# Tagged Questions

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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### Inverting a Particular Integral Operator

Consider trying to find a function $f \in L^2(0,1)$ satisfying $$a_n = \int_0^1 f(x)x^n dx$$ Where $n$ is a nonnegative integer. Is there any method to go about doing this in general for any ...
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### Definite integral of trigonometric functions with complicated arguments

I came across beautiful integral (where $n$ is integer) $I(n, z) = \int_0^{\pi} \cos(nx) \sin(z \cos(x) ) \mathrm{d}x$ According to Gradshteyn and Ryzhik (p 414, Sec. 3.715, Eq. 13), solution is ...
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### Find the function whose Fourier sine transform is $\frac{s}{1+s^2}$? (Please check my solution)

There needs to a $\sqrt{\frac{2}{\pi}}$ before that integral, which I have left by mistake. I have used this result : $$\int_{0}^{\infty}\frac{\sin{a x}}{x}dx = \frac{\pi}{2}$$ where $a > 0$. My ...
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### A problem about Laplace transform and Parseval–Plancherel theorem

I am reading a paper about fractional differential equation. One of the piece said as follow: By applying the Parseval–Plancherel theorem we may show: \int_0^\infty v(t)B_\...
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### Intuition behind chosing coordinates

Once given the right coordinates (polar, sphere, cylindric) I am able to determine the value of a given integral. But how do I know, if the coordinates are not explicitly given, which coordinates to ...
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### Abel transformation - sources

I'd like to study the Abel transformation, that is, $$Af(x) = \int\limits_x^\infty\frac{f(t)t}{\sqrt{t^2 - x^2}}\ \mathrm{dt},\quad x\in(0,\infty).$$ I'm especially interested in estimates/...
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### Transforming a PDE to an ODE by discretizing the size-axis of the integro-partial differential equations

I have this issue on modelling a process. It is a system of ODEs except for one equation related to the population balance of crystals in a mixture that is a PDE. The author of the publication ...
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### Differentiation under integral sign?

I am trying to understand the following argument given in a text book: Suppose $f \in L^1(\mathbb R^n)$, consider the function $\hat{f}(\zeta)= \int_{\mathbb R^n} \exp(-2\pi i X.\zeta)f(X)dX$. ...
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### Fourier transform of integral with isotropic kernel

The textbook I'm reading claims that this integral: $$A = \int_V \,d\mathbf{r} \int_V\,d\mathbf{r}' f(\mathbf{r}) K (| \mathbf{r} - \mathbf{r}'| ) f(\mathbf{r}')$$ can be written in Fourier ...
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### Riesz transform does not preserve continuity

I've read somewhere that the Riesz operator $R_j$ defined by $$R_j f(t) := c(n) \, \text{pv} \int_{\mathbb{R}^n} \frac{x_j}{|x|^{n+1}} f(t-x) \, dx$$doesn't preserve the continuity, but I can't ...
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### Complex inversion of a function

I am trying to find the function whose laplace transform is below using the complex inversion formula: $$f(s)= \frac{se^s}{(s-2)^3}$$ My attempt below seems to be giving me the wrong answer but I'm ...
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### Transform and domain such that phase shift of the original function results in shift in transformed domain variable

It is known that the Fourier transform of a phase-shifted function results in a constant shift of the dependent variable of the phase spectrum: If $F(x(t)) = X(w) = |A(w)| \cdot e^{-i \cdot \phi(w)}$...
Let $\varphi \in L_1(c+i\mathbb R)$, where $c > 0$. Then we can define the Mellin-Barnes transform (or the inverse Mellin transform) of the function $\varphi$ by the formula $$\mathcal M_c^{-1}... 0answers 26 views ### Why M.G.F transform is injective a.s.? We always use the theorem that If we know a random variable's MGF, we can determine its Pdf, which means the map from Pdf to Mgf is injective almost surely. And I just wanna know why this is ture. 2answers 73 views ### Inverse Laplace Transform of e^{\frac{1}{s}-s} doing some work on a PDE system I have stumbled across a Laplace transform which I'm not sure how to invert:$$ F(s) = e^{\frac{1}{s}-s} $$I can't find it in any table and the strong singular growth ... 3answers 219 views ### How to learn Integral Transform? I major in Electronic Engineering when back in college. I learned the Fourier Transform, Laplace Transform, Z Transform and wavelet Transform. But I always feel a lack of thorough understanding of ... 0answers 35 views ### Solving differential equation by transform with contour integral In my notes, I have an example for solving for the Airy function in the equation:$$\frac{d^2y}{dx^2}-xy=0 So he uses the contour integral to represent the solution (with $C$ as a yet unspecified ...
I have a Laplace tranform in the form given below $\mathcal{L}_I(s)=\text{exp}(-\pi\lambda \Gamma(1+\frac{2}{\alpha})\Gamma(1-\frac{2}{\alpha})P^{2/\alpha}s^{2/\alpha})$ Can some one help me to find ...