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.

Integral transformations have been successfully used for almost two centuries in solving many problems in applied mathematics, mathematical physics, and engineering science. Historically, the origin of the integral transforms including the Laplace and Fourier transforms can be traced back to celebrated work of P. S. Laplace (1749–1827) on probability theory in the 1780s and to monumental treatise of Joseph Fourier (1768–1830) on La Théorie Analytique de la Chaleur published in 1822.

The integral transform of a function $~f(x)~$ defined in $~a ≤ x ≤ b~$ is denoted by $~\mathcal I \{f(x)\} = F(p)~$, and defined by $$~\mathcal I \{f(x)\} = F(p)~=\int_a^bf(x)~K(x,p)~dx$$where $~K(x,t)~$ is called the integral kernel of the transform. The operator $~\mathcal I~$ is usually called an integral transform operator or simply an integral transformation. The transform function $~F(p)~$ is often referred to as the image of the given object function $~f(x)~$ , and $~p~$ is called the transform variable.

Similarly, the integral transform of a function of several variables is defined by $$~\mathcal I \{f(x)\} = F(p)~=\int_Sf(x)~K(x,p)~dx$$where $~x=(x_1,~\cdots~,~x_n)~$,$~~p=(p_1,~\cdots~,p_n)~$, and $~S ⊂ \mathbb R^n~$.

A mathematical theory of transformations of this type can be developed by using the properties of Banach spaces. From a mathematical point of view, such a program would be of great interest, but it may not be useful for practical applications.

References:

https://en.wikipedia.org/wiki/Integral_transform

"Integral Transforms and Their Applications" by Lokenath Debnath, Dambaru Bhatta

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Integrate $\int\frac{x+1}{x^{4}+6x^{2}+4x}dx$

This gives a cubic polynomial in denominator and nonfactorisable It looks so simple but i just am not able to solve it
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2D Poisson equation and Bessel Functions

If I were to solve a inhomogeneous 2D Laplace's equation (in polar coordinates), of the form: $$\nabla^2 f= J_2(r)$$ How would I use the Hankel or the Mellin transform to find a solution? I couldn't find any references on the subject. Thanks!
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trouble with non-homogeneous ODE system... which method shall I use?

I am an undergrad statistics student and I am having troubles with non-homogeneous ODE systems. During my classes I went over just three methods for solving odes: Laplace transform, Fourier transform and power series; so I should use one of those to…
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Interchanging the order of integration.

Can someone explain why the following works please? $\int^t_0 f(s) \int^s_0 f(u) \,du\,ds = \int^t_0 f(s) \int^t_s f(u) \,du\,ds $ EDIT: All I know is that $f(x)$ is an integrable function. This is to do with non-homogeneous Poisson process and…
user1066113
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Graphing Fourier transforms on a frequency versus intensity plot (how to deal with complex numbers)

I am trying to understand how Fourier transforms are used to make HNMR plots. HNMR basically consists of hitting some molecules with some radiation and listening to the radio signal that results. This signal is then analyzed using the FFT and this…
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Quick Question on Zeroes of Transfer Function

Sorry for not providing context here. Suppose I have an output $Y(z)=\frac{z-1}{z}$ and input $X(z)=\frac{z^2+3z+2}{z^2}$ to yield a transfer…
JP McCarthy
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Why is this laplace identity true $\int_{\Bbb{R}^+}\frac{f(t)}{t}\,dt = \int_{\Bbb{R}^+}\mathcal{L}\{f\}$?

I was wondering why this laplace identity is true? Does it follow from definition? $$\int_{\Bbb{R}^+}\frac{f(t)}{t}\,dt = \int_{\Bbb{R}^+}\mathcal{L}\{f\}$$ I'm trying to understand the first solution here how do you do this integral from fourier…
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Transform an Integral bounds from -inf, inf to 0 to 1

Good day, If i have an integral from -infinity to infinity, how do I change the bounds/limits to 0 to 1? I don't want to give exact question since this is part of an assignment. I know how to figure out the integral but I am suppose to use Monte…
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2-Dimensional FOURIER TRANSFORM

How can I do to calculate the Inverse Fourier Transform of: $$G(w,y)=e^{-|w|y}$$ where w is real (w is the transform of x). I want to have $g(x,y)$, where $G$ is the Fourier Transform of $g$ Thanks
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Positivity of a Sine transform of a positive function

Consider a function $f(t)$ with $f(t>0)>0$ and $f(-t)=-f(t)$. Can I make any statement about the positivity of the Sine transform $$\hat{f}(\omega) = \int_{0}^{\infty} \sin(\omega t) f(t) \mathrm{d}t$$ of such a function? I guess that…
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Why is the Z-transform of $e^{at}$, t = kT, different from Laplace transform of $e^{at}$

The Laplace transform of $e^{at}$ takes a well known form of $\frac{1}{s-a}$ The Z transform of $e^{at} = e^{akT} $ T is the sampling period takes the form of $\frac{z}{z-e^{aT}}$ Does anyone know why the Z-transform takes a different form compared…
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Kernels of integral transform and linear transformation

Is there any relation between the $kernel$ of an $integral \ transform$ and the $kernel$ of a $linear \ transformation$?
Sidd
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Hilbert transform of a Gaussian wave packet

For the following function $f$, function from real number to real number, with $\mu$ real, $k$ real, $\sigma$ real strictly positif, defined by: \begin{equation} f(x)=cos(k x ) e^{\frac{-(x-\mu)^2}{2\sigma^2}} \end{equation} The Hilbert…
ucsky
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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}}{\sqrt a} & \text{if $|x|\le a$,} \\ 0 &\text{if…
Sherry
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What's wrong with this Laplace transform?

The following operators keep the area under the convergent integrals unchanged: $$\int_0^\infty f(x)\,dx=\int_0^\infty\mathcal{L}_t[t f(t)](x) \, dx=\int_0^\infty\frac1x\mathcal{L}^{-1}_t[ f(t)](x)\,dx$$ I conjectured that this should still work…
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