# 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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### To evaluate $\int_0^{+\infty} \frac{\;\mathrm dx}{\sqrt[3]{x^3+a^3}\sqrt[3]{x^3+b^3}\sqrt[3]{x^3+c^3}}$

$$f(a,b)=\int_0^{+\infty} \frac{\;\mathrm dx}{\sqrt{x^2+a^2}\sqrt{x^2+b^2}}$$ To use Landen's transformation $$f(a,b)=\int_0^{+\infty} \frac{\;\mathrm dx}{\sqrt{x^2+(\frac{a+b}{2})^2}\sqrt{x^2+ab}}$$...
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### Fourier, Laplace, … and other Integral-transformations

I know Laplace, Fourier and Mellin-Transformation. Is there a general theory of transformations? My main interest is about classification of transformations satisfying specified properties like ...
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### Integral transform Laguerre function

Given the following integral transform $$g(m)= \int_{0}^{\infty}dxe^{-x}f(x)L_{m}(x),$$ then how could we obtain $f(x)$ from $g(m)$ ?? I have thought that for a continuum '$m$' like in our ...
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### What should I call integrals of the form $\int_{-a}^{b} f(t) e^{zt}dt$?

I've been studying integrals of the form $$\int_{-a}^{b} f(t)e^{zt}dt$$ where $z$ is a complex variable. So far I've been calling them "exponential integrals" following the practice of P. Miller ...
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### Integral transform of gaussian function

I am looking for an integral transform of $f(x)=\exp(-\frac{x^2}{2\sigma^2})$ such that: $$(Tf)(u)=\int_{x_1}^{x_2}\, K(x,u)f(x)\,dx\,\stackrel{??}{=}\,\frac{a}{g(u)}$$ with $g(u)$ having zeros ...
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### Laplace, Legendre, Fourier, Hankel, Mellin, Hilbert, Borel, Z…: unified treatment of transforms?

I understand "transform methods" as recipes, but beyond this they are a big mystery to me. There are two aspects of them I find bewildering. One is the sheer number of them. Is there a unified ...
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### transform that is invariant under rotation

We know that the magnitude of the Fourier transform (resp. Mellin transform) of a shifted (resp. scaled) function is identical to the magnitude of the original function. I wonder if there is a ...
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### Uniqueness for Integral Transform

What can be said about the uniqueness of the following integral transformation: $(Tf)(u) = \int_0^{\infty} f(t)G(tu)dt$ defined for all $u\geq 0$, where the kernel $G(z) \in [0,1]$ for all $z\geq0$,...
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### Fourier transform of $\text{sinc}^3 {\pi t}$

$$f(t)=\frac{\sin^3(\pi t)}{(\pi t)^3}$$ I want to calculate the Fourier transform. I can't calculate this integral: $$\int_0^\infty\frac{\sin^3(\pi t)}{(\pi t)^3}\cos(ut)\,\mathrm{d}t$$
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### On using fourier transforms to solve the root of a convolution

In continuation of Lower bounds of laplace transform of characteristic functions. My question is: Can anyone point out where i'm going wrong in the derivation below. It's been a while ...
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### proving a z transform

I am having trouble demonstrating the Z transform of $a^{n-1}u(n-1)$ is $\frac{1}{(z-a)}$, as it says in this table. I try using the definition of the z transform, but it comes out different than ...
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### What's the connection between Banach/Hilbert spaces and tools like power series, Fourier transforms etc.

I've learned, abstractly, about Banach and Hilbert spaces, and more concretely about $l^p$ and $L^p$ spaces. I also understand these ideas have something to do with a variety of tools that are useful ...
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### fancy about some properties of kernel functions at infinity

Consider the two common types of kernel functions $\sum\limits_{t=a}^bf(t)K(x,t)$ and $\int_a^bf(t)K(x,t)~dt$ , prove whether the following properties are correct or not: $1.$ If $K(x,t)$ is bounded ...
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### Existence of zeros of Mellin transform and properties of function to be transformed

Mellin transform of function $f(x)$ defined for $x\geqslant 0$ is given by $$f^\ast(z) =\int\limits_0^\infty x^{z} f(x) \frac{dx}{x}.$$ I consider only exponentially decreasing (there exist such ...
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### Fourier transform of $g(x)=x\frac{\partial f}{\partial x}$?

I have a problem with the Fourier transform of the function $g(x)=x\frac{\partial f}{\partial x}$. I need the transform to be itself a function of the Fourier transform of $f(x)$ and I don't know how ...
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### Continuity of the inverse Laplace Transform

If I know $Y(s)$, can I predict when $\mathscr{L}^{-1}[Y(s)]=y(t)$ will be continuous or continuously differentiable or even stronger conditions? For example; I'm solving an ODE with the Laplace ...
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### Fourier Transform of a function under an arbitrary coordinate transform [duplicate]

Consider a function $f(x)$ and its Fourier Transform $\tilde{f}(k)$ given by $$\tilde{f}(k) = \int_\mathbb{R}\!\!\!dx\; e^{-ikx}f(x).$$ Now, lets have the coordinate transform $\xi = \tau(x)$ and, ...
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### Dirac delta questions form Mellin transform

We know that $$f(s)= \int_{-\infty}^{\infty}f(x)\delta (x-s) d x$$ however, is there a similar delta function so for the Mellin transform $$f(s)=\int_{0}^{\infty}f(x)m(xs) d x$$ ? That is a ...
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