# Questions tagged [integral-transforms]

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 transforms.

528 questions
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### Connection between Fourier transform and Taylor series

Both Fourier transform and Taylor series are means to represent functions in a different form. What is the connection between these two? Is there a way to get from one to the other (and back again)? ...
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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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### 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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### Laplace transformations for dummies

Is there a simple explanation of what the Laplace transformations do exactly and how they work? Reading my math book has left me in a foggy haze of proofs that I don't completely understand. I'm ...
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### Usage of inverse Laplace transform

At my current study level in college, use of inverse Laplace transform is not mentioned well - textbooks say "use tables." So, can anyone show me how to use inverse Lapalce transform? And also proof? ...
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### Where to submit a small work of applied math written by a physicist?

Working in physics, I recently discovered a mathematical identity useful to solve a particular partial differential equation. Using the same idea, I found several other identities but I do not know ...
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### Integrate $\int\frac{x^{6}-2x^{3}}{\left(x^{3}+1\right)^{3}}dx$ .

$$\int\frac{x^{6}-2x^{3}}{\left(x^{3}+1\right)^{3}}dx$$ I tried adding and subtracting 1 to bring a square expression with numerator as $(x^3-1)^2 -1$ but always going to partial fraction which ...
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### 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?
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### Convolution square root of $\delta$

I want to somehow classify the distributional solutions of the equation $$f \ast f = \delta$$ where $\delta = \delta _0$ is the Dirac delta distribution. Clearly, by Fourier transformation, we ...
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### 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 ...
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### An alternating series identity with a hidden hyperbolic tangent

How does one use the inverse Mellin transform to prove that the following identity holds? $$\sum_{n=1}^{\infty}\frac{(-1)^n}{n(e^{n\pi} + 1)} = \frac{1}{8}(\pi - 5\log(2))$$ The identity follows ...
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### Fourier transform using principal value

Can anyone help me compute the Fourier transform of $1/|x|^{n-\alpha}$ in $\mathbb{R}^n$ where $0 < \alpha < n$ ? Somehow it becomes the principal value of $1/|x|^\alpha$ which I can't ...
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### Meaning of polynomially bounded

I am currently reading through some lecture notes on Fourier Transform and Distributions on my own, and came upon the notion of a polynomially bounded function. I am not sure I understand this concept,...
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### Stieltjes Transform

Stieltjes transform for a distribution $F(x)$ is defined as $$m(z)=\int \frac{dF(x)}{x−z}$$ where z is complex with positive imaginary parts and $F(x)$ is a distribution function. Basically, I am ...