2
votes
1answer
26 views

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 ...
2
votes
0answers
53 views

Fourier transform of a sinusoidal function

Let us consider following table which I want to calculate myself $$ x(t)=\frac{\sin(\omega_bt)}{\pi t}\quad\iff\quad X(j\omega)= \begin{cases} 1 & \text{if $|\omega|<\omega_b$}, ...
1
vote
1answer
27 views

Does the integral in the formal 2D Fourier transform of the logarithm converge?

If $k$ is a nonzero vector in $\mathbb R^2$, how to interpret this integral: $$\int_{\mathbb R^2}e^{ik\cdot x}\ln{|x|}dx$$ Does it converge and in what sense? Thanks in advance.
0
votes
1answer
33 views

How to calculate the value of $\int_{-\infty}^{\infty} y(t)dt$?

For a function $g(t)$, $\int_{-\infty}^{\infty} g(t)e^{-j\omega t}dt=\omega e^{-2\omega ^2}$ for any real value $\omega$. If $y(t)=\int_{-\infty}^{t}g(\tau)d\tau$ then how to calculate the value of ...
1
vote
0answers
35 views

Is this function square-integrable? Able to be Fourier expanded?

I want to do a 3-dimensional Fourier series expansion on this function$$\frac{\cos (x) \cos (y) \cos (z)-\sin (x) \sin (y) \sin (z)}{\left[(a+\sin (y)+\cos (z))^2+(b+\cos (x)+\sin (z))^2+(c+\sin ...
3
votes
1answer
84 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 ...
1
vote
0answers
29 views

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, ...
3
votes
2answers
160 views

The Fourier transform of a power of the absolute value function (and a related integral)

What (Fourier-analytic?) methods would I use to compute the following two integrals? $\displaystyle\int_{\mathbb{R}} e^{2 \pi i t} |t|^a dt \:\:\:\:\:\:\: \:\:\:\:\:\:\: \text{ and } ...
1
vote
1answer
42 views

Integral from inverse Fouriertransform of 1/(1+p^2)^2

In a calculation I end up with the following integral $$\int_0 ^\infty \frac{p \sin (pr)}{(1+p^2)^2}dp , $$ could someone give me a hint how to evaluate this one? (This integral comes from the ...
2
votes
3answers
592 views

Does $f(x)\in L^1$ imply that $\lim_{\omega \to \infty } \, \int_{-\infty }^{+\infty } f(x) e^{-i x \omega } \, dx=0$?

Suppose that $f(x)$ is $L^1$ and R- integrable function, problem is to resolve if it is possible existence of such a $f(x)$ that: $$\lim_{\omega \to \infty } \, \int_{-\infty }^{+\infty } f(x) e^{-i x ...
3
votes
1answer
71 views

Ques from exam: sequence of functions and improper integrals

$P_n(x):R\rightarrow R$ is a sequence of functions defined by: $$P_n(x)= \frac{n}{1+n^2x^2}$$ f:R->C is continuous and 2pi periodic. We define: $$f_n(x)=\frac{1}{\pi}\int ...
2
votes
1answer
244 views

Use Fourier transform to calculate double integral of harmonic function

Let $$P_y(x)=\dfrac{1}{2\pi}\int_{-\infty}^\infty e^{-y|t|}e^{ixt}dt=\dfrac{1}{\pi}\dfrac{y}{x^2+y^2}.$$ Then $P_y(x)$ is harmonic in the upper half-plane $y>0$ and for $f\in L^1(\mathbb{R})$, ...
1
vote
2answers
177 views

Integral through Fourier Transform and Parseval's Identity

$$ \int_{-\infty}^{\infty}{\rm sinc}^{4}\left(\pi t\right)\,{\rm d}t\,. $$ Can you help me evaluate this integral with the help of Fourier Transform and Parseval Identity. I could not see how it is ...
3
votes
3answers
169 views

Compute $\int_0^{\infty}\frac{\cos(\pi t/2)}{1-t^2}dt$

Compute $$\int_0^{\infty}\frac{\cos(\pi t/2)}{1-t^2}dt$$ The answer is $\pi/2$. The discontinuities at $\pm1$ are removable since the limit exists at those points.
4
votes
3answers
160 views

Evaluate $\int_{-\infty}^\infty x\exp(-x^2/2)\sin(\xi x)\ \mathrm dx$

Evaluate $\int_{-\infty}^\infty x\exp(-x^2/2)\sin(\xi x)\ \mathrm dx$ The answer given by Wolfram Alpha is $\sqrt{2\pi}\xi\exp(-\xi^2/2)$. Observe how this is related to the Fourier transform of ...
0
votes
1answer
38 views

Showing Airy's Integral (Fourier Transform of e^{-ip^3/3}) Converges

Airy's integral (a.k.a. $\widehat{e^{-ip^3/3}}$ times some constant multiple) is given by $\displaystyle\text{Ai}(x)=\frac{1}{2\pi}\int_{-\infty}^\infty e^{-ipx}e^{-ip^3/3}dp$. Although it looks ...
2
votes
3answers
189 views

Complex Integral with exponential

I've been struggling with this: $$\int_{0}^{\infty }\frac{e^{-px}}{x^{2}+1}\mathrm{d}x, \; \; p\ge 0.$$
4
votes
2answers
268 views

Complex-valued Fourier integral: $ \int_{ - \infty }^{ + \infty } {\frac{{\cos (ax)}}{{{x^2} + 1}}{e^{ - ibx}}\,\mathrm dx} $

I'm working on the Fourier transform, but I don't know how to evaluate the integral: $$I = \int_{ - \infty }^{ + \infty } {\frac{{\cos (ax)}}{{{x^2} + 1}}{e^{ - ibx}}\,\mathrm dx} $$
0
votes
1answer
149 views

Problem evaluating an improper integral $\int_0^{\infty} \frac{(\sin{2x}-2x\cos{2x})^2}{x^6}$ using fourier transform

This is a question from one of the past papers of my university which I am unable to do. I am not being able to do question 2 from below. Let $f(x)= a^2-x^2 \,\,\,\,\, |x|<a ...
0
votes
2answers
2k views

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.
3
votes
3answers
219 views

Two improper log integrals

Evaluate $$\int_0^{\frac{\pi}{2}}\ln ^2(\tan x)\text{d}x$$ $$\int_0^{\frac{\pi}{2}}\ln ^2(\sin x)\text{d}x$$
4
votes
1answer
805 views

Fourier transform of Cauchy principal value

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: $$ ...
3
votes
1answer
156 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 ...
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 ...
2
votes
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 ...
3
votes
0answers
231 views

2 dimensional Fourier transform integral

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 ...