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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73 views

Laguerre transform function

For continuous indices 'n' and 'm' is it possible to have $$ \int_{0}^{\infty}dx e^{-x}L_{n}(x)L_{m}(x) = \delta (n-m)? $$ Another question: let $g(m)$ be the function defined via the transform $$ ...
8
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2answers
419 views

Fourier series is to Fourier transform what Laurent series is to …?

Since the coefficients $$a_k = \frac1{2\pi i}\oint_C\frac{f(z)}{(z-c)^{k+1}}\,dz$$ for the Laurent series $$f(z)\Big|_{r\le|z|\le R} = \sum_{k=-\infty}^{\infty}a_k\cdot(z-c)^k $$ of a function ...
7
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1answer
135 views

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 ...
3
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2answers
406 views

Continuity of integral function

How to show that the following function is right continuous at $0$ (that is, when $a\to0+$): $I(a) = \int_0^{\infty}\frac{\sin x}{x}e^{-ax}dx$? I know that Lebesgue integral $I(0) = \frac{\pi}{2}$. ...
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1answer
222 views

Fourier cosine and sine transforms of 1

What is the Fourier sine and cosine transform of $f(x)=1$? I have seen some sources refer to the transform of $f=1$ involving the Dirac Delta function, but this goes against the integral definition ...
3
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0answers
139 views

Understand integral from Gradshteyn and Ryzhik book “Table of integrals, series, products”

I was checking useful integrals in this book. I have found one (6.298) that is what I need, but I don't understand how every step towards the final result works. ...
3
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1answer
107 views

What should I call integrals of the form $\int_{-a}^{b} f(t) e^{zt}dt$?

I've been studying integrals of the form $$ \int_{-a}^{b} f(t)e^{zt}dt $$ where $z$ is a complex variable. So far I've been calling them "exponential integrals" following the practice of P. Miller ...
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0answers
78 views

Integral transform of gaussian function

I am looking for an integral transform of $f(x)=\exp(-\frac{x^2}{2\sigma^2})$ such that: $$(Tf)(u)=\int_{x_1}^{x_2}\, K(x,u)f(x)\,dx\,\stackrel{??}{=}\,\frac{a}{g(u)}$$ with $g(u)$ having zeros ...
43
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3answers
856 views

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 multiplicity of such transforms. Is ...
3
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0answers
65 views

transform that is invariant under rotation

We know that the magnitude of the Fourier transform (resp. Mellin transform) of a shifted (resp. scaled) function is identical to the magnitude of the original function. I wonder if there is a ...
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55 views

Uniqueness for Integral Transform

What can be said about the uniqueness of the following integral transformation: $ (Tf)(u) = \int_0^{\infty} f(t)G(tu)dt$ defined for all $u\geq 0$, where the kernel $G(z) \in [0,1]$ for all ...
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1answer
1k views

Fourier transform of $\text{sinc}^3 {\pi t}$

$$f(t)=\frac{\sin^3(\pi t)}{(\pi t)^3}$$ I want to calculate the Fourier transform. I can't calculate this integral: $$\int_0^\infty\frac{\sin^3(\pi t)}{(\pi t)^3}\cos(ut)\,\mathrm{d}t$$
0
votes
0answers
110 views

Fourier, the Fourier transform

Could You help me? Where $g(t)$ is Cantor function: $$G(\omega)= \int_0^1 e^{2\pi i\omega t}dg(t)$$ Show, that $G(\omega)\not\to0$, if $\omega\to\infty$
2
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1answer
123 views

On using fourier transforms to solve the root of a convolution

In continuation of Lower bounds of laplace transform of characteristic functions. My question is: Can anyone point out where i'm going wrong in the derivation below. It's been a while ...
0
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1answer
36 views

proving a z transform

I am having trouble demonstrating the Z transform of $a^{n-1}u(n-1)$ is $\frac{1}{(z-a)}$, as it says in this table. I try using the definition of the z transform, but it comes out different than ...
2
votes
1answer
119 views

What's the connection between Banach/Hilbert spaces and tools like power series, Fourier transforms etc.

I've learned, abstractly, about Banach and Hilbert spaces, and more concretely about $l^p$ and $L^p$ spaces. I also understand these ideas have something to do with a variety of tools that are useful ...
2
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1answer
53 views

Practical applications of the Fantappiè transform

The Fantappiè transform of function $f(x_1,x_2,\ldots,x_n)$, $x_1 \geqslant 0, \ldots, x_n\geqslant 0$, is defined by the formula $$ (\Phi f)(y) = \int\limits_{\mathbb R^n_+} ...
0
votes
1answer
151 views

Problem evaluating an improper integral $\int_0^{\infty} \frac{(\sin{2x}-2x\cos{2x})^2}{x^6}$ using fourier transform

This is a question from one of the past papers of my university which I am unable to do. I am not being able to do question 2 from below. Let $f(x)= a^2-x^2 \,\,\,\,\, |x|<a ...
2
votes
1answer
674 views

Evaluating improper integrals using laplace transform

I want to calculate the following improper integral using Laplace and transforms (and laplace transforms only). $$\int_0^{\infty} x e^{-3x} \sin{x}\, dx$$ I propose the following method. I plan to ...
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1answer
659 views

Condition for the inverse laplace transform of a function to exist and bromwich integral

Given any function, is there any way of determining from the nature of the function, if it is the laplace transform of a piecewise continuous function of exponential order? For e.g. say the function ...
4
votes
1answer
1k views

Solving an initial value ODE problem using fourier transform

I am a physics undergrad and studying some transform methods. The question is as follows: $y^{\prime \prime} - 2 y^{\prime}+y=\cos{x}\,\,\,\,y(0)=y^{\prime}(0)=0\,\,\, x>0$ I am having some ...
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2answers
717 views

Solving an integral equations using fourier transform

I have to solve the equation $\int_0^{\infty} f(x) \cos{(\alpha x)}\, dx=\frac{\sin{\alpha }}{\alpha}$ Using fourier transform. I know this is half of the usual fourier cosine transform, and so ...
0
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1answer
167 views

Probability integral transform: Is it integral transform? Can it be for discrete distribution?

From Wikipedia the probability integral transform or transformation relates to the result that data values that are modelled as being random variables from any given continuous distribution can be ...
0
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1answer
37 views

Lambert transform of monomials

The Lambert transform of a function $f(x)$, is given by: $$\int_{0}^{\infty}\frac{f(x)}{e^{xt}-1}dx\;\;\;\;(\Re(t)>0)$$ We wish for a closed form of the transform : ...
3
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0answers
36 views

fancy about some properties of kernel functions at infinity

Consider the two common types of kernel functions $\sum\limits_{t=a}^bf(t)K(x,t)$ and $\int_a^bf(t)K(x,t)~dt$ , prove whether the following properties are correct or not: $1.$ If $K(x,t)$ is bounded ...
2
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0answers
83 views

Existence of zeros of Mellin transform and properties of function to be transformed

Mellin transform of function $f(x)$ defined for $x\geqslant 0$ is given by $$ f^\ast(z) =\int\limits_0^\infty x^{z} f(x) \frac{dx}{x}. $$ I consider only exponentially decreasing (there exist such ...
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3answers
218 views

Fourier transform of $g(x)=x\frac{\partial f}{\partial x}$?

I have a problem with the Fourier transform of the function $g(x)=x\frac{\partial f}{\partial x}$. I need the transform to be itself a function of the Fourier transform of $f(x)$ and I don't know how ...
2
votes
0answers
91 views

Continuity of the inverse Laplace Transform

If I know $Y(s)$, can I predict when $\mathscr{L}^{-1}[Y(s)]=y(t)$ will be continuous or continuously differentiable or even stronger conditions? For example; I'm solving an ODE with the Laplace ...
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0answers
239 views

Laplace transform of a product of functions

While trying to compute the Laplace transform of a certain product, part of the calculation leaves me with a Bromwich integral which has the form: ...
0
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1answer
304 views

Solving a recurrence relation using Z transform

I'm trying to solve the following recurrence using Z transforms: For $n\in \mathbb{N}^{*}$ $T(n)=1\ for\ n< 4$ $T(n)=T(\lfloor \frac{n}{4} \rfloor)+T(\lfloor \frac{3n}{4} \rfloor)+n\ for\ n\geq ...
3
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0answers
125 views

Showing a Hölder continuous function acted on by a singular integral operator is Hölder continuous

Consider the following function defined by a singular integral \begin{equation} F(x)= \lim_{\epsilon \rightarrow 0} \int_{|x-y| \geq \epsilon} \partial_k \partial_j k_i(x-y) \left(Y_k(x)- Y_k(y) ...
2
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0answers
70 views

Inverse Laplace transform is required

I shall be very very thankful if some one can find the inverse Laplace of the function given below. I really need it as early as possible. $$ \frac{1}{s-a}\exp ...
2
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0answers
112 views

Mellin tranform of $\cos x$ using Ramanujan's master theorem

I've been messing around with Ramanujan's master theorem. $\displaystyle \int_{0}^{\infty} x^{s-1} \cos x \ dx = \int_{0}^{\infty} x^{s-1} {}_0F_1 \left(-;\frac{1}{2};\frac{-x^{2}}{4} \right) \ dx $ ...
7
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1answer
210 views

Mellin transform of digamma function

what is the Mellin trasnform of the Digamma function ?? from Ramanujan master theorem http://mathworld.wolfram.com/RamanujansMasterTheorem.html y believe it should be equal to $$ ...
1
vote
1answer
95 views

Fourier Transform of a function under an arbitrary coordinate transform [duplicate]

Consider a function $f(x)$ and its Fourier Transform $\tilde{f}(k)$ given by $$ \tilde{f}(k) = \int_\mathbb{R}\!\!\!dx\; e^{-ikx}f(x). $$ Now, lets have the coordinate transform $\xi = \tau(x)$ and, ...
1
vote
1answer
103 views

Dirac delta questions form Mellin transform

We know that $$ f(s)= \int_{-\infty}^{\infty}f(x)\delta (x-s) d x$$ however, is there a similar delta function so for the Mellin transform $$ f(s)=\int_{0}^{\infty}f(x)m(xs) d x$$ ? That is a ...
5
votes
4answers
135 views

Change of variables Double integral

I have $$\iint_A \frac{1}{(x^2+y^2)^2}\,dx\,dy.$$ $A$ is bounded by the conditions $x^2 + y^2 \leq 1$ and $x+y \geq 1$. I initially thought to make the switch the polar coordinates, but the line ...
2
votes
1answer
482 views

Inversion formula for the Abel transform

I need an inversion formula for the Abel transform $$ F(y) = 2\int_y^\infty\frac{f(r)r\,dr}{\sqrt{r^2-y^2}}. $$ Hint: The inversion formula found on Wikipedia appears to be incorrect. The ...
5
votes
1answer
913 views

Hilbert transform and Fourier transform

Assume the following relationship between the Hilbert and Fourier transforms: $$ \mathcal{H}(f) = {\mathcal{F}^{-1}}(-i ~ \text{sgn}(\cdot) \cdot \mathcal{F}(f)), $$ where $ \displaystyle ...
4
votes
0answers
58 views

Some properties of an analogue of the integral Fourier operator

Let $\phi(x,\theta)$ be an infinitely differentiable function in $X^2 = X \times X$, where $X = \mathop{\mathsf{int}}\mathbb{R}^n_{+}$, and let$\phi(\lambda x, \theta) = \phi(x,\lambda \theta) = ...
0
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0answers
182 views

Triple Product Integral on Real Spherical Harmonics Basis Functions

Okay I know that Real Spherical Harmonics are given by If $m \lt 0$ $~$ then $\sqrt{2}$ $~$ $Im(\text{SphericalHarmonicY}[l,|m|])$ If $m=0$ $~$ then $~$ $\text{SphericalHarmonicY}[l,0]$ If $m \gt ...
2
votes
0answers
83 views

Theta series and Riemann Hypothesis

in the paper http://www.fuchs-braun.com/media/dd209bf5c2203a87ffff80a3ffffffef.pdf section 2 ' Hilbert-Polya space' page: 180 the author introduce the Theta series $$ F(\phi(x))= ...
0
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0answers
65 views

nonlinear integral equation

let be the integral equation for two functions $ f(x) $ and $ g(x) $ in the form $$ g(s)= \int_{0}^{s}\sqrt{s-f(x)}dx $$ is valid to accept that in the sense of fractional calculus, the ONLY ...
3
votes
1answer
98 views

Double Integral involving modifed bessel function

I'm try to derive a closed form of the following double integral: $\int\limits_0^x {\int\limits_0^x {{e^{ - {K_1}uv}}{I_0}\left( {2{K_1}\sqrt {uv} } \right)du} dv}$; where $K_1$ is a constant. Do you ...
1
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1answer
92 views

arguing away - complex analysis

Probably a trivial question but I can't understand how to argue away the value of integrals in complex analysis. I am trying to find the inverse Laplace transform of $F(s)=\frac{1}{s(s+1)}$. The ...
2
votes
1answer
329 views

An inverse definite integral problem

I am seeking a function $f(x)$ that satisfies this condition: $\int_{0}^{\infty }f(x)x^ndx=\sqrt{n!}$ where n is an integer. I guess that $f$ will contain $e^{-\alpha x^2}$ as one of its factors, ...
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1answer
35 views

When to use other transforms?

maple code int(g*f, x=-infinity..infinity) when $g$ is $\large exp^{i*t*x}$, Fourier transform between density function and characteristic function If $g$ are $x^t$, $|x^{t}|$, $t^{x}$, what do they ...
2
votes
2answers
124 views

Can an integral operator with negative kernel have a positive eigenvalue?

While reading "Integral equations - a reference text" (Zabreiko et al. eds) I came up with this question I cannot answer: Suppose $A:L^2(\mathbb{R})\to L^2(\mathbb{R})$ is a bounded linear integral ...
0
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1answer
120 views

Question about L2 Inner Product and Integrals

Does exists $f\in L_2(\mathbb{R}^d)$ such that for all $g\in L_2(\mathbb{R}^d)$ which is not identically zero: ...
6
votes
2answers
245 views

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? ...