Questions tagged [laplace-transform]

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51 votes
6 answers
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What does Mellin inversion "really mean"?

Given a function $f: \mathbb{R}^+ \rightarrow \mathbb{C}$ satisfying suitable conditions (exponential decay at infinity, continuous, and bounded variation) is good enough, its Mellin transform is ...
Frank Thorne's user avatar
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47 votes
6 answers
11k views

Intuition for Integral Transforms

It is well known that the operations of differentiation and integration are reduced to multiplication and division after being transformed by an integral transform (like e.g. Fourier or Laplace ...
vonjd's user avatar
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5 votes
1 answer
507 views

Closed form for $\int_0^T e^{-x}\frac{I_n(\alpha x)}{x}dx$

EDIT: Some additional details and corrections, I would appreciate any information about the highlighted expression. I try to solve $\int_0^T e^{-x}\frac{I_n(\alpha x)}{x}dx$ where $I_n(x)$ is the ...
Alexandre's user avatar
  • 368
4 votes
1 answer
394 views

Using a quadratic kernel instead of a linear kernel in the Laplace transform

Suppose $f$ is a bounded continuous function on $[0,\infty)$ such that $\int_0^\infty f(t) \exp(-xt) \: dt \rightarrow 0$ as $x \rightarrow 0^+$. Does it follow that $\int_0^\infty f(t) \exp(-xt^2) \: ...
James Propp's user avatar
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74 votes
9 answers
25k views

Motivating the Laplace transform definition

In undergraduate differential equations it's usual to deal with the Laplace transform to reduce the differential equation problem to an algebraic problem. The Laplace transform of a function $f(t)$, ...
elaichi's user avatar
  • 883
21 votes
4 answers
6k views

When I can safely assume that a function is a Laplace transform of other function?

If I have a function and I want to represent it as being the Laplace transform of another, that is, I want to be sure that there is $\hat{f}(s)$ such that my function $f(x)$ can be written as: $$f(x) =...
Rorsa's user avatar
  • 873
6 votes
1 answer
742 views

Variations on the Mellin and Dirichlet transforms

There are a number of variations on the Laplace transform that turn up all over math. Some examples: $\int_{-\infty}^{\infty} f(t)e^{-st} dt$ - The Laplace transform $\sum_{-\infty}^{\infty} f(t)z^{-...
Mike Battaglia's user avatar
6 votes
2 answers
325 views

On frequency decay of an integral transform of a function

Suppose $f \in C^{\infty}_c((-1,1))$ and assume that there exists constants $a,b>0$ such that $$ \bigg|\int_{\mathbb R} f(t) \,e^{\tau t^2+i\tau t}\,dt\bigg| \leq a\,e^{-b|\tau|},$$ for all $\tau \...
Ali's user avatar
  • 4,077
5 votes
2 answers
179 views

Limit of the extremal process of i.i.d. Gaussians see from the tip

I'm trying to calculate the weak limit of $\mathcal{E}_N(x)=\sum_{k=1}^{2^N}\delta_{x-Z_k}$ , with $Z_k=X_k-\max_{k\leq 2^N}X_k$, $\{X_k\}$ being $2^N$ copies of i.i.d. Gaussians with mean zero and ...
MikeG's user avatar
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4 votes
1 answer
374 views

Relations involving Stirling numbers of second kind

While inverting a Laplace transform using Post's inversion formula I found the following expression: $$ \sum_{k=1}^n S^n_k \ x^k(\alpha)_k $$ where $S^n_k$ is a Stirling number of second kind and $(\...
guaraqe's user avatar
  • 157
4 votes
1 answer
605 views

Characterization of the Laplace Transform

One of the main properties of the Laplace transform is given by the convolution theorem. $$\mathcal{L}(f*g)=\mathcal{L}(f)\cdot\mathcal{L}(g)$$ Question: Is there a full characterization of the ...
Henrique de Oliveira's user avatar
3 votes
1 answer
601 views

Is inverse Laplace Transform of a power of $s$ a positive function?

It's trivial that the Laplace Transform of a positive function is a positive function on $s$ domain. What about the inverse thought? What can we say about the positiveness of the inverse Laplace ...
Quiet_waters's user avatar
3 votes
0 answers
140 views

Monotonicity of a function defined by an integral

The question below is motivated by the related question Integral of a function changing sign and the associated answer: Can we study the monotonicity of the following function on $(0,1)$? $$\small f(x)...
Migalobe's user avatar
  • 395
2 votes
0 answers
164 views

Regularization of the area under hyperbola

So, I am trying to find the regularized value of the divergent integral $I=\int_1^\infty \sqrt{x^2-1}dx$. Since the area of $\int_0^1 \sqrt{1-x^2}dx=\frac\pi4$, I wonder whether the area under ...
Anixx's user avatar
  • 9,306
2 votes
2 answers
3k views

a limit of the laplace transform and its derivative

If $\phi(s)$ is the Laplace tranfrom of $f(t)$, then $\lim_{s\rightarrow \infty} s\phi(s) = f(0^+)$. and also $\lim_{\rightarrow \infty} s\phi'(s) = \lim_{t\rightarrow 0^+}tf(t)$ since $\phi'(s)$ is ...
Jose M. Del Castillo's user avatar
1 vote
2 answers
901 views

Solving a general, constant-coefficient, first-order, two-indep-variable system of PDEs

I have the following system of PDEs that I want to solve as "analytically" as possible: $$\left(\partial_t + A\partial_x + B\right)\mathbf{u}(t, x) = 0,$$ where $A$ and $B$ are constant, ...
Josh Burkart's user avatar
1 vote
2 answers
273 views

Can we meaningfully ascribe values to these divergent integrals?

My gut feeling is that $\int_0^\infty (1-\frac1{x^2})dx=0$ $\int_0^\infty (x-\frac2{x^3})dx=0$ $\int_0^\infty (x^2-\frac6{x^4})dx=0,$ etc, and in general, $\int_0^\infty (x^k-(k+1)!x^{-(k+2)})dx=0,$ ...
Anixx's user avatar
  • 9,306
1 vote
1 answer
1k views

Inverse Laplace transform to get CDF

I have the following problem. If I can get some help, I would greatly appreciate it. I am trying to replicate a particular research paper and came across a problem: Suppose $X$ is a birth death ...
turtle_in_mind's user avatar
0 votes
1 answer
202 views

Laplace transform injectivity for different values of $p$

Let $y\in L^{2}(0,1)$ and let $\widetilde{y}$ be its extension on $(0,\infty ).$ Assume that there exist $p_{0},p_{1}\in %TCIMACRO{\U{2102} }% %BeginExpansion \mathbb{C} %EndExpansion ,$ $p_{0}\neq ...
Gustave's user avatar
  • 545
0 votes
1 answer
211 views

What is a sufficient condition for summability of formel power series? [closed]

There are several kind of summability , i accrossed differents conditions for applying for example Borel summation or laplace transform which let me mixed and confused , really i don't know if i have ...
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