Tagged Questions
2
votes
2answers
57 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:
$$I = \int_{ - \infty }^{ + \infty } {\frac{{\cos (ax)}}{{{x^2} + 1}}{e^{ - ibx}}\,\mathrm dx} $$
1
vote
1answer
50 views
Fourier transform of $\frac{1}{1+x^2}$
we know that the Fourier transform of $\frac{1}{1+x^2}$ is $f(y) = \int_{-\infty}^{\infty} \frac{1}{1+x^2} e^{-2\pi i x y} dx $ . Here is the idea used in my textbook, for y<0 :
We calculate the ...
1
vote
1answer
38 views
Calculating Inverse Fourier Transform
I can't quite get an inverse Fourier Transform to match up with a statement in my textbook. At one point, my textbook writes:
"If $g$ is a function that is one on the interval $(- \pi, \pi]$ and ...
3
votes
1answer
56 views
Highly Oscillating Integrals
I'd like to know the behavior of integrals of the form:
$$
\int_0^1 f(x) \cos(k x) dx
$$
as $ k \rightarrow \infty $ where f is a smooth function. It is easy to see, by expanding f in power series, ...
0
votes
1answer
68 views
Why aren't these two question equal?
Firstly I doubt whether the 12 is right in Q1.If it is right,please give a proof.
Secondly why (1) is not equal to (2) in Q2?
0
votes
1answer
65 views
I can't understand the last second step.
It is known that $f(x)=\sum_{n=-\infty}^{\infty}c_{n}e^{inx}$, with $c_{n}:=\frac{1}{2\pi}\int_{-\pi}^{\pi}f(x)e^{-inx}\:dx$, for $n\in\mathbb{Z}$.
To prove: ...
1
vote
0answers
26 views
Complex Fourier series of a function [duplicate]
I need to find the complex Fourier series of this function, and I'm having problems calculating these integers:
$$|a|<1$$
$$x\in [-\pi,\pi]$$
$$f(x)=\frac{1-a\cos(x)}{1-2a\cos(x)+a^2}$$
...
2
votes
2answers
183 views
Complex Fourier series
I need to find the complex Fourier series of this function, and I'm having problems calculating these integers:
$$|a|<1$$
$$x\in [-\pi,\pi]$$
$$f(x)=\frac{1-a\cos(x)}{1-2a\cos(x)+a^2}$$
...
6
votes
2answers
97 views
Arnold's Trivium problem 51
Calculate $$ f(k) = \int_{-\infty}^{+\infty} e^{ikx}\frac{1 - e^x}{1+e^x}dx.$$
As far as I know, this is not a function but rather the Fourier transform in tempered distributions.
1) What is ...
0
votes
0answers
25 views
Fourier transform of given characteristic function
I have a function $$g(l) = E [ e^{iuX}|X>l ] - Prob (X>l) $$ and i need to derive how its Fourier transform is: $$F_{l,v}(g(l)) = \frac{\phi_X(u+v)-\phi_X(v)}{iv}$$.
This gets down to ...
2
votes
1answer
42 views
How to recover a measure from its Fourier transform?
Let $f$ be the complex function defined on $\mathbb{R}$ by
$$
f(t)=\frac{1-it}{1+it}.
$$
1) Does there exist a complex bounded measure $\mu \in M(\mathbb{R})$ such that $\hat{\mu}=f$ (where $\hat{}$ ...
0
votes
1answer
31 views
Fourier transform of $\mathrm{rec}(x) =\begin{cases} 1 & \text{if }|x| < 0.5,\\ 0.5& \text{if }|x| = 0.5,\\ 0& \text{if }|x| > 0.5 \end{cases}$
$$\mathrm{rec}(x) =\begin{cases}
1 & \text{if }|x| < 0.5,\\
0.5& \text{if }|x| = 0.5,\\
0& \text{if }|x| > 0.5
\end{cases}$$
The Fourier transform of this function is ...
5
votes
0answers
275 views
Show that the function is constant
Let $S^n$ be an $n$-dimentional unit sphere.
Consider $f: S^n \longrightarrow R_+$ even continuous function.
Denote
$$
...
0
votes
2answers
249 views
1
vote
1answer
34 views
Why does $\sin{\alpha}\cdot i\sin{\alpha x}$ disappear from this integral?
In a section on fourier transforms, my textbook contains these steps for an example:
$$f(x) = \int_{-\infty}^\infty \frac{\sin{\alpha}}{\pi \alpha}e^{i\alpha x}d\alpha$$
$$= ...
3
votes
3answers
168 views
Two improper log integrals
Evaluate
$$\int_0^{\frac{\pi}{2}}\ln ^2(\tan x)\text{d}x$$
$$\int_0^{\frac{\pi}{2}}\ln ^2(\sin x)\text{d}x$$
1
vote
1answer
58 views
Do we have a general form for this integral?
Is there a general formula or recursion for this integral?
$$\int_0^1\left(\frac{\arcsin x}{x}\right)^n\text{d}x,\ \ n\in\mathbb{N}$$
1
vote
3answers
122 views
A integral with polygamma
I was doing a integral, the last part is
$$\int_0^{\frac{\pi}{2}}x^3\csc x\text{d}x$$
I ran this on Maple, it turns into polygammas...How we evaluate this?
I think there should be a way to evaluate ...
2
votes
2answers
124 views
Another integral with Catalan
Show that:
$$\int_0^1\frac{\arcsin^3 x}{x^2}\text{d}x=6\pi G-\frac{\pi^3}{8}-\frac{21}{2}\zeta(3)$$
I evaluated this by some Fourier series. Is there any other method?
Start with substitution of ...
1
vote
1answer
41 views
Need help with a integral
I was evaluating
$$\int_0^{\frac{\pi}{2}}x\ln \cos x \, \text{d}x$$
I like to try with the fourier series
$$\int_0^{\frac{\pi}{2}}\left(\sum_{k=1}^\infty\frac{(-1)^{k-1}\cos (2kx)}{k}x-x\ln 2\right) ...
3
votes
1answer
118 views
Fourier's method for PDE
We have $$U_{tt}=U_{xx} \quad 0<x<{\pi \over 2}, \quad t>0$$
$$U(x,0)=U(0,t)=U_x({\pi \over 2},t)=0$$
$$U_t(x,0)=cos(5x)\cdot sin(x)$$
We are looking for a solution in the form ...
8
votes
3answers
332 views
A log improper integral
Evaluate :
$$\int_0^{\frac{\pi}{2}}\ln ^2\left(\cos ^2x\right)\text{d}x$$
I found it can be simplified to
$$\int_0^{\frac{\pi}{2}}4\ln ^2\left(\cos x\right)\text{d}x$$
I found the exact value in the ...
2
votes
1answer
93 views
More on the generalized integral
Refer to my previous topic: A generalized integral need help
I think we get this :
$$\frac{\sin \theta}{1-2\cos \theta x+x^2}=\sum_{k=1}^{\infty}\sin (k\theta )x^{k-1}$$
Then $$\int_0^1\frac{\ln ...
2
votes
0answers
92 views
Do we have closed form for these series?
Continuous on the previous question, can we get a closed form for these?
$$\begin{align}
& \sum\limits_{n=1}^{\infty }{\frac{n+3}{{{n}^{3}}+\ln n}} \\
& \sum\limits_{n=1}^{\infty }{\left( ...
2
votes
1answer
430 views
3D Fourier Transform
I'm trying to calculate the inverse of the following 3D Fourier transform.
$$
\widetilde{f}= \frac{1}{(k^6-\alpha*k^2-\alpha*k_3^2)}
$$
where $k = (k_1^2+k_2^2+k_3^2)^{1/2}$
the fourier transform is ...
4
votes
0answers
149 views
How to find the inverse Fourier Transform of the product of two bessel functions of the first kind and a complex exponential function?
I am attempting to find a closed form or symbolic expression of the inverse Fourier transform of the product of two Bessel functions of the first kind and a complex exponential, e.g.
$P(t) = IFT_w \{ ...
6
votes
2answers
240 views
log sin and log cos integral, maybe relate to fourier series
I try to use the method of differentiation under integral sign for the first one
And integrate it back, but I failed to find the constant $c$ ....
Anyone hav other method?
$$\begin{align}
& ...
2
votes
0answers
115 views
Is there a way to Fourier transform Cos[Sin[x]]
In my physics problem, I encountered a solution has the form like Cos[Sin[t]],
and I need to do the Fourier transform to this solution. Is there a way to do the Fourier transform analytically to ...
2
votes
1answer
181 views
Heisenberg uncertainty principle in $d$ dimensions.
Suppose $f(x)$ is a $d$-dimensional real function and $\int_{R^{d}}|f(x)|^2dx=1$. Show that
$$ (\int_{R^{d}}|x|^2|f(x)|^2dx)(\int_{R^{d}}|\xi|^2|\hat f(\xi)|^2d\xi)\geq\frac{d^2}{16\pi^2}$$
I ...
0
votes
1answer
67 views
Calculate $\int_{R^{3}}e^{-2\pi ix \cdot \xi}\left(\frac{1}{4\pi}\int_{S^2}f(x-\gamma t)d\sigma(\gamma)\right)dx$
I should derive
$$\int_{R^{3}}e^{-2\pi ix \cdot \xi}\left(\frac{1}{4\pi}\int_{S^2}f(x-\gamma t)d\sigma(\gamma)\right)dx=\hat{f}(\xi)\frac{\sin(2\pi |\xi| t)}{2\pi |\xi|t}$$
I already calculate ...
7
votes
2answers
252 views
Fourier transform of fourier transform?
I have the definition of Fourier transform $$\hat f(\lambda) = \int_{\infty}^\infty f(t) \exp(- i \lambda t) dt$$ and have proved the following lemmas:
$\hat E(x) = \sqrt{2 \pi} E(x)$ where $E(x) = ...
6
votes
2answers
396 views
Recursive Integration over Piecewise Polynomials: Closed form?
Is there a closed form to the following recursive integration?
$$
f_0(x) =
\begin{cases}
1/2 & |x|<1 \\
0 & |x|\geq1
\end{cases}
\\
f_n(x) = ...
1
vote
2answers
47 views
Help solving $\frac{1}{{2\pi}}\int_{-\infty}^{+\infty}{{e^{-{{\left({\frac{t}{2}} \right)}^2}}}{e^{-i\omega t}}dt}$
I need help with what seems like a pretty simple integral for a Fourier Transformation. I need to transform $\psi \left( {0,t} \right) = {\exp^{ - {{\left( {\frac{t}{2}} \right)}^2}}}$ into ...
0
votes
1answer
66 views
Fourier Series with Signals
So the question is:
Determine the fourier series representations for the following signal:
Here the formula for the fourier series
$$C_k=\frac{1}{T}\int_T \! x(t)e^\frac{-j2\pi kt}{T} \, \mathrm{d} ...
2
votes
1answer
72 views
Fourier transform of characteristic function in a sphere
A similar question was asked before for an interval in $\mathbb{R}$. I wonder how to do it for a characteristic function of $\{x\in\mathbb{R}^3:|x|<r\}$ i.e. I want to calculate $$ ...
0
votes
0answers
47 views
Fourier Series for Signals
So my question requires the picture of a graph so here it is. I'm trying to do part $(a)$ and I have worked out all the way up to this part:
$C_k=\frac{1}{2} \int_T \! te^\frac{-j2\pi kt}{T} \, ...
0
votes
1answer
40 views
how to prove the convolution formular?
let $\overset{\backsim} {g}(x)=g(-x)$;
suppose $u,\phi,\psi$ always make the integral significant,$E_n$ is the n-dimensional euclidean space. Then how to prove
...
2
votes
1answer
109 views
Path integrals using Fourier transformation
While going through a book named Mirror Symmetry, I came across a path integral,
$$Z(\beta) = \int\limits_{X(t+\beta) = X(t)} DX(t) \exp\left(-\int\frac{1}{2}( \dot{X}^2 + X^2)dt\right)dt $$
where ...
1
vote
1answer
102 views
is there a nice way to find the fourier transform of…
I am looking for a nice way to calculate the FT of the following function
$f(x)=\biggl(\sum_{n=1}^{c}~a_n~e^{-\frac{i}{2}~x~b_n}\biggr)^d$,
where $d,c>0$, $a_n$ and $b_n$ are real coefficients, ...
3
votes
1answer
145 views
Bounds on integral
I am calculating Fourier coefficients for certain functions and have come across an integral of the form
$$I=\int_0^{2\pi} \int_0^1 r^2e^{2\pi i r(m\cos\theta+n\sin\theta)}drd\theta,$$
where ...
2
votes
2answers
206 views
How to find this moment generating function
I am trying to find the moment generating function of a random variable $X$, which has probability density function given by
$$f_{X}\left( x\right) =\dfrac {\lambda ^{2}x} {e^{\lambda x}}$$
Where ...
2
votes
1answer
38 views
equivalence in Fourier space
I have a comprehension problem regarding Fourier transforms. So far I know, the Fourier transform can be defined on the whole Schwartz space $\mathcal{S}(\mathbb{R})$ and is bijective on it. So I have ...
7
votes
4answers
326 views
Computing the Gaussian integral with Fourier methods?
There are many proofs that
$$\int_{-\infty}^\infty e^{-x^2} \, \mathrm dx = \sqrt{\pi}.$$
For example, using a change to polar coordinates, differentiation under the integral sign, and the theory ...
1
vote
1answer
116 views
Boundedness of supremum of an Integral operator
I am trying to find an $L_2$ - bound on a certain class of operators, and on my way I produced an estimate for which I need to show that
\begin{equation}
\sup_{x \in \mathbb{R}^n} \, ...
2
votes
1answer
116 views
the integral of the inverse of a Fourier series
Let $\{a_h\}$ be a double-sided complex sequence such that $\sum_{h=-\infty}^{\infty} |a_i| <\infty$ with $a_{0}\neq0$.
Set $f(x) := \sum_{h=-\infty}^{\infty} a_h \exp(ixh)$ and
assume that ...
1
vote
0answers
109 views
Do Fourier transforms of $\min$ and $\max$ exist (in closed form)?
I am wondering if there are Fourier transforms of $\min(x,a)$ and $\max(x,a)$ functions. Please forgive me if this is a dumb question, I don't normally use Fourier transforms.
I attempted to simply ...
2
votes
1answer
150 views
Fourier transform of function involving $\log$
I found the following problem which I am unable to solve.
Calculate the following integral
$$\int_{\mathbb{R}} \frac{d\omega}{2\pi} \log (1 + i a/\omega ) e^{-i \omega t}$$
for $a>0$ and ...
1
vote
0answers
83 views
Dealing with integrals and Fourier transforms.
I have the following expression: $$\sum_{k}\left(\int_{-\infty}^{\infty}e^{-ikx}\, f(k')dk'-\int ...
2
votes
1answer
330 views
Fourier transform in Mathematica
When I calculate the Fourier transform of the function $$f(t) = \mathrm e^{-|t|/\tau} \text{ with } \tau >0$$ in Mathematica once via the function FourierTransform and once by hand, I get different ...
3
votes
2answers
113 views
Solving this Fourier transform?
Is there any way to compute in closed form (in terms of known functions) the Fourier integral
$$ \int_{-\infty}^{\infty} \frac{\cos(ux)}{(x^{2}+a^{2})^{s}} dx$$
where $u$ and $a$ are real positive ...
