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
votes
6answers
13k views

Connection between Fourier transform and Taylor series

Both Fourier transform and Taylor series are means to represent functions in a different form. My question: What is the connection between these two? Is there a way to get from one to the other (and ...
61
votes
4answers
2k 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 number of them. Is there a unified ...
47
votes
1answer
1k views

To evaluate $\int_0^{+\infty} \frac{\;\mathrm dx}{\sqrt[3]{x^3+a^3}\sqrt[3]{x^3+b^3}\sqrt[3]{x^3+c^3}}$

$$f(a,b)=\int_0^{+\infty} \frac{\;\mathrm dx}{\sqrt{x^2+a^2}\sqrt{x^2+b^2}}$$ To use Landen's transformation $$f(a,b)=\int_0^{+\infty} \frac{\;\mathrm dx}{\sqrt{x^2+(\frac{a+b}{2})^2}\sqrt{x^2+ab}}$$...
31
votes
1answer
3k views

Does a Fourier transformation on a (pseudo-)Riemannian manifold make sense?

the Fourier transformation of a scalar function with respect to one variable might be defined as $\mathcal{F}\left[w\right](\omega )\equiv \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}w(t)e^{-\mathrm{...
28
votes
2answers
6k views

Why do engineers use the Z-transform and mathematicians use generating functions?

For a (complex valued) sequence $(a_n)_{n\in\mathbb{N}}$ there is the associated generating function $$ f(z) = \sum_{n=0}^\infty a_nz^n$$ and the $z$-Transform $$ Z(a)(z) = \sum_{n=0}^\infty a_nz^{-n}$...
28
votes
7answers
26k views

Laplace transformations for dummies

Is there a simple explanation of what the Laplace transformations do exactly and how they work? Reading my math book has left me in a foggy haze of proofs that I don't completely understand. I'm ...
21
votes
1answer
231 views

Exponential integral $ \int_0^\infty \frac{x^t}{\Gamma(t+1)}\text dt$

Now since the sum $$ \sum_{n=0}^\infty \frac{x^n}{n!},\quad x\in\Bbb R, $$ does have some relatively nice properties, is the same true for its analogues integral? If we take the gamma function to be a ...
18
votes
1answer
554 views

Is Fourier transform characterized by its diagonalization properties?

Let us fix the following convention for the Fourier transform in $L^1(\mathbb{R})$ space: $$\hat{f}(\xi)=\int_{-\infty}^\infty f(x)\, e^{-2\pi i x\xi}\, dx.$$ We then have the following properties: \...
14
votes
2answers
855 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 $f\...
12
votes
1answer
780 views

Are Laplace Transforms a Special Case of Fourier Transforms?

A Laplace Transform is based on the integral: $F(\xi) = \int_0^{\infty} f(x) e^ {-\xi x}\,dx.$ In a roundabout way, a Fourier transform can get to $\hat{f}(\xi) = \int_{-\infty}^{\infty} f(x)\ e^{- ...
11
votes
1answer
224 views

A particular case of Truesdell's unified theory of special functions

I'm reading through Clifford Truesdell's "An essay toward a unified theory of special functions", Princeton Univ. Press, 1948. All his exposition is based on the functional equation $$\frac{\partial}{...
11
votes
3answers
2k views

What's the connection between the Laplace transform and the Fourier transform?

Both the Laplace transform and the Fourier transform in some sense decode the "spectrum" of a function. The Laplace transform gives a power-series decomposition whereas the Fourier transform gives a ...
11
votes
1answer
114 views

Extending the domains of densely defined bounded integral transforms on $L^2(\Bbb R)$

This is a question I've contemplated for quite some time since it's pretty closely related to Fourier theory (particularly choosing the "right" space to define the Fourier transform on). However I've ...
10
votes
4answers
401 views

Image of closed ball under degenerate integral operator is a closed set

I will be putting a bounty on this problem as soon as it lets me. For those who want to understand where the problem came from I encourage reading the edits, as I cut out several failed attempts and ...
10
votes
1answer
416 views

What exactly is an integral kernel?

I am not sure if I have seen integral transforms in the right way, but given a transform like Fourier transform - it's actually a basis transformation right ? $$ F(y) = \int K(x,y) f(x) \text{d}x $$ ...
9
votes
1answer
128 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 ...
9
votes
0answers
147 views

What plays the role of the identity for the generalized convolution associated to the Fourier-Bessel transform?

In traditional Fourier theory, the Dirac delta plays the role of an "identity" for the $L^1$ algebra with respect to the usual convolution. The convolution is traditionally built out of group ...
8
votes
1answer
396 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 $$ \int_{0}^{\infty}...
8
votes
1answer
127 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 \...
8
votes
1answer
96 views

Where to submit a small work of applied math written by a physicist?

Working in physics, I recently discovered a mathematical identity useful to solve a particular partial differential equation. Using the same idea, I found several other identities but I do not know ...
7
votes
2answers
556 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? ...
7
votes
2answers
140 views

Mellin Transform of the floor function

I have from integral transform tables that: $$\operatorname{Mellin}(\lfloor x\rfloor) = -\dfrac{\zeta(-z)}{z},\quad \operatorname{Re}(z)<-1$$ How can this be proved?
7
votes
1answer
1k views

Bessel function integral and Mellin transform

Gradshteyn&Ryzhik 6.635.3 provides the following integral, with the usual constraints on $\nu,\alpha,\beta$, $$\int\limits_0^\infty \exp\left(-\frac{\alpha}{x}-\beta x\right)J_\nu(\gamma x)\frac{\...
7
votes
2answers
337 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 ...
7
votes
1answer
219 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 ...
7
votes
1answer
934 views

Meaning of polynomially bounded

I am currently reading through some lecture notes on Fourier Transform and Distributions on my own, and came upon the notion of a polynomially bounded function. I am not sure I understand this concept,...
7
votes
2answers
2k 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 [\mathcal{H}...
7
votes
1answer
287 views

Evaluate $\int\limits_{\mathbb{R}^2} \frac{e^{i \langle \xi, x \rangle} d\xi}{ \langle\xi,\theta\rangle}$

The task is to evalute $$ \int\limits_{\mathbb{R}^2} \frac{e^{i \langle \xi, x \rangle} d\xi}{ \langle\xi,\theta\rangle}, \;\;\; \theta \in \mathbb{C}^2 \setminus ( \mathbb{S}^1 \cup \left\{ 0 \...
7
votes
0answers
14k 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 ...
6
votes
1answer
79 views

Integrate $\int\frac{x^{6}-2x^{3}}{\left(x^{3}+1\right)^{3}}dx$ .

$$\int\frac{x^{6}-2x^{3}}{\left(x^{3}+1\right)^{3}}dx$$ I tried adding and subtracting 1 to bring a square expression with numerator as $(x^3-1)^2 -1$ but always going to partial fraction which ...
6
votes
1answer
2k 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: $$\int_0^\infty\frac{\sin^3(\pi t)}{(\pi t)^3}\cos(ut)\,\mathrm{d}t$$
6
votes
4answers
155 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 $x+...
6
votes
3answers
221 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 ...
6
votes
1answer
352 views

Convolution square root of $\delta $

I want to somehow classify the distributional solutions of the equation $$ f \ast f = \delta $$ where $\delta = \delta _0$ is the Dirac delta distribution. Clearly, by Fourier transformation, we ...
6
votes
2answers
283 views

Stieltjes Transform

Stieltjes transform for a distribution $F(x)$ is defined as $$m(z)=\int \frac{dF(x)}{x−z}$$ where z is complex with positive imaginary parts and $F(x)$ is a distribution function. Basically, I am ...
6
votes
1answer
478 views

Multiple Fourier Integrals involving Heaviside Theta Function

I want to evaluate the integral: $$I=\int_{-\infty}^{\infty}dx_1 \int_{-\infty}^{\infty}dx_2 \ \Theta(x_1-x_2) \ e^{i(ax_1+bx_2)}$$ where $\Theta(x)$ is the Heaviside function. What I was doing now ...
6
votes
1answer
70 views

Integral transforms and uncertainty products

Heisenberg's uncertainty principle is well-studied and has become a bit of a pop science phenomenon due to its widespread implications in quantum mechanics. (Though interpretations are often ...
6
votes
1answer
185 views

How fast does the function $\displaystyle f(x)=\lim_{\epsilon\to0}\int_\epsilon^{\infty} \dfrac{e^{xt}}{t^t} \, dt $ grow?

Let $x$ be a positive real number and $f(x):=\lim_{\epsilon\to0}\int_\epsilon^{\infty} \dfrac{e^{xt}}{t^t} \, dt $. How fast does this function grow ? In other words can we find a good asymptote for $...
6
votes
1answer
436 views

Qualitative interpretation of Hilbert transform

the well-known Kramers-Kronig relations state that for a function satisfying certain conditions, its imaginary part is the Hilbert transform of its real part. This often comes up in physics, where ...
6
votes
0answers
91 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 \...
5
votes
2answers
10k views

Fourier transform of Bessel functions

I'm curious as to how the Fourier transform of the various types of Bessel functions would be calculated. The Wikipedia page on the Fourier transform gives the transform of $J_o(x)$ as being $\frac{...
5
votes
2answers
602 views

integral transforms: why do roots in frequency domain correspond to eigenvalues in time domain (and how does it help solve differential equations)?

In Wikipedia you can read about integral transforms, esp. the Laplace transform which maps a differential equation in the time domain into a polynomial equation in the complex frequency domain: ...
5
votes
6answers
2k views

What does it mean when two functions are “orthogonal”, why is it important?

I have often come across the concept of orthogonality and orthogonal functions e.g in fourier series the basis functions are cos and sine, and they are orthogonal. For vectors being orthogonal means ...
5
votes
2answers
4k 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 ...
5
votes
1answer
490 views

A Parseval-like theorem for Mellin transforms

A particular case of Parseval's theorem for Fourier transforms says that if $f$ is square integrable on $\mathbb{R}$, then $$ \int_{-\infty}^{\infty} |f(t)|^{2} \ dt = \int_{-\infty}^{\infty} |\hat{f}...
5
votes
1answer
6k views

What exactly is the Probability Integral Transform?

I've been going back over my notes from Stats class and came across the Probability Integral Transform. From my limited understanding, the basic idea is that from a cdf in terms of one variable, can ...
5
votes
1answer
81 views

What is the advantage of the Fourier Transform over the Hartley Transform?

The Hartley_transform is defined as $$ H(\omega) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^\infty f(t) \, \mbox{cas}(\omega t) \mathrm{d}t, $$ with $\mbox{cas}(\omega t) = \cos(\omega t) + \sin(\omega t)$...
5
votes
1answer
46 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) = \int_0^{\...
5
votes
0answers
71 views

Riemannian generalization/adaption of the Hubbard–Stratonovich transformation

I'd like to write the Hubbard–Stratonovich (HS) transformation of a scalar function on a Riemannian manifold (curved space). This transformation is quite simple in Euclidean space. One can consider ...
4
votes
2answers
166 views

Series around $s=1$ for $F(s)=\int_{1}^{\infty}\text{Li}(x)\,x^{-s-1}\,dx$

Consider the function $$F(s)=\int_{1}^{\infty}\frac{\text{Li}(x)}{x^{s+1}}dx$$ where $\text{Li}(x)=\int_2^x \frac{1}{\log t}dt$ is the logarithmic integral. What is the series expansion around $s=1$? ...