This covers all transformations of functions by integrals, including but not limited to the Radon, X-Ray, Hilbert, Mellin transforms. Use (wavelet-transform), (laplace-transform) for those respective transforms, and use (fourier-analysis) for questions about the Fourier and Fourier-sine/cosine ...

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2
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1answer
61 views

Dirichlet Series and Asymptotic Expansions

Consider the Dirichlet series $\tilde{f}(s)= \sum_{n=1}^{\infty} f(n) n^{-s}$. In the page "Zeta Function Regularization" of Wikipedia http://en.wikipedia.org/wiki/Zeta_function_regularization I ...
6
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0answers
89 views

Estimating the (double) Riesz transform.

I'm trying to verify the following estimate, which appears in a paper I'm reading. It seems I'm missing something easy, I just can't figure this out. $\textbf{Background}:$ For a function $f \in \...
0
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2answers
50 views

How to calculate the Fourier transform?

If the Fourier transform is defined by $\hat f( \xi)=\int_{-\infty}^{\infty}e^{-ix \xi}f(x)dx$. How to calculate the Fourier transform of $$\begin{equation*} f(x)= \begin{cases} \frac{e^{ibx}}{\...
2
votes
2answers
37 views

Inverse laplace transform excercise

I want to find the inverse transform of $$\frac{1}{(2s-1)^3}$$ I first applied a shifting theorem to get $$(e^t)\mathcal{L}^{-1}\left( \frac{1}{(2s)^3} \right)$$ I am just wondering is it possible ...
2
votes
2answers
90 views

Fourier Transform of Sine

I'm having trouble calculating the Fourier Transform of the sin function. Specifically, the function $ G(\omega)=\int _{-\infty}^{\infty} g(t)\ e^{-i \omega t} dt $ For the fourier transform of $ ...
2
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0answers
43 views

Can this divergent integral transform be regularized?

The integral $$\int_0^{\infty} e^u \ K_{i t}(u) du$$ is the adjoint Kontorovich-Lebedev transform of the increasing exponential function, but unfortunately this integral is divergent because $$e^u \ ...
9
votes
1answer
126 views

What are the “right” spaces for the Laplace transform

There are for example several canonical spaces to define the Fourier transform (i.e. Schwartz's space). Is there also a particularly suitable space to define the Laplace transform, so that the Laplace ...
2
votes
1answer
67 views

Express $f(x)=\sin{x}$ as an even function

Express $f(x)=\sin{(x)}$, with $(0 < x< \pi )$ as an even function, $f(x+ 2\pi)=f(x)$ The topic is on Fourier Series. I have the following so far: Since $f(x)$ must be an even function, we ...
3
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1answer
51 views

Proving that $(2 \pi i)^{-1} \int e(\pi v^2/y^2) x^v y^{-1} dv = e(-\pi (\log x)^2 y^2 /4)$

I've seen a particular integral transform (an inverse Mellin Transform) used a few times, but I don't know how it's proved. In particular, I'm trying to prove $$\frac{1}{2\pi i} \int_{(2)} e^{\pi ...
1
vote
1answer
60 views

Connection between $\nu=\frac{N}{2}-1^\text{th}$ order Hankel transforms and hyperspherically symmetric functions?

In The Transforms And Applications Handbook 2nd edition chapter 9 (Hankel Transforms), Piessens briefly mentions that the Fourier transform of an $N$-dimensional hyperspherically symmetric function ...
2
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1answer
113 views

What's the inverse of the Weierstrass-Mittag-Leffler-Transform $\exp\left[g(z) + \int_\mathbb C f(y)\ln(z-y)\,dy\right]$?

As mentioned in another post, as a consequence of Mittag-Leffler's theorem combined with the Weierstrass factorization theorem, after reducing to the common denominator, any meromorphic function can ...
0
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1answer
60 views

How to find the coefficients in the Fourier series solution of a 1-D heat equation?

I am trying to use Fourier's method to solve a problem. $u(x,t) = \sum \limits_{n=1}^\infty B_ne^{-(n\pi C / L)^2 t}\sin\left(\frac{n\pi x}{L}\right), B_n=\frac2L\int_0^L \sin\left(\frac{n\pi x}{L}\...
0
votes
1answer
211 views

PDEs for string deflection.

Okay, I have to find $u(x,t)$ for the string of length $L=\pi$ when $c^2=1$. I know: $$\text{wave equation}: \frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2}$$ $$u(x,0)=\frac ...
0
votes
1answer
329 views

How to solve this Laplace transform? $f(t)=e^{-2t}\cos^2 3t - 3t^2 e^{3t}$

Find the laplace transform of $$f(t)=e^{-2t}\cos^2 3t - 3t^2 e^{3t}$$ The answer is $$\frac{1}{2(s+2)}+ \frac{1}{2} \frac{s+2}{s^2 + 4s + 40} - \frac{6}{(s-3)^3}.$$ This took me about an hour to ...
2
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0answers
197 views

What is the inverse kernel to this integral transform

What is the associated inverse kernel to the integral transform $T$ defined by \begin{align*} (Tf)(u) & = \int_{-\infty}^{0} \hat{f}(s)\exp((2i\pi+c)us)\ ds + \int_{0}^{+\infty} \hat{f}(s)\exp((2i\...
2
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0answers
56 views

finding solution to a partial integro differential equation

I want to find a function (or a set of functions) such that $u(x,t)$ satisfies the following partial integro-differential equation with singular kernel \begin{eqnarray} &&u_x(0,t) = \int_0^t \...
3
votes
2answers
234 views

Kernel of an integral operator with Gaussian kernel function

Suppose we have the integral operator $T$ defined by $$Tf(y) = \int_{-\infty}^{\infty} e^{-\frac{x^2}{2}}f(xy)\,dx,$$ where $f$ is assumed to be continuous and of polynomial growth at most (just to ...
0
votes
2answers
33 views

Mellin Transform of $e^{-2 \pi t}$

I need to show that if $f(\tau)=e^{-2 \pi \tau}$ then: $$\{\mathcal{M}\,f\}(s)=(2 \pi)^{-s} \Gamma(s)$$ where : $$\Gamma(s)=\int_{0}^{+\infty} e^{-t}t^{s}\frac{dt}{t}$$ and $$\{\mathcal{M}\,f\}(...
4
votes
1answer
110 views

How to develop the Fourier Transform in my mind now that I know the Fourier Seires?

I know that we can represent some function $f$ in this way: $$f(t) = \frac{a_0}{2} + \sum_{n=1}^\infty a_n\cos\left(\frac{n\pi t}{L}\right) + \sum_{n=1}^\infty b_n\sin\left(\frac{n\pi t}{L}\right)$$ ...
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0answers
23 views

Sturm-Liouville equation with rational coefficient

I am trying to solve a regular Sturm-Liouville type Ordinary Differential Equation (ODE) with the following form: \begin{equation} \frac{d}{dy} [q(y) \frac{dW(y)}{dy}]+p(y) W(y) = -\lambda W(y) \end{...
2
votes
2answers
62 views

Laplace transform convolution attempt

I can't seem to get this Laplace working using the convolution method. $H(s) = \frac{1}{s^2(s+2)}$ Which I can't get to work using convolution. So I am separating it into $\frac{1}{s^2} * \frac{1}{s+...
0
votes
1answer
30 views

Laplace transform on a non-standard sort of problem

I don't know where a laplace comes into play here: $\ddot{a}+2a=0,a(0)=b_1,\dot{a}(0)=b_2$ I am meant to solve the above using a Laplace transform, but I don't see how I would use it here? I ...
1
vote
1answer
24 views

Inverse Laplace Transform Table, Absolution of Form

Do I need to ensure I don't stray from the transform in the table? $\frac{-2}{s-1}$ this looks like $-2*\frac{a}{s^2-a^2},$ for $a=1$ Does this yield $-2\sinh(t)$, or should it fit perfectly to ...
1
vote
1answer
51 views

Laplace Transform assistance

Find the inverse laplace transform of: $\frac{25}{(s-1)^2(s^2+4)}$ $\frac{25}{(s-1)^2(s^2+4)}=\frac{A}{s-1}+\frac{B}{(s-1)^2}+\frac{C}{s^2 + 4}$ $$25=A(s^2+4)(s-1)+B(s^2+4)+C(s-1)^2$$ $\frac{25}{(s-1)...
2
votes
1answer
128 views

Can integral transforms be viewed as change of basis formulas?

Forgive any lack of rigor. If you have a countable orthonormal basis $B$ for a Hilbert space $H$ , then any function $f \in H$ can be expressed as $$ f(t) = \sum\limits_{g \, \in \, B} \langle f, \,...
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0answers
63 views

Fourier transform and inverse transorm

I have to prove that for $f\in L^1(\mathbb{R})$ $$ \check{\hat{f}}=\hat{\check{f}}, $$ where $$ \hat{f}(\xi):=\int\limits_{\mathbb{R}}e^{-i\xi x}f(x)\mathrm{d}x $$ and $$\check{f}(\xi):=\frac{1}{2\pi}\...
2
votes
1answer
42 views

Fourier transform dependent upon a parameter and $L^2$ convergence

Suppose I know the Fourier transform of a function depending upon a parameter, call it $f_\epsilon(x)$, and that I want to know the Fourier transform of a function $f(x)$. Furthermore, suppose I know ...
1
vote
1answer
58 views

Integral transforms and scaling properties

It's well-known that the Fourier transform plays nicely with scaling. Particularly if we define, for $\alpha >0$, $D_{\alpha}$ by $D_{\alpha}f(x) = \alpha^{-1/2} f(x/\alpha)$, then (for suitable ...
3
votes
2answers
5k views

What is difference between Fourier Transform and Fast Fourier Transform?

If you think about Fourier Transform, in the classical cases, say on the real line, what it is? Just a waded sum. Right? You take a function $f$, and you take it's Fourier Transform at particular ...
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2answers
35 views

Fourier Transform-1

I am trying to solve a Fourier transform problem and I am stuck. The problem is: $$f(t)= \frac{\sin(2t)}{e^{|t|}}.$$ I have used integration, but the answer that I come up with is different than ...
4
votes
2answers
82 views

Computation of the fourier transformation of a function with a matrix

I want to compute the Fourier transformation of the following function: \begin{align} f:& \mathbb R^n \rightarrow \mathbb R \\ & x \mapsto \exp(-\left<Ax,x\right>) \end{align} where $A$...
6
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1answer
70 views

Integral transforms and uncertainty products

Heisenberg's uncertainty principle is well-studied and has become a bit of a pop science phenomenon due to its widespread implications in quantum mechanics. (Though interpretations are often ...
3
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1answer
206 views

Fourier transform of $te^{-t^2}$?

How can I find the Fourier transform of: $$f(t) = te^{-t^2}$$
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0answers
48 views

Solution to recursion relation using Mellin transform

I have been trying to solve the following recursive equation for $0<x_c<1$ for few a days: $$ P(x) = 2\mathbf{1}_{0\leq x\leq x_c} + 2\int_x^1 dy P\left(\frac{x}{y}\right)y^{-1} \mathbf{1}_{x_c&...
0
votes
2answers
68 views

Laplace transform of $L({1-e^{-t}\over t})$

I have to find the Laplace transform of $$\mathcal{L}\left[\dfrac{1-e^{-t}}t\right],$$ then this is equivalent to $$\mathcal{L}\left[\dfrac{1}t\right]-\mathcal{L}\left[\dfrac{e^{-t}}t\right]$$ But $\...
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0answers
57 views

how to solve this inverse mellin transform

We know from the Perron's formula that $$ \int _{c-i\infty}^{c+i\infty}\frac{\zeta(s)}{2\pi i s}x^{s}ds=[x] $$ $$ \int _{c-i\infty}^{c+i\infty}\frac{\zeta(s)^{2}}{2\pi i s}x^{s}ds=\sum_{n\le x}d(n) $...
21
votes
1answer
230 views

Exponential integral $ \int_0^\infty \frac{x^t}{\Gamma(t+1)}\text dt$

Now since the sum $$ \sum_{n=0}^\infty \frac{x^n}{n!},\quad x\in\Bbb R, $$ does have some relatively nice properties, is the same true for its analogues integral? If we take the gamma function to be a ...
3
votes
1answer
278 views

Contradiction in inverse Laplace transform problem with Mellin's inverse formula?

Let say we have to solve a given differential equation $$ty''+y'+ty=0$$ $$y(0)=1,\ y'(0)=0$$ (which is Bessel equation with the solution $y=J_0 (t)$, of course) with the Laplace transform. Then we ...
3
votes
1answer
77 views

Inverting the integral $f(x)=\int_x^a \frac{g(t)}{\sqrt{t-x}}dt$

I am curious if there is a way to invert the integral $$f(x)=\int_x^a \frac{g(t)}{\sqrt{t-x}}dt$$ to solve for g(x) when f(x) is a known function. The integral from x to a makes this problem seem a ...
0
votes
2answers
68 views

How do I take inverse Laplace transform of $\frac{-2s+3}{s^2-2s+2}$?

How do I take inverse Laplace transform of $\frac{-2s+3}{s^2-2s+2}$? I have checked my transform table and there is not a suitable case for this expression.
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0answers
33 views

Splitting Mellin Transforms

Let's say we have two functions, $f(x)$ and $g(x)$, such that the Mellin transform of $f(x)$ converges on the strip $a < x < b$ and the Mellin transform of $g(x)$ coverges on the strip $c < ...
3
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0answers
31 views

Why do you need an integral to invert the Z-Transform?

With integral transforms both the transform and its inverse are integrals. In the case of the Z-Transform the transform is a sum. My question Why do you need an integral (instead of another sum) to ...
1
vote
1answer
106 views

Example for finite dimensional analog of integral transforms

I understand that integral transforms are generalisations of the dot product of functions that could be interpreted as infinite dimensional vectors. The most significant advantage then is that ...
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1answer
201 views

Fourier-Laplace Transform of Heaviside Step function multiplied to Sine

In a Advanced Solid State lecture I encountered the following assertion- Fourier Transform of $\Theta(t)\sin(\omega_0 t)$ is $\frac{1}{\omega+\omega_0}-\frac{1}{\omega-\omega_0}+i\pi\delta(\omega+\...
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0answers
56 views

Clarification on the absolute convergence of Mellin transform

I have a question I haven't been able to find a direct answer to that I presume is true but I am unable to show. We know these two following results on the mellin transform. If $$\int_0^\infty |f(x)|...
7
votes
2answers
137 views

Mellin Transform of the floor function

I have from integral transform tables that: $$\operatorname{Mellin}(\lfloor x\rfloor) = -\dfrac{\zeta(-z)}{z},\quad \operatorname{Re}(z)<-1$$ How can this be proved?
3
votes
1answer
201 views

Bessel function integral

How to solve the integral for $J_1{(2x\sin{\frac{\theta}{2}})}$ at $[0,\pi]$? If solving by Matlab, please provide me the source. Thank you!
3
votes
1answer
63 views

“Reduction of Dirichlet series into power series”

In a paper of Riemann, he states to following formal identity. If $f(s)=\sum\limits_{k=1}^{\infty}\frac{a_k}{k^s}$ and $F(x)=\sum\limits_{k=1}^{\infty}a_kx^k$ then $$\Gamma(s)f(s)=\int\limits_{0}^{\...
3
votes
2answers
594 views

Delta function with imaginary argument

We have an integral representation for the Dirac delta function as $\delta(x) = \frac{1}{2\pi} \int_{-\infty}^\infty dk e^{ikx}$ . On the other hand, we have for delta function the property: $\delta(...
0
votes
1answer
429 views

Compatibility Condition of the Poisson Equation with Neumann Boundary Conditions

I am trying to solve the following general Poisson equation with homogeneous Neumann boundary conditions in a rectangular domain ($0 \le x \le L$ and $0 \le y \le H$). $$ \frac{\partial^2 p(x,y)}{\...