# Tagged Questions

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### fourier transform of scaled function

let us consider following example one thing which i did not understand is where absolute value of $a$ came from?ok if we have $\int^{\infty}_{-\infty} x(a*t)*e^{-j\omega*t}dt$ then we may have ...
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### 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 ...
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### The Fourier sine transform of $f(x)/\sin x$

Is the following result $$\lim_{\lambda \to \infty} \frac{2}{\pi} \int_0^\infty \frac{f(x)}{\sin x} \sin(\lambda x) \, dx = f(0) + 2\sum_{k = 1}^\infty f(k\pi),$$ where $\lambda$ is an odd integer, ...
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### Calculate the Fourier transform of ${\rm b}\left(x\right) = 1/\left(x^{2} +a^{2}\right)$

I need help to calculate the Fourier transform of this funcion $${\rm b}\left(x\right)=\frac{1}{x^{2} + a^{2}}\,,\qquad a > 0$$ Thanks.
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Evaluate $$\int_0^{\frac{\pi}{2}}\ln ^2(\tan x)\text{d}x$$ $$\int_0^{\frac{\pi}{2}}\ln ^2(\sin x)\text{d}x$$
I try to understand the direct computation of the Fourier transform of the distribution `Cauchy principal value' $v.p \frac{1}{x}$. I don't understand the following change of order of integration: $$... 1answer 161 views ### integral evaluation of an exponential let be the function$$ e^{-a|x|^{b}} $$with  a,b  positive numbers bigger than zero then how could i evaluate this 2 integrals ?$$ \int_{-\infty}^{\infty}dxe^{-a|x|^{b}}e^{cx}$$here 'c' can ... 1answer 1k views ### Calculating the Fourier transform of \frac{\sinh(kx)}{\sinh(x)} I'm trying to compute$$\int_{-\infty}^\infty \frac{\sinh(kx)}{\sinh(x)}e^{-i\omega x} \ dx$$i.e. the Fourier transform of x\mapsto \frac{\sinh(kx)}{\sinh(x)}, where 0<k<1 is fixed. But ... 1answer 115 views ### Asymptotics of an improper integral I have to show that if x \to \infty, then$$ \int\limits_{\mathbb{R}^d} \frac{e^{i\xi x}}{\xi^2 + 2k\xi}d\xi = O\left(|x|^{-\frac{d-1}{2}} \right) \;\;\; \; d\geqslant2, \;\;\; k\in \mathbb{C}^d ...
I'm trying to calculate the two dimensional Fourier integral $$\iint \mathrm d\vec{R} \; (x^2 + y^2) \; e^{-2 \sqrt{ x^2 + y^2 + z^2}} \; e^{i\vec{K}\vec{R}} \;,$$ with $\vec{R}=(x,y)$. Switching to ...