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4
votes
2answers
310 views

Expectation of $(c+e^{N(0,\sigma^2)})^{-n},\, n>0$

I would like to know if there's a way to compute or approximate the following expectation: $$\mathbb{E}[(c+e^X)^{-n}]$$ where $X=N(0,\sigma^2)$ and $n,c>0$ (you can also assume that $n$ is a ...
1
vote
1answer
280 views

Solving Stokes Equations using 3D Fourier transforms

How do you calculate the inverse Fourier transform of $\frac{k_ik_j}{k^4}$. I know it has to be a matrix of the form $=δ_{ij}A(r)+r_ir_jB(r)$, but how do you calculate the functions A(r) and B(r)? I ...
0
votes
1answer
290 views

nonnegative Fourier Transform

Let $\widehat{f}(\xi)$ be Fourier transform of $f$ given by \begin{align} \widehat{f}(\xi)=\int_{\mathbb{R}^n} e^{-ix\cdot\xi}f(x)dx. \end{align} Suppose that $\widehat{f}(\xi)$ is nonnegative and ...
2
votes
1answer
496 views

Integral solving request

Dear all, please help me solve the following integral. I need to solve this integral for one of my problems. $$(\frac{1}{2\pi})^2\int_0^\infty\int_{-\infty}^\infty \frac{J_0(\rho R_0)J_0(\rho ...
1
vote
1answer
542 views

Space of Bandlimited Functions

Hi, I was asking myself about some necessary and/or sufficient conditions for a function to be bandlimited (i.e. its Fourier transform is zero t residing out of [-B,B] for some B>0). Of course, if a ...
2
votes
2answers
376 views

Fourier transform of $e^{it|\xi|^{\alpha}}$

Consider the fourier transform of $e^{it|\xi|^{2\alpha}}$ ($\alpha>0$)in $\mathbb{R}^n$,let $K_{\alpha}=\mathcal{F}(e^{it|\xi|^{\alpha}})$,so $K$ is a tempered distribution.Now I want to know if ...
-2
votes
1answer
345 views

derivative of a function of time using its inverse fourier transform

What would be the bounds on the derivative of a function using its inverse fourier transform representation. Furthermore what would be the bounds on the absolute value of the function itself?
0
votes
0answers
249 views

Theta functions and Fourier transforms

Let $T_\tau$ be the 2-dimensional torus, with the complex structure induced by the lattice generated by $1$ and $\tau$. Then for a line bundle $L_k$ over $T$ with level $k$, there is an orthonormal ...
9
votes
6answers
31k views

Fourier vs Laplace transforms

In solving a linear system, when would I use a Fourier transform versus a Laplace transform? I am not a mathematician, so the little intuition I have tells me that it could be related to the boundary ...