# 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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### Time Settings of the $z$ and Laplace Transforms.

I'm aware that the $z$-transform and the Laplace Transform have an analogous relationship but I want to be doubly-sure that the $z$-transform only works in discrete-time and that the Laplace transform ...
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### Mellin transform of $e^{iat}$

When doing the change of variables $v=-iat$, shouldn't the limits be reversed? Or is it because its the same as $v=\frac{at}{i}$ I cant see why this is not the case
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### Laplace transform $\mathcal{L}[t^\alpha f(t)]$

I am interested in the Laplace transform $\mathcal{L}[t^\alpha f(t)]$ where $f(t)$ is an arbitrary function and $\alpha$ is non integer. I know that for $\alpha=n$ integer, this is the n-th ...
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### Inverse integral transform of $\cos(t-u)$

I have the following integral transform $$f(u) = \int_0^{2\pi} g(t)\cos(u-t)\,dt$$ where I know what $f(u)$ is (I have raw data rather than an analytical form) and I need to reconstruct $g(t)$. ...
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### Relationship Between The Z-Transform And The Laplace Transform

Below I've quoted Wikipedia's entry that relates the Z-Transform to the Laplace Transform. The part I don't understand is $z \ \stackrel{\mathrm{def}}{=}\ e^{s T}$; I thought $z$ was actually an ...
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### Fourier Transform for triangular wave

Could someone tell me if I've worked this out right? I'm unsure of the process, especially the final parts where I convert it to a sinc function. Please let me know if I've made mistakes anywhere ...
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### Fourier Transform Help Needed

I need help with a Fourier Transform problem for a composite waveform for an assignment. I'm stumped with how to approach this one. The only way I could think of to solve this was by considering it ...
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### Need help with a Fourier Transform Question

I need an way to solve this Fourier transform problem. $$f(t)= \begin{cases} \cosh(t) & \text{ For } |t|<1\\ 0 & \text{ For }|t|>1 \end{cases}$$ The given answer for the ...
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### Proof of inverse Laplace transform

Why is $$f(t) = \frac{1}{2πj}\int_{\sigma-j\infty}^{\sigma+j\infty} F(s) e^{st} \, ds,$$ provided that $$F(s) = \int_{0}^{\infty} f(t) e^{-st} \, dt \ ?$$ I tried to find out myself, or searched ...
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### Proof that $\frac{e^{st}}{2\pi i}$ is an orthogonal basis.

I was studying the Linear Algebra perspective about the Laplace Transform. We know that the Laplace Transform is given by: $$F(s) = \int_{0}^{\infty}f(t)e^{-st}dt$$ Where $e^{-st}$ is the integral ...
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### Hilbert Transform: limit of xHf(x)

In Terence Tao's notes page 1, cited below, he mentions that it is easy to see that $\lim_{|x| \to \infty} xHf(x) = \frac{1}{\pi}\int f$ where $f$ is a Schwartz function and $H$ is the Hilbert ...
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### Laplace Transform of an integral function of a convolution

Making suitable assumptions wherever necessary, what is the Laplace Transform $\mathcal{L}(S(t))$ where $S(t)=\int_{0}^{t}\int_{0}^{t}f(t-s_1,t-s_2)g(s_1)h(s_2)ds_1ds_2$. I tried using the Double ...
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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_\alpha ...
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### How to disprove an equality involving a double integral

I want to show that the following equality does not hold: \label{at3} \frac{\lambda^2-1}{2}x^2-\int_{-\infty}^{\infty}\!\!\int_{-\infty}^{\infty}K(y_1,y_2,x)\ln g(y_1,y_2)dy_2dy_1=...
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### Is there a name for the square of a function plus the square of its Hilbert transform?

Given a real-valued analytic function $f$ defined on the whole real line, and its Hilbert transform ${\cal H}f$, it seems that the quantity $f(x)^2+{\cal H}f(x)^2$ should have some kind of importance ...
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### How can I solve this integral equation with the inverse Laplace Transform?

This question is related to Solving an integral equation with inverse Laplace transform. Let $\alpha,\beta,\mu>0$ with $\alpha/\beta>\mu$ and $X\sim\operatorname{Gamma}(\alpha,\beta)$. I am ...
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### Boundary to boundary transformation of an integral

In my textbook "Mathematical analysis I" we saw something called "Boundary to boundary transformation of an integral" (Note that my textbook is a Dutch textbook, I've tried to translate the name the ...
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### Binomial sum for an arbitrary function

I'm looking for some known results for sum of this type but I can't find anything. The sum is defined as: $$S(x,a,b,n)=\sum_{k=0}^n \binom{n}{k} (-1)^{k} f((a(n-k)+bk)x)$$ where $f$ is an arbitrary ...
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### Bessel integral invovling algebraic and hyperbolic functions

I am desperate in evaluating the following Hankel transform $$\int_{0}^{\infty} \frac{J_0(kr)}{k^2+\xi^2} \frac{\cosh(ky)}{\cosh(k)} k\mathrm{d} k,$$ where $J_0(kr)$ is the Bessel function of ...
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### Seeking Methods to solve $\int_{0}^{\frac{\pi}{2}} \ln\left|\sec^2(x) + \tan^4(x) \right|\:dx$

After weeks of going back and forth I've been able to solve the following definite integral: $$I = \int_{0}^{\frac{\pi}{2}} \ln\left|\sec^2(x) + \tan^4(x) \right|\:dx$$ To solve this I employ ...
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### Reference Request: n-dimensional Laplace Transform

I am looking for a reference, where the conditions for the existence of the n-dimensional Laplace transform are proven, i.e. when the laplace transform F(\lambda_1, ..., \lambda_n) = ...
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 ...
### About $F(u) = \int_{-\pi/2}^{+\pi/2} \ln(g(x) + u) dx$
We know for $u > 1$ $$\int_{-\pi/2}^{+\pi/2} \ln(\sin(x) + u) dx = \pi \left(\ln\left(u + \sqrt{u^2 -1}\right) - \ln(2)\right)$$ Usually this is shown by using differentiation under the ...