Consider the following Poisson's equation;
$${\nabla}^2\phi=-f(\mathbf{r})$$
How can I solve for $\phi$ using integral transform?
I tried beginning with inverse fourier transform but just after I wrote few lines I do not know how to proceed..
$$f(\mathbf x)=∫dkf(\mathbf k)e^{ik⋅\mathbf x}$$
$$∇^2f(\mathbf x)=∇^2∫dkf(\mathbf k)e^{ik⋅\mathbf x}=∫dkf(\mathbf k)∇^2e^{ik⋅\mathbf x}=-k^2∫dkf(\mathbf k)e^{ik⋅\mathbf x}$$
Can someone help me please.
Assuming three-dimensional Cartesian coordinates, I would transform from space ($x$,$y$,$z$) to Fourier Domain ($X$,$Y$,$Z$) \begin{equation} (iX)^2\, \Phi + (iY)^2\, \Phi + (iZ)^2\, \Phi = -F(X,Y,Z) \end{equation} with \begin{equation} F(X,Y,Z)=\int_{-\infty}^\infty \int_{-\infty}^\infty \int_{-\infty}^\infty f(x,y,z)\, e^{-ixX} e^{-iyY} e^{-izZ}\,\text{d}x\,\text{d}y\,\text{d}z. \end{equation} And then solve for \begin{equation} \Phi(X,Y,Z) = \frac{F(X,Y,Z)}{X^2 + Y^2 + Z^2} \end{equation} and finally hope to find the inverse transform \begin{equation} \phi(x,y,z) = \frac{1}{2\pi} \frac{1}{2\pi} \frac{1}{2\pi} \int_{-\infty}^\infty \int_{-\infty}^\infty \int_{-\infty}^\infty \Phi(X,Y,Z)\, e^{ixX} e^{iyY} e^{izZ} \,\text{d}X\,\text{d}Y\,\text{d}Z. \end{equation} by analytical integration, transform tables (can $\Phi$ be found as entry or product of entries?) or numerical approximations.