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 ...

learn more… | top users | synonyms

0
votes
0answers
22 views

understanding uv coordinates integral bounds [on hold]

When integrating using $u$,$v$-coordinates, cases where you're dealing with trigonometric functions for example: Integrate $f(x,y) = x$ over $R$ where $R$ is the region bounded by $x=0$, $y=0$ and ...
1
vote
1answer
34 views

Is the Fourier Transform of the limit the limit of the Fourier Transform?

Assume you want to compute the Fourier transform of a function $f_\epsilon(x)$ given by \begin{align} \mathcal{F}(f_\epsilon)(k) = \int f_\epsilon(x) e^{-ikx}\, dx \end{align} Further assume, that ...
2
votes
5answers
43 views

“eigenfunction” of a transformation

Fourier transform of a gaussian is another gaussian. Fourier/Laplace transforms of $\frac{1}{\sqrt t}$ is something like $\frac{1}{\sqrt \omega}$. I realize that we can't call these eigenfunctions ...
0
votes
2answers
51 views

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 ...
1
vote
1answer
30 views

Smoothness of Fourier transform of $\frac{1}{|x|^p}$

Consider the "function" (more precisely it is a tempered distribution) given by $f : \mathbb{R}^n \to \mathbb{R}$, $f(x) = \frac{1}{|x|^p}$, where $0 < p < n$. It can be calculated that the ...
2
votes
0answers
50 views

Inverse Fourier transform of $\frac{1}{\sqrt{\xi}}$

Does talking about the inverse Fourier transform of $\frac{1}{\sqrt{\xi}}$ even make sense? If it does, how can we conclude about the decay properties, support and smoothness of the inverse Fourier ...
-2
votes
1answer
24 views

Identifying a function is even or odd or not even and odd. [closed]

Here I have a very confusing problem. I'm right now solving Fourier transform. In which different formulas has to be applied according to the nature of the function wether it is odd or even or not ...
0
votes
1answer
21 views

Probability transformation

I have a question regarding probability transformations. Could someone tell me wether I am doing it good or not? Consider $f_{X}(x)=3/2e^{-3x}+3e^{-6x}, x \geq 0$, 1) Calculate the pdf of ...
3
votes
3answers
189 views

A more elegant way to find the Fourier transform

Let $f$ be defined analytically as : $$f(x)=\arccos \left ( \sin \left ( 2x \right ) \right ), x \in\left (0,10 \right ], f(x)=0, x\notin\left ( 0,10 \right ]$$ Here is a graph of the above ...
0
votes
0answers
7 views

Inverse Radon Transform of a function in the Schwartz class

This question comes from reading through Stein and Shakarchi's Fourier Analysis, page 206. Consider the two Schwartz spaces $\mathcal{S}(\mathbb{R}^3)$ and $\mathcal{S}(\mathbb{R}\times S^2)$, where ...
0
votes
0answers
24 views

Geometric Interpretation of Fourier Transforms

I'm interested in tsunami wave science and I've already got an engineering degree and a basic knowledge of signal processing. The courses I took were intensively computational and taught some skills ...
2
votes
0answers
6 views

Consistency/range conditions for (integral) transform mapping into higher-dimensional space

I am interested in learning more about what the (formal) implications are when transforming a $n$-dimensional function space (e.g., the space of all $\mathbb{R}^n\to\mathbb{R}$) to a higherdimensional ...
1
vote
0answers
25 views

Singularities in the Gauss Hypergeometric Function

I am evaluating the following term in a series: $$I_k = \int\!x^{-3(2k+1)}(1+\lambda x^4)^{-1/2}\,\mathrm dx$$ When I plug this into WolframAlpha, I get the following result: $$I_k = ...
0
votes
0answers
13 views

Identify functional form of expectation value

I have the equation: $g(z) = {\int_{\mathbb{R}^n}}\;f(x)\exp(-c(x)^Tz)dx = \mathbb{E}_X[\exp(-c(X)^Tz)]$ where $c:\mathbb{R}^n \to \mathbb{R}_+^m$, $z \in \mathbb{R}_+^m$ and $f(x)$ is a ...
2
votes
0answers
45 views

A Mellin Transform of a generating function

I am trying to find the Mellin transform of the function $$ G(z) = \sum_{k \ge 1} C_k\left( 1- \exp \left( \frac{-z}{4^k} \right )\right), $$ where $C_k$ denotes the $k$-th Catalan number ($C_k = ...
1
vote
0answers
34 views

Solving initial value problem using Laplace transforms, one other method, and comparing results

So for my solution using characteristic equations I get (fixed a typo for first coefficient) $$\frac{11}{30} e^{-3t} - \frac{21}{20} e^{-2t} + \frac{21}{20} e^{2t} - \frac{11}{30} e^{3t}$$ For the ...
0
votes
0answers
17 views

Transforming a function - general methods

I have some rather general questions about transforming a function from one form to another. I have tried reading about integral transforms, but I wish to know more about the theory in general, as ...
0
votes
1answer
18 views

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. ...
0
votes
1answer
22 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
votes
1answer
94 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} ...
0
votes
1answer
31 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 ...
0
votes
1answer
37 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 ...
3
votes
1answer
56 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 ...
4
votes
3answers
91 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 ...
1
vote
1answer
36 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
71 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) ...
0
votes
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
0
votes
1answer
23 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) ...
0
votes
1answer
43 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
votes
0answers
28 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
155 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 ...
0
votes
1answer
35 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
votes
1answer
163 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
votes
0answers
69 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),\, ...
1
vote
0answers
24 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 ...
0
votes
0answers
42 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
300 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 ...
0
votes
1answer
24 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 ...
0
votes
1answer
39 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
votes
1answer
38 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 ...
0
votes
1answer
59 views

Kernels of integral transform and linear transformation

Is there any relation between the $kernel$ of an $integral \ transform$ and the $kernel$ of a $linear \ transformation$?
0
votes
1answer
76 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
votes
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
votes
1answer
69 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
votes
0answers
29 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 ...
0
votes
0answers
56 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
votes
1answer
52 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 ...
0
votes
0answers
37 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} ...
5
votes
0answers
55 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 ...