4
votes
1answer
73 views

Complex Integration: theorem of Residues

I am trying to prove a theorem that is doing my head in a bit. I have tried to simplify the problem as much as possible and leave out the details, even though it might look a bit too big. The ...
3
votes
1answer
87 views

Solve $\mathscr{F}^{-1} [ \cot{a \omega} \times \mathscr{F} \{ U(t) \sin{\omega_0 t} \} ] $ using contour integration

I wish to evaluate $y(t) = \mathscr{F}^{-1} [ \cot{a \omega} \times \mathscr{F} \{ U(t) \sin{\omega_0 t} \} ] $, where $\mathscr{F}$ represents the Fourier transform, and U(t) represents the ...
4
votes
0answers
38 views

Inverse Fourier transform using Residues for a ratio of hyperbolic functions.

I'm new and glad to be here. I have a problem relating to an inverse Fourier transform. I have $$g(w)= \frac{\sinh{w(a-b)}}{w \cosh{wa}}$$ and want to find $$G(t)$$. I cannot find this in tables so I ...
5
votes
2answers
114 views

Find the Fourier Transform of $\dfrac{x}{x^4+4}$

I have a problem here that becomes quite difficult to manage. I have to find the fourier transform of: $$f(x)=\frac{x}{x^4+4}$$ I'm sure there will be many ways to do this and I'll post my method ...
7
votes
3answers
367 views

Evaluate $\int_{0}^{\infty}\dfrac{\mathrm dx}{(e^{\pi x}+e^{-\pi x})(16+x^2)}$

Find the integral $$I=\int_{0}^{\infty}\dfrac{1}{(e^{\pi x}+e^{-\pi x})(16+x^2)}dx$$ My try:let $x=-t$ $$I=\int_{-\infty}^{0}\dfrac{1}{(e^{\pi x}+e^{-\pi x})(16+x^2)}dx$$ so ...
5
votes
1answer
88 views

Looking for guidance on a Fourier integral

Working with a Fourier transform problem, I've encountered the following integral: $$ \int_{-\infty}^{\infty}\frac{\exp\left(-a^2x^2+ibx\right)}{x^2+c^2}dx $$ where $a$, $b$, and $c$ are real ...
2
votes
2answers
110 views

Residue theorem, $\int_{-\infty}^{\infty} e^{-ikx}(1-ika^2)^{-m} dk$ with integer $m$

I am trying to solve this integral $\int e^{-ikx}(1-ika^2)^{-m} dk$ using the residue theorem, but I cannot find the residue of the function. $$\frac{1}{(1-ika^2)^{-m}}=\sum (ika^2)^n(-1)^n ...
5
votes
1answer
528 views

Fourier transform of $ \frac{\sinh(kx)}{\sinh(x)}$ [duplicate]

Possible Duplicate: Calculating the Fourier transform of $\frac{\sinh(kx)}{\sinh(x)}$ Im trying to compute the integral of $$I = \int_{-\infty}^\infty ...
1
vote
1answer
626 views

Finding the Fourier transform of $f(x) = \frac{a}{\pi} \frac{1}{a^2 + x^2}$ with the residue theorem

I keep getting the wrong answer for this problem! Find the Fourier transform of $f(x) = \frac{a}{\pi} \frac{1}{a^2 + x^2}$ using the residue theorem. Well, by definition: $$\hat f(k) = ...
8
votes
1answer
1k views

Calculating the Fourier transform of $\frac{\sinh(kx)}{\sinh(x)}$

I'm trying to compute $$\int_{-\infty}^\infty \frac{\sinh(kx)}{\sinh(x)}e^{-i\omega x} \ dx$$ i.e. the Fourier transform of $x\mapsto \frac{\sinh(kx)}{\sinh(x)}$, where $0<k<1$ is fixed. But ...