} \iint \frac{d^2\mathbf{k}}{(2\pi)^2} e^{i\mathbf{k}\cdot\boldsymbol{\rho}} \frac{ik_{x}}{k^2-\tilde{k}^2} \end{equation} numerically I start to get issues and the integral does not agree with the analytic expression. Where are things going wrong? If I take yet another derivative, the Fourier components approach a constant in some direction in $(k_x, k_y)$ space. What is going on? Are there delta function contributions from the origin that are missing? Is the TM case in E&M representable in this Fourier basis but TE isn't?
TLDR: If I want something like $F(x,y)$ where \begin{equation} F(x,y) \equiv -\frac{i\tilde{k}}{4}H_{1}^{(1)}(\tilde{k}\sqrt{x^2+y^2})\frac{x}{\sqrt{x^2+y^2}} \stackrel{?}{=} \iint \frac{d^2\mathbf{k}}{(2\pi)^2} e^{i\mathbf{k}\cdot\boldsymbol{\rho}} (?) \end{equation} how do I figure out what must go in the $(?)$ to make the two sides agree for $x^2+y^2 \neq 0?$