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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1answer
18 views

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 function ...
8
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1answer
70 views

The inverse Laplace transform of $ s^{3/2}-a-bs \over s^{3/2}+a+bs$

How can I solve the inverse Laplace transform as below: $$\mathscr{L}^{-1}\left( s^{3/2}-a-bs \over s^{3/2}+a+bs \right) $$ where a and b are constants. Hint: we can consider $${ s^{3/2}-a-bs ...
1
vote
1answer
19 views

Find a function in terms of another

I need to express an in terms of f(x). I did it but I'm not sure if it is right. Consider \begin{cases} -u_{xx}+u=f(x), &0<x<\ell\\[0.5em] u_x(0)=0,\;u_x(l)=0& \end{cases} ...
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1answer
30 views

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 ...
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1answer
35 views

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 ...
2
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1answer
41 views

Is the Gamma Function multivalued??

Consider the definition of the Gamma function $$ \Gamma(s) = \int_{0}^{\infty}\left[x^{s-1}e^{-x} \right] dx $$ Clearly: $x^{s-1}$ may have multiple defined values for $s$ if $s-1$ is rational or ...
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3answers
71 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 ...
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1answer
32 views

Fourier transform of a random variable

During my research i'm dealing with a stochastic partial differential equation. The random term appearing in my equation is a tensorial random variable: $\boldsymbol{\sigma}(\boldsymbol{x},t)$ Which ...
2
votes
1answer
60 views

Inverse transform of a modified Abel transform

I have been struggling for 6 months on finding the analytical inverse transform of a transformation below: $$F(y,k) = 2 \int_y^{\infty}\cos\left(ka\sqrt{r^2-y^2}\right) f(r,k) ...
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3answers
33 views

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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1answer
22 views

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) ...
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1answer
26 views

Fourier Transform - Laplace Transform - Which variable transform?

I need to know when do I have to transform $x$ and when $y$ in a PDE in Fourier Transform and Laplace Transform. In an exercise of Fourier Transform, I have to solve a Laplace Equation, where ...
3
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0answers
24 views

Mellin transform defined for function on group $(\mathbb{R}^{+}, \times)$

In the article on the $\Gamma$ function in the Princeton Companion to Mathematics, the author states The Mellin transform is a type of Fourier transform, but it is defined for functions on the ...
2
votes
1answer
104 views

Fourier COSINE Transform (solving PDE - Laplace Equation)

I'm trying to solve Laplace equation using Fourier Cosine Transform (I have to use that), but I don't know if I'm doing everything OK (if I'm doing everything OK, the exercise is wrong and I don't ...
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1answer
26 views

Fourier cosine and sine transform of $\exp{(-ax)}(1+bx)^{-1}$ and $\exp{(-ax)}(1+bx)^{-2}$

As stated in the title I should calculate the cosine and sine Fourier transform of: $$f_1(x)=\exp{(-ax)}(1+bx)^{-1}$$ and $$f_2(x)=\exp{(-ax)}(1+bx)^{-2}$$ That obviously means calculating: ...
3
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1answer
105 views

Relation between Hankel transform and Fourier transform

As a physics student, I ran into the following problem. I left out a lot of context, if anything is unclear please ask me. I quote: The statistic that is observable is the angular correlation ...
2
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0answers
51 views

About Mellin transform and harmonic series

Let $\mathfrak{M}\left(*\right)$ the Mellin transform. We know that holds this identity$$\mathfrak{M}\left(\underset{k\geq1}{\sum}\lambda_{k}g\left(\mu_{k}x\right),\, ...
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0answers
23 views

Can wavelets be used for texture discrimination?

I've recently been studying wavelet analysis with a view to differentiating certain areas of texture images where the texture differs from the background pattern (which is quite random); for example a ...
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0answers
35 views

Check my Solution: Possible kernels for an integral transformation

I solved a problem and would like to know if my solution is correct. Feel free to comment. Problem: Given the integral equation $$\frac{1}{2\pi}\int dk e^{-ikx'}\int dx e^{ikx} u(k,x) f(x)=f(x')$$ ...
7
votes
2answers
288 views

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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1answer
17 views

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 ...
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1answer
32 views

Laplace-Fourier transform issue

Given a function $f:\mathbb{R}\rightarrow\mathbb{R}$ we take the generalised Fourier transform $\hat{f}(w)=\int_{-\infty}^{+\infty}e^{iwx}f(x)dx$ where $w\in \mathbb{C}$. Now assume, this transform ...
3
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1answer
33 views

Prove that the Laplace transform of $I_0\left(2\sqrt t\right)$ is $\exp\left(1/s\right)/s$

Wolfram Alpha gave me the answer to this, but unfortunately Wolfram Alpha doesn't show its work, I can't find a proof anywhere else, and my feeble attempts to show it myself went nowhere. How can it ...
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1answer
40 views

Kernels of integral transform and linear transformation

Is there any relation between the $kernel$ of an $integral \ transform$ and the $kernel$ of a $linear \ transformation$?
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1answer
67 views

Good recommendations for solving PDE's by integral transforms

I look for good books on solving partial diffrential equations (PDE's) using integral transforms specially Fourier and laplace transforms. Do you have any recommendations for such books? I don't ...
5
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1answer
44 views

Can $f(x+1) = f(x)^{\ln(x)}$ be expressed as integral transform $\int g(x,t) dt $?

Let $x$ be a real number. Can some real-analytic function $f$ that satisfies for $x>3$ :$f(x+1) = f(x)^{\ln(x)}$ be expressed by standard functions as an integral transform : $$f(x) = ...
2
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1answer
62 views

dummy variable in Fourier transform confusion

In this text, why is it using different dummy variable for the integral of coefficients $a_n$ and $b_n$? I know that choosing the dummy variable does not affect integral but over here since we are ...
2
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0answers
27 views

Solving String Vibration Using Integral Transform

$$U_{tt} - c^2 U_{xx}= -g$$ where BC: $U_{x}(0,t)=a\sin(ωt)$ IC: $U(x,0)=0$, $U_{t}(x,0)=0$ where $c, g, A$ and $ω$ are positive constants Normally I wouldn't post for help here but I am ...
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0answers
38 views

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 ) ...
2
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1answer
47 views

Dirichlet Series and Asymptotic Expansions

Consider the Dirichlet series $\tilde{f}(s)= \sum_{n=1}^{\infty} f(n) n^{-s}$. In the page "Zeta Function Regularization" of Wikipedia http://en.wikipedia.org/wiki/Zeta_function_regularization I ...
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0answers
33 views

Weber Transform

During my studies I meet the Weber Transform of the free space potential function, that is: $$\int _{\rho }^{\infty }\exp(-i \text{$\key} t) (Y_0(\text{$\lambda $p} \rho ) J_0(\text{$\lambda $p} ...
4
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0answers
46 views

Estimating the (double) Riesz transform.

I'm trying to verify the following estimate, which appears in a paper I'm reading. It seems I'm missing something easy, I just can't figure this out. $\textbf{Background}:$ For a function $f \in ...
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2answers
41 views

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} ...
2
votes
2answers
26 views

Inverse laplace transform excercise

I want to find the inverse transform of $$\frac{1}{(2s-1)^3}$$ I first applied a shifting theorem to get $$(e^t)\mathcal{L}^{-1}\left( \frac{1}{(2s)^3} \right)$$ I am just wondering is it possible ...
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2answers
50 views

Fourier Transform of Sine

I'm having trouble calculating the Fourier Transform of the sin function. Specifically, the function $ G(\omega)=\int _{-\infty}^{\infty} g(t)\ e^{-i \omega t} dt $ For the fourier transform of $ ...
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0answers
22 views

Equivalence of two functionals

Fix a certain $x \in \mathbb{R}^{n}.$ Let us denote by $\omega_{n}$ the surface area of the unit sphere. Let $g(\pi)$ be a function defined in the set of hyperplanes, $\mathcal{P}$. Such a function ...
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0answers
33 views

Finding the best transformation for a triangle (Jacobian)

A triangle in xy-plane has following vertices: (0,0) (2,3) (3,0) Book gave the following transformation in uv-plane and it works out nicely, but I am not sure ...
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0answers
19 views

Is the Schwarz-Christoffel transformation minimizing the modified Liao functional?

I am using the sctoolbox for Matlab from T. Driscoll to transform the upper complex halfplane onto an area given by the three points x1=-1 and x2=1i. The thrid point is at infinity. This works fine ...
2
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0answers
24 views

Can this divergent integral transform be regularized?

The integral $$\int_0^{\infty} e^u \ K_{i t}(u) du$$ is the adjoint Kontorovich-Lebedev transform of the increasing exponential function, but unfortunately this integral is divergent because $$e^u \ ...
9
votes
1answer
78 views

What are the “right” spaces for the Laplace transform

There are for example several canonical spaces to define the Fourier transform (i.e. Schwartz's space). Is there also a particularly suitable space to define the Laplace transform, so that the Laplace ...
2
votes
1answer
58 views

Express $f(x)=\sin{x}$ as an even function

Express $f(x)=\sin{(x)}$, with $(0 < x< \pi )$ as an even function, $f(x+ 2\pi)=f(x)$ The topic is on Fourier Series. I have the following so far: Since $f(x)$ must be an even function, ...
4
votes
1answer
35 views

Proving that $(2 \pi i)^{-1} \int e(\pi v^2/y^2) x^v y^{-1} dv = e(-\pi (\log x)^2 y^2 /4)$

I've seen a particular integral transform (an inverse Mellin Transform) used a few times, but I don't know how it's proved. In particular, I'm trying to prove $$\frac{1}{2\pi i} \int_{(2)} e^{\pi ...
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0answers
12 views

Hilbert transform of the product of functions

Let $H[g]$ denotes a Hilbert transform of function $g$. What would be the constant $C$ in the following inequality: $$ \|H[(\cos n)(\cos{1/(2n))}f](x)\|_{L_2}\leq C\|f\|_2? $$
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1answer
52 views

Connection between $\nu=\frac{N}{2}-1^\text{th}$ order Hankel transforms and hyperspherically symmetric functions?

In The Transforms And Applications Handbook 2nd edition chapter 9 (Hankel Transforms), Piessens briefly mentions that the Fourier transform of an $N$-dimensional hyperspherically symmetric function ...
2
votes
1answer
76 views

What's the inverse of the Weierstrass-Mittag-Leffler-Transform $\exp\left[g(z) + \int_\mathbb C f(y)\ln(z-y)\,dy\right]$?

As mentioned in another post, as a consequence of Mittag-Leffler's theorem combined with the Weierstrass factorization theorem, after reducing to the common denominator, any meromorphic function can ...
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0answers
27 views

Inverse fourier transform using Parseval relation.

Please, if someone can help with this question I will be grateful. Considering Parseval's relation, show that the Inverse Fourier Transform can be written as $$ f(t)=\int_{-\infty}^\infty ...
0
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0answers
28 views

Existence and uniqueness of Volterra integral equations of the first kind with vanished kernel [duplicate]

$$ \int_0^t k(s,t)f(s)ds=g(t) $$ To prove the existence and uniqueness of solutions of Volterra integral equation(VIE) of the first kind, we usually differentiate it and convert to the VIE of the ...
0
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1answer
36 views

How to find the coefficients in the Fourier series solution of a 1-D heat equation?

I am trying to use Fourier's method to solve a problem. $u(x,t) = \sum \limits_{n=1}^\infty B_ne^{-(n\pi C / L)^2 t}\sin\left(\frac{n\pi x}{L}\right), B_n=\frac2L\int_0^L \sin\left(\frac{n\pi ...
0
votes
1answer
78 views

PDEs for string deflection.

Okay, I have to find $u(x,t)$ for the string of length $L=\pi$ when $c^2=1$. I know: $$\text{wave equation}: \frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2}$$ $$u(x,0)=\frac ...
0
votes
1answer
51 views

How to solve this Laplace transform? $f(t)=e^{-2t}\cos^2 3t - 3t^2 e^{3t}$

Find the laplace transform of $$f(t)=e^{-2t}\cos^2 3t - 3t^2 e^{3t}$$ The answer is $$\frac{1}{2(s+2)}+ \frac{1}{2} \frac{s+2}{s^2 + 4s + 40} - \frac{6}{(s-3)^3}.$$ This took me about an hour to ...