How to compute multidimensional inverse Fourier transform

[Hanyu Ye]

Hello, everybody. I am currently working on deriving solutions for Stokes flows. I encounter a multidimensional inverse Fourier transform. I already known the Fourier transform of the pressure field:
$\tilde{p}=-\frac{i}{{{k}^{2}}}\mathbf{F}\centerdot \mathbf{k}$
where $i$ is the imaginary unit, $\mathbf{k}$ is the frequency vector, $k$ is the length of $\mathbf{k}$ (That is, $k=\left\| \mathbf{k} \right\|$), and $\mathbf{F}$ is a constant vector. I don't know how to perform the inverse transform, although I have found the final answer in some references, which reads
$p=\frac{\mathbf{F}\centerdot \mathbf{x}}{4\pi {{r}^{3}}}$
where $r=\left\| \mathbf{x} \right\|$
Does anybody has an idea? Thanks a lot.

 

[Evo]

Hello, everybody. I am currently working on deriving solutions for Stokes flows. I encounter a multidimensional inverse Fourier transform. I already known the Fourier transform of the pressure field:
$\tilde{p}=-\frac{i}{{{k}^{2}}}\mathbf{F}\centerdot \mathbf{k}$
where $i$ is the imaginary unit, $\mathbf{k}$ is the frequency vector, $k$ is the length of $\mathbf{k}$ (That is, $k=\left\| \mathbf{k} \right\|$), and $\mathbf{F}$ is a constant vector. I don't know how to perform the inverse transform, although I have found the final answer in some references, which reads
$p=\frac{\mathbf{F}\centerdot \mathbf{x}}{4\pi {{r}^{3}}}$
where $r=\left\| \mathbf{x} \right\|$
Does anybody has an idea? Thanks a lot.

If you could show what you fear is the problem you have in solving these problems, it could help us understand where you need help.

 

[Hanyu Ye]

Hello, everybody. I am currently working on deriving solutions for Stokes flows. I encounter a multidimensional inverse Fourier transform. I already known the Fourier transform of the pressure field:
$\tilde{p}=-\frac{i}{{{k}^{2}}}\mathbf{F}\centerdot \mathbf{k}$
where $i$ is the imaginary unit, $\mathbf{k}$ is the frequency vector, $k$ is the length of $\mathbf{k}$ (That is, $k=\left\| \mathbf{k} \right\|$), and $\mathbf{F}$ is a constant vector. I don't know how to perform the inverse transform, although I have found the final answer in some references, which reads
$p=\frac{\mathbf{F}\centerdot \mathbf{x}}{4\pi {{r}^{3}}}$
where $r=\left\| \mathbf{x} \right\|$
Does anybody has an idea? Thanks a lot.

Oh, I have found the answer. It is presented in the following link:
http://www.fuw.edu.pl/~mklis/publications/Hydro/oseen.pdf