# Tagged Questions

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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### 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 ...
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### How to establish the smoothness class

Consider a smooth function $f(x)$ on $\mathbb{R}^{n}$ from some smoothness class $S_1$ and define $$F(y) = \int\limits_{g(x,y)\leqslant 0}f(x)dx$$ where $y \in \Omega$, $\Omega$ is some domain in ...
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### 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 ...
1answer
73 views

### can asymptotic of a Mellin (or laplace inverse ) be evaluated?

i mean given the Mellin inverse integral $\int_{c-i\infty}^{c+i\infty}dsF(s)x^{-s}$ can we evaluate this integral, at least as $x \rightarrow \infty$ can the same be made for \$ ...
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### 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) = ...
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### 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 ...
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51 views

### Can I solve for a unique integral kernel?

Consider, for $\mathbf{v},\mathbf{w} \in \mathbb{R^3}$, $$f(\mathbf{w}) := \int K(\mathbf{w,\mathbf{v}}) g(\mathbf{v}) \, d\mathbf{v} \, .$$ Is it possible to solve for the integral kernel, ...
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### Properties of Eigenfunctions of a Kernel

I'm a newbie and may be this question is bit simple for you but pardon me if it's too simple and provide me some references. I've and Kernel function $K(x,y)$ ...
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169 views

### Do kernel functions of integral transforms have any special properties?

From the Wikipedia page on integral transforms, it states that: ...an integral transform is any transform $T$ of the following form: $$(Tf)(u)=\int^{t_2}_{t_1}K(t,u)f(t)dt$$ ...There are ...
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2k views

### 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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199 views

### Series of nested double integrals

This is kind of a follow-up of my previous question. I'm investigating the following infinite series of nested two-dimensional integrals \sigma(t,t^\prime) = 1 - \int_{t^\prime}^t\mathrm dt_1 ...
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### 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) ...
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### 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  ...
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52 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))= ...
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49 views

### Gram's series for integral equation

The prime counting function $\pi(x)$ satisfies the integral equation $$\log\zeta (s)= s\int_{0}^{\infty}dx \frac{ \pi (e^{t})}{e^{st}-1} \tag{0}$$ and it has the solution in terms of Gram's ...
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### Is this formula for $\sum_{n} (n^{2}+z^{2})^{-s}$ correct?

I would like to know if this formula is true: $$\sum_{n=1}^{\infty}\frac{1}{(z^{2}+n^{2})^s}=\frac{1}{\Gamma(s)} \sum_{n=0}^{\infty}\Gamma(s+n)\zeta(2s+2n)\frac{ (-z^2)^n}{n!}.$$ I have used the ...
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127 views

### Mellin inverse transform

would be possible to evaluate the Mellin inverse transform $\int_{c-i\infty}^{c+i\infty}ds \frac{\zeta(s)}{2i \pi s}$ in terms of the zeros of the RIemann Zeta ??? i know how to compute the invers ...
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270 views

### Fourier Transform of Bessel function with square root argument

Fourier Transform of the following function: ...
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147 views

### Integral Transform

I am having some trouble with knowing the best route to go about evaluating this integral for the I.F.T. It states the following. $w(t)$ has the Fourier Transform \$W(f)= \dfrac{(j\pi f)}{(1+j2\pi ...
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116 views

### What does it mean to carry out calculations modulo $S^{- \infty}$

I have trouble making sense of the following remark that is given in a set of lecture notes I am currently going through on my own: " Oscillatory integrals are used for the study of singularities of ...
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72 views

### Evaluating the limit $y \to 0^+$

Given $t \in \mathbb{R}$ and $z = x + iy$ and $y>0$. $\lim_{y\to0^+} \frac{1}{t - z} = \frac{1}{t-x} + \pi i \delta(t-x)$ This limit is given in the book Integral Transforms and Their ...
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21 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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71 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: ...
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41 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 ...
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54 views

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

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

Assume $\omega , \mu \in \mathcal{D}'(\mathbb{R})$ are distributions with $\operatorname{supp}\mu$ compact, that are related according to  \omega = \varphi \ast \mu = \int (x-y)^{1/2}_+ \mu (y) \, ...
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26 views

### Do integral transforms with meromorphic kernels always have analytic continuation?

Do integral transforms with meromorphic kernels always have analytic continuation ? I think so, but I do not know how to prove it. For clarity with analytic continuation I assume it was already ...
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63 views

### Gelfand-Levitan-Marchenko equation

how can one solve the integral $$f(x,y)+K(x,y)+\int_{0}^{x}K(x,t)f(t,y)dt =0$$ (1) so $$q(x)= 2\frac{d}{dx}K(x,x)$$ (2) $$-y''(x)+q(x)y(x)=0$$ (3) $$y(0)=0=y(\infty)$$ $q(x)$ here is ...
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### Let $f(x+1) = P(f(x),x)$ where P is a polynomial. Express $f(x)$ as an integral.

Let $x$ be a real number and $f(x)$ a real analytic function such that $f(x+1) = P(f(x),x)$ where $P$ is a given real polynomial. Express $f(x)$ as an integral from $0$ to $\infty$. As an example we ...
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71 views

### relationship between Connes trace formula and Weil's trace formula

Connes trace formula $$Tr{U(h)}=2h(1)log\Lambda + \sum_{v} \int d^{*}x \frac{h(u^{-1})}{|1-u|}$$ Weil's trace  \int_{C}h(u)|u|d^{*}u- ...
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84 views

### An integral transform.

Let's consider a complex function that can be represented in the following form: $$K(z)=\int_{-\infty}^{\infty}A(\alpha)z^\alpha d\alpha$$ Writing $z=re^{i\theta}$, we get:  ...
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### Different proofs of support theorem for Radon Transform

Let $\mathcal{H}$ be a set of all hyperplanes in $\mathbb{R}^n$. Radon transform of function $f(x) \colon \mathbb{R}^n \to \mathbb{R}$ is defined as function $R[f] \colon \mathcal{H} \to \mathbb{R}$ ...
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128 views

### Inverting an integral transform

This question is either very obvious or by nature unsolvable in a general case. I Google'd around for a solution to no avail. Given an integral transform of kernel K across some interval I as a ...
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134 views

### Hankel Transform - Eigenfunctions and Inverse

I was reading Akhiezer's Lectures on Integral Transforms and in chapter nine, The Hankel Transform, he says that because the kernel of the Hankel transform is symmetric, its eigenfunctions ...
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235 views

### What is the difference between resolvent kernel and iterative kernel of an integral equation?

As we have different methods to find resolvent kernel, which is more suitable among all those methods? And what is the difference between resolvent and iterative kernels?
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### Fourier transform integral

I'm trying to calculate the 3D fourier transform of this function: $$\frac{1}{(x^2+y^2+z^2)^{1/2}}$$ Any help would be appreciated, thanks.
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136 views

### Limits in Double Integration Question

I’m really having problems getting suitable limits when using double integration. For example: Let E be the region defined by {(x,y): y-x <= 2, x + y >= 4, 2x + y <= 8} Sketch the region E. ...
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### 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_+} ...
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14 views

### Lower estimates on Mellin transform

Let $f(t)$ be a smooth decreasing function on $[0,+\infty)$. Its Mellin transform is the function $f^\ast(z)$ given by $$f^\ast(z) = \int\limits_0^\infty x^{z-1} f(x) \, \mathrm dx.$$ What are ...
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66 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 ...
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40 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 ...
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68 views

### Inverse laplace transform - infinite residues

I need to compute the inverse transform of the following, $f(s)= \dfrac{\sinh(k(l-x))}{\sinh(kl)}\dfrac{\omega}{\omega^2+s^2}$ where $k=\sqrt{\dfrac{s^2}{c^2}+n^2\pi^2},\ 0\leq x\leq l$. I used what ...
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130 views

### Integral Transform with Hyperbolic Functions

I am at it with understanding the nitty-gritty of the integral transform suggested in a previous question of mine: Length of a Parabolic Curve To solve this integral, you can use the substitution ...
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34 views

### Mathematical definitions of frequency transform and time-frequency transform?

I was wondering if there are mathematical definitions for an integral transform $$(Tf)(u) = \int \limits_{t_1}^{t_2} K(t, u)\, f(t)\, dt$$ to be frequency transform or time-frequency transform? An ...
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722 views

### Square of the sinc function integral

Considering the pulse function $\text{sinc}(t)$, show that : $$\int_{-\infty}^\infty \text{sinc}^2(t) dt = 1$$
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