0
votes
1answer
14 views

Apply the Fourier Transform to $A\cdot e^{-a|k - k_0|}$

I have the following problem: The task is to show that $$f^*(k) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^\infty f(k) e^{ik(x-vt)} dk$$ with $f(k) = A\cdot e^{-a|k - k_0|}$ equals $$f^*(k) = ...
2
votes
1answer
71 views
+50

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 } ...
0
votes
0answers
13 views

Obtain the complex Fourier Series of the following function:

$$f(t)=t^3 \;\;\;\;\;\;\;\;\;\;\;\; -3/2<t\leq 3/2 $$ $$f(t)=f(t+3)$$ I've tried setting up an integral for $C_n$ coefficients using the formula $$C_n = \frac{1}{L} \int^{L/2}_{-L/2} f(t) ...
1
vote
1answer
35 views

Integral equation, Fourier transform

Find all functions $ f : \mathbb{R} \rightarrow\mathbb{R} $, that solve $\int_{-\infty}^{\infty} f(t-x)f(x) dx =e^{-t^2}$, $ t\in \mathbb{R}$ How do I solve this? I know that the left part is the ...
0
votes
1answer
31 views

Integral-Fourier sum

I am trying to prove the following relation in (3) where $\alpha,\beta,\gamma,\delta,\omega \in \mathbb{R}$. Given the integral $$ I=\frac{1}{2}\int_0^\alpha dx \left( \beta ...
0
votes
2answers
40 views

How to calculate this basic Fourier Transform?

I am trying to calculate the Fourier Transform of $g(t)=e^{-\alpha|t|}$, where $\alpha > 0$. Because there's an absolute value around $t$, that makes $g(t)$ an even function, correct? If that's ...
0
votes
2answers
71 views

fourier transform of sinc function

let us consider fourier transform of sinc function,as i know it is equal to rectangular function in frequency domain and i want to get it myself,i know there is a lot of material about this,but i want ...
1
vote
1answer
39 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 ...
0
votes
1answer
20 views

Existence of a subsequence and convergence to $0$ of function solving heat equation

Let $f^j(x)$ be a sequence of integrable functions on the circle such that $$\int_{-\pi}^{\pi}|f^j (x) |^2 dx = 1.$$ ALso, let $u^j(x,t)$ solve the heat equation on the circle with initial data ...
3
votes
0answers
46 views

Prove the converse of convolution theorem

I am trying to prove the converse of convolution theorem: $$ \mathscr{F}[f(x)g(x)]=\frac{1}{\sqrt{2\pi}}\,\widetilde{f}(\omega)*\widetilde{g}(\omega)$$ I try to apply the definition of convolution ...
0
votes
0answers
17 views

How to find the inverse fourier transform of the fourier transform of $\delta(x-x_0)$ [duplicate]

I know that $F(\delta (x-x_0 )) = e^{j\omega x_0} $ and so therefore $F^{-1}(e^{j\omega x_0 }) = \delta(x-x_0) $ but when I try to do the integral for the inverse fourier transform: I get ...
1
vote
2answers
65 views

Fourier transform of function

What is Fourier transform of $$f(x)=\frac{e^{-|x|}}{\sqrt{|x|}}?$$ I tried to calculate it using $$F(e^{-|x|})=\sqrt{\frac{\pi}{2}}e^{-|a|}$$ and $$F(\frac{1}{\sqrt{|x|}})=\frac{1}{\sqrt{|a|}}$$ and ...
18
votes
1answer
309 views

Fourier transform of $\operatorname{erfc}^3\left|x\right|$

(this is a follow-up on my another question) Could you please help me to find the Fourier transform of $$f(x)=\operatorname{erfc}^3\left|x\right|,$$ where $\operatorname{erfc}z$ denotes the the ...
0
votes
0answers
63 views

What's the Fourier Transform of an Error Function?

What is the Fourier transform of $\displaystyle \operatorname{Erf}\left[a+bx^{2}\right] $? I need this in order to evaluate $$ \int_{-\infty}^{\infty}e^{-\beta x^{2}}Erf\left[a+bx^{2}\right]dx $$ ...
2
votes
0answers
26 views

Invariant functions under integral transforms

We all know Fourier transform has invariants such as $e^{-x^2}$, and another MSE post has shown the non-existence of invariant function under Hilbert transform using Fourier transform. I am wondering ...
1
vote
0answers
62 views

On $\int_{-\infty}^{+\infty} {\frac{\tan(t-t_0)}{\cosh^2(t-t_0)} \cos(\omega t) \,\mathrm{d}t}$

How to count this? $$ \int_{-\infty}^{+\infty} {\frac{\tan(t-t_0)}{\cosh^2(t-t_0)} \cos(\omega t) \,\mathrm{d}t} $$ Can we use residue formula?
1
vote
1answer
79 views

Integral $ \int_{-\infty}^\infty \frac{e^{ikx}}{x^{3/2}}dx$

Hi I'm trying to solve this integral Fourier Transform $$ \int_{-\infty}^\infty \frac{e^{ikx}}{x^{3/2}}dx=\sqrt{2\pi|k|}(1+i) (-1+\text{sgn}(k)) $$ where sgn(k)$=1$ for k>1 and $-1$ for k<1. I am ...
2
votes
2answers
55 views

How can this integral be rewritten with convolutions?

I've got $f:\mathbb{R}\rightarrow\mathbb{R}$ bounded and I'm trying to write `$\mathtt{f}$,' a discrete version of $f$, where each element in the domain takes on the average of the corresponding ...
4
votes
2answers
131 views

Fourier transform of $\operatorname{erfc}^2\left|x\right|$

Could you please help me to find the Fourier transform of $$f(x)=\operatorname{erfc}^2\left|x\right|,$$ where $\operatorname{erfc}z$ denotes the the complementary error function.
4
votes
1answer
68 views

A Parseval-like theorem for Mellin transforms

A particular case of Parseval's theorem for Fourier transforms says that if $f$ is square integrable on $\mathbb{R}$, then $$ \int_{-\infty}^{\infty} |f(t)|^{2} \ dt = \int_{-\infty}^{\infty} ...
1
vote
1answer
23 views

time integration property of fourier transform

I am having some trouble with some Fourier transform, Suppose that $F(\omega)$ is the Fourier transform of $f(x)$, i.e. where $$F(\omega)=\int_{-\infty}^{\infty}f(x)e^{-i\omega x}\,dx.$$ What is ...
2
votes
1answer
42 views

Riemann-Lebesgue application

By the Riemann-Lebesgue lemma, I have shown that for any finite interval measurable set $I$ of finite measure, any $h \in \mathbb{R}$, $$\lim_{n \to\infty}\int_I \cos (n(x+h)) \mathop{dx} = 0.$$ I ...
0
votes
0answers
14 views

Relating Fourier Transform to an Integral involving Sin(vt)

I have data for a function $S(Q)$ and I'm trying to find values for a different function $g(r)$ Now I know $g(r) = \int_0^{\infty} Q(S(Q)-1) \sin(Qr)\, dQ$ This is closely related to the sine ...
0
votes
1answer
54 views

Inverse Fourier transform problem

Can anybody please guide me how to compute the following inverse Fourier Transform ? $$ p(x) = \frac{1}{2\pi} \int_{-\infty}^{\infty} \frac{1}{(1-j\omega\bar{x})^K}e^{-j\omega x}d\omega $$ I shall ...
0
votes
1answer
42 views

Fourier transform real and imaginary part question?

I have to find the fourier transform of $f(t)=e^{-a^*t}*u(t)$ For a>0 the signal has an infinite value therefore doesnt have a Fourier transform.For a>0 we have: ...
1
vote
1answer
76 views

Laplace transform of $g_n(t)=\begin{cases}\frac{(1-e^{-t})^n}{t^n}&:t>0,\\0&:t\le0.\end{cases}$

Find Laplace transform for this function "$g$" $$g_n(t)=\begin{cases}\frac{(1-e^{-t})^n}{t^n}&:t>0,\\0&:t\le0.\end{cases}$$ Then Take advantage of it to calculate the following ...
0
votes
1answer
33 views

Integral of a function is not affected by altering the function values at zero-measure set

I'm studying about Fourier analysis from a book Fourier analysis and its applications, Folland 1992 and I have one point in the source I need clarification about: On page 69 it is stated that: "The ...
1
vote
2answers
37 views

Problem with the Fourier transform of a function

I'm having some troubles with this one: $$\mathcal F(e^{-|x|} +|x|e^{-|x|})$$ I know that $\displaystyle\mathcal F(e^{-|x|})={1\over \pi (1+w^2)}$ but the second part is where I get stuck.
0
votes
0answers
38 views

Fourier transform of a function involving $\sec(\omega)$

The summary of my question is: What should I make of $\mathcal{F}^{-1}\left[\frac{\csc(\omega)}{\omega^2-\beta^2}\right]$ where $\mathcal{F}^{-1}$ is the inverse fourier transform (taking $F(\omega)$ ...
1
vote
1answer
41 views

Complex Fourier integral

Why is the $\omega$ in the solution for this integral written in absolute value? $$\int_{-\infty}^{\infty} \frac{x e^{i\omega x}}{(x^2+1)^2}dx = \frac{\pi \omega}{2}e^{-|\omega|}$$
0
votes
2answers
112 views

A contour integral for Fourier transform

How does one show the following, preferably with contour integral on the complex plane? $$\frac{\Gamma(\alpha)}{2\pi}\int_{-\infty}^\infty (ik)^{-\alpha}e^{-ikx}dk = (-x)_+^{\alpha-1},$$ where $x$ is ...
1
vote
1answer
31 views

Limits of a Fourier transform

Consider a function $f(t)$ satisfying the following properties $$ \lim_{t\to\pm\infty} f(t) = f_0,~~~~~\lim_{t\to\pm\infty} f'(t) = 0 \tag1 $$ Consider now the Fourier transform of this function $$ ...
3
votes
0answers
46 views

Equivalent definitions of Fourier transform of a measure

For me the fourier transform of a measure $\mu\in\mathcal{S}'(\mathbb{R})$ is defined by $\hat{\mu}(\varphi)=\mu(\hat{\varphi})$ where $\varphi\in\mathcal{S}(\mathbb{R})$. My question is: if one has ...
1
vote
2answers
51 views

How to estimate (compute) Fourier transform?

Let $f:\mathbb R \to \mathbb R$ such that $$f(x)= \frac{\sin \pi x}{x (x^{2}-1)}$$ for $x\in \mathbb R - \{ 0, -1, 1 \}$ and $f(x):= \pi $ for $x=0$ and $f(x)=-\frac{\pi}{2}$ for $x= -1, 1$. ...
2
votes
2answers
96 views

Finding the complex fourier series of the function $x^2sin(x)$ in the interval $[{-\pi}, \pi]$?

This forms part of a project I am doing and I wish to see how well complex fourier series approximates a smooth curve such as this one. After tedious integration by parts, I have attained an answer ...
1
vote
1answer
107 views

Contour Integral of Exponential

I want to show the following for $a > 0$: $$e^{-a} = \int_{0}^{\infty}{\frac{e^{-x}}{\sqrt{x}}e^{-a^{2}/(4x)}dx}.$$
2
votes
2answers
106 views

Understanding Dirac delta integrals?

I'm confused as to how exactly to integrate using the Dirac delta function. I have the following example: $$\int \delta (x-4)(x^3-4x^2-3x+4)dx$$ and am told this evaluates to 8. Can anyone please ...
0
votes
3answers
83 views

Calculating this integral?

I'm trying to calculate $$\int\limits_{-\pi}^0e^{-x}\cos(nx)\,\mathrm{d}x$$ as part of a Fourier series calculation. My problem is the calculations seem to loop endlessly - I'm integrating by parts ...
4
votes
1answer
161 views

Fourier transform of $f(x)=\frac{1}{e^x+e^{-x}+2}$

Let $$f(x)=\large \frac{1}{e^x+e^{-x}+2}$$ Compute the Fourier transform of $f$. We can factor the denominator to get $$f(x)=\frac1{(\exp(x/2)+\exp(-x/2))^2}=\frac1{(2\cosh(x/2))^2}$$ I'm thinking ...
2
votes
1answer
220 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 $f\in L^1(\mathbb{R})$, ...
5
votes
2answers
76 views

Inequality associated with Fourier transform

Suppose $$\int_{I_1} x^2|f(x)|^2dx\ge\frac12\int_\Bbb Rx^2|f(x)|^2dx$$ and $$\int_{I_2} x^2|\hat f(x)|^2dx\ge\frac12\int_\Bbb Rx^2|\hat f(x)|^2dx$$ for interval $I_1, I_2$ centered at origin and ...
4
votes
1answer
92 views

Show $\lim_{N\to\infty}\int_0^\pi\left(\frac1{\sin\frac{x}2}-\frac2x\right)\sin\left((N+\frac12)x\right)dx=0$

Prove that the function $\csc(x/2)-2/x$ is integrable on $(0,\pi)$. In fact, prove that it is bounded. In fact, prove that it tends to zero as $x\to0$. Use this to show that ...
1
vote
1answer
71 views

Derivative of the Fourier integrals calculated in a point

I have considered the following definitions for the Fourier integral pairs: $\Phi(\omega) = \frac{1}{(2\pi)^3} \int_{\mathbb{R}^3} e^{ix\omega \cos\theta}R(x)x^2 dx \sin\theta d\theta d\phi$ $R(x) = ...
1
vote
1answer
63 views

Evaluating $\int\,\cos(x)\cos(\omega x)\,dx$ using trigonometric addition formulas

I'm looking to solve the integral $$\frac{1}{\pi}\int_{-\frac{\pi}{2}}^\frac{\pi}{2}\, \cos(x)\cos(\omega x)\,dx$$ by rewriting the terms using the trigonometric addition formulaes. It should end ...
0
votes
1answer
48 views

convolution of two functions and relations with their p-norms

Let $f\in L^p(\mathbb{R})$, $g\in L^1(\mathbb{R})$, $1\leq p< \infty$. Then I have proved the convolution $f\ast g\in L^p(\mathbb{R})$ and $||f\ast g||_p\leq ||f||_p||g||_1$. Does $f\ast g$ ...
1
vote
3answers
192 views

Closed form for integral $ \int_0^{\pi} \frac{\sin (m \phi)}{(1 + r \cos \phi)^n} d\phi$

Is there a closed form for $n>0$ integer, $m\neq 0$ integer, and $|r|<1$ real?
2
votes
1answer
118 views

Calculate the Fourier transform on $\mathbb{R}$ of $\frac1{(1+x^2)^2}$ and $\frac{x}{(1+x^2)^2}$.

Calculate the Fourier transform on $\mathbb{R}$ of $\frac1{(1+x^2)^2}$ and $\frac{x}{(1+x^2)^2}$. Also calculate the Fourier transforms of $\frac1{(1+ix)^2}$ and $\frac{\cos(\pi x/2)}{1-x^2}$. ...
5
votes
1answer
117 views

Fourier Transform of $\frac{1}{(1+x^2)^2}$

I need to find the Fourier Transform of $f(x) = \frac{1}{(1+x^2)^2}$ Where the Fourier Tranform is of $f$ is denoted as $\hat{f}$, where $\hat{f}$ is defined as ...
2
votes
1answer
149 views

Fourier transform on Hermite polynomial

Let $h_0(x)=e^{-x^2/2}$ and $h_k=B^kh_0$, where $B=-\dfrac{d}{dx}+x$. Define a transformation $T$ as $$Tf(y)=\dfrac{1}{\sqrt{2\pi}}\int_\mathbb{R}f(x)e^{-ixy}dx$$ How can I find the ...
2
votes
1answer
63 views

Inverse of Short-time Fourier transform

The Gabor transform (i.e. Short-time Fourier transform with some Gaussian window) can be defined by (see http://en.wikipedia.org/wiki/Gabor_transform) : $G_x(t_0,\xi) = \int_{-\infty}^\infty ...