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Inverse Laplace transform dependent on a parameter

I have to evaluate the inverse Laplace transform of a function of the type $F(s-a)/F(s)$. Clearly, if $a=0$ the solution is the impulse function. If $a$ is not equal to zero, I can compute the ...
Analytic's user avatar
0 votes
2 answers
129 views

Is a signed measure $\mu$ on $\mathbb{R}^d$ characterized by the transform $\mathcal{L}_\mu (\lambda ):=\int e^{\langle \lambda,x\rangle }\mu (dx)$?

In the book "Probability Theory" by Achim Klenke there's the following theorem: a finite measure $\mu$ on $[0,\infty )$ is characterized by its Laplace transform $\mathcal{L}_\mu(\lambda):=\...
noob's user avatar
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1 vote
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105 views

Inverse Laplace transform of the Gaussian hypergeometric function $_{2}F_{1}(a,b,;c;x)$

I want to calculate the inverse Laplace transform of the Gaussian hypergeometric function $_{2}F_{1}(a+p,b,;c;-\omega)$ in which $p$ is the Laplace variable. The inverse Laplace transform is given by $...
Dante's user avatar
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10 votes
3 answers
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What is the intuition behind applying the Mellin Transform to prime distribution?

I am an undergraduate student writing an expository thesis on the complex-analytic proof of the Prime Number Theorem. I understand that applying the Mellin Transform to the partial sum of the van ...
onionbread's user avatar
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0 answers
119 views

Counterexamples in Laplace transforms

Of all the examples I know of (bilateral) Laplace transforms $F$ defined on their maximal vertical strips $V_{a,b}=\{ z \in \mathbf{C} : a < \operatorname{Re}(z) < b \}$ with $-\infty < a <...
proofromthebook's user avatar
1 vote
1 answer
99 views

Examining the Hilbert transform of functions over the positive real line

$\DeclareMathOperator\supp{supp}$Let $H:L^{2}(\mathbb{R})\to L^{2}(\mathbb{R})$ be the Hilbert transform. Let suppose we have a compaclty supported function $f \in L^{2}(\mathbb{R})$ such that $\supp(...
Gabriel Palau's user avatar
1 vote
0 answers
151 views

Laplace transform

\begin{equation} \begin{cases}\mathbb{D}_t^\beta u(x, y, t)=-a(x)\left(u_x(x, y, t)+u_y(x, y, t)\right)+\ell(x, y, t, u(x, y, t)), & x>0, y>0, t>0 \\ u(x, y, 0)=0, & x>0, y>0 \\ ...
TUHOATAI's user avatar
33 votes
8 answers
3k views

Motivation and physical interpretation of the Laplace transform

Concerning the one-sided Laplace transform, $$\mathcal{L}\{f\}(s) = \int_0^\infty f(t)e^{-st} dt$$ what is a motivation to come up with that formula? I am particularly interested in "physical&...
AlpinistKitten's user avatar
2 votes
0 answers
142 views

Beyond Watson's lemma

Suppose $f:[0,1]\rightarrow \mathbb{C}$ is a smooth function, which I wish to approximate near $0$. Watson's Lemma implies that I can find a smooth function $a:[0,1]\rightarrow \mathbb{C}$ such that: $...
SnowRabbit's user avatar
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Computing the Laplace transform of an expression

I would like to find the Laplace transform of the following expression with respect to the Laplace parameter s $ \int_{z=u}^{\infty} e^{-az/c} g^{'}(\dfrac{z-u}{c}) \int_{x=0}^{\infty} \varphi(z-x)dF(...
Rosy's user avatar
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0 votes
1 answer
368 views

Necessary conditions for convergence of convolution

In math.SE, I've asked a question about the convergence of convolution of two functions which have bilateral Laplace transform and also have disjoint Region Of Convergence (ROC) but the question didn'...
S.H.W's user avatar
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5 votes
1 answer
372 views

How to evaluate inverse Laplace transform of $e^{- \sqrt{s}} $ using series?

I tried to find an inverse Laplace transform by series as follows $$ f(t)=L^{-1}_s\left(e^{-\sqrt{s}}\right)(t)=L^{-1}_s\left(\sum_{k=0}^{\infty}\frac{(-1)^k}{k!} s^{\frac{k}{2}}\right)(t)$$ and by ...
Faoler's user avatar
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1 answer
245 views

Laplace transform of Brownian motion functional

Let $(B_r,r\geq 0)$ be a standard Brownian motion on $\mathbb{R}$ started at $0$. I am interested in the quantity $$g(s,t) = \mathbb{E}_0\left[ \exp \left(- \beta \int_s^t \left\vert \frac{B_r}{r}\...
David Geldbach's user avatar
0 votes
1 answer
112 views

Explanation for Tauberian theorems for Laplace transform

I am struggling with the following theorem in Feller's book "Probability Theory and its Applications". The tauberian theorem is written as follow : Let $F : [0,\infty) \to \mathbb{R}$ of ...
NancyBoy's user avatar
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0 answers
86 views

Inverse Laplace of the Complex conjugate of the Laplace transform

Let the Laplace transform of f(t) be F(s) and let the inverse Laplace transform of F(s*) be g(t). is there a theorem relating f(t) and g(t)? Basically, looking for a way to calculate g(t) from f(t) ...
cortadisimo's user avatar
3 votes
1 answer
121 views

Thinning of (mixed) binomial point process

Let $N= \sum_{i=1}^M \delta_{X_i}$ be a mixed Binomial process over $(\mathbb X, \mathcal X)$. I.e., $M$ is a $\mathbb Z_+$ valued random variable with probability mass function $q_M(m)$, $m=0, 1, \...
mariob6's user avatar
  • 133
1 vote
0 answers
60 views

Infinite dimensional version of the Laplace transform and Gaussian integrals

This question is somehow related to my previous one Convergence of the Gaussian integral on $\mathcal{E}'$ for a mapping supported on $L^2$ Let $F : L^2(S^1) \to L^2(S^1)$ be a (nonlinear) Borel-...
Isaac's user avatar
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1 vote
1 answer
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Possibility of bounding one functional by another functional (under certain constraints)

Suppose that we consider a class of $L^2(\mathbb{R}_+)$ functions $h$ such that $h$ can be expressed as a difference of two cumulative distribution functions $F$ and $G$ (whose corresponding densities ...
Fei Cao's user avatar
  • 720
0 votes
1 answer
267 views

Equivalence between the $L^2$ norm and the $L^2$ norm of Laplace transform

It is well-known that the Laplace transform, defined by $$\mathcal{L} \colon f(x) \in L^2(\mathbb{R}_+) \to \hat{f}(\xi) \in L^2(\mathbb{R}_+)$$ via $$\hat{f}(\xi) = \int_{\mathbb{R}_+} f(x)\,\mathrm{...
Fei Cao's user avatar
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0 answers
44 views

A condition for complete monotonicity

In D. V. Widder, The Laplace Transform, Chapter III, The Moment Problem, given a sequence $(\mu_n)_{n=0}^\infty$, it is defined that $$\lambda_{k,m}:= {k\choose m}(-1)^{k-m}\Delta^{k-m}\mu_m, \, \...
Hans's user avatar
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0 answers
110 views

Laplace transform of a stochastic process

Let $R := (R_1, R_2)$ be a two-dimensional diffusion process defined by the following SDE: $$\mathrm{d}R_{1,t} = -\lambda_1 R_{1,t} \, \mathrm{d}t + \lambda_1 \sigma(R_{1,t}, R_{2,t}) \, \mathrm{d}W_t$...
Greyearl's user avatar
1 vote
0 answers
111 views

Relating $f(x)$ to its Laplace Transform for values other than $x=0$?

Suppose $X\in (0,1]$ is a random variable where $f(x)$ is its CDF and $g(t)$ is the Laplace Transform of $f(x)$. Tauberian theorems (Theorem 2.3 in Coqueret's "Approximation of probabilistic ...
Yaroslav Bulatov's user avatar
1 vote
0 answers
118 views

Contour integral with two essential singularity

I'm solving problems on the Gamma random variables and there is this question where it wants me to calculate the Mellin transform of sum of two independent Gamma variables from their moment generating ...
K.K.McDonald's user avatar
5 votes
1 answer
317 views

Long tail property of Laplace transforms

A function $F: \mathbb R_+ \rightarrow \mathbb R_+$ is said to be long tailed if $F(\infty)=0$ and for all $y \geq 0$ $$\frac{F(x+y)}{F(x)} \rightarrow 1, \quad x\rightarrow \infty.$$ Let $\mu$ be a ...
Mr_3_7's user avatar
  • 105
5 votes
1 answer
206 views

Upper bound for an inverse Laplace transform

Can anyone see how to get a tight upper bound for the function defined in terms of the inverse Laplace transform below? $$g_\text{sgd}(t)=\mathcal{L}^{-1}\left[\left(\frac{s}{2}+\frac{5 \sqrt{\frac{2}{...
Yaroslav Bulatov's user avatar
2 votes
1 answer
186 views

2D lattice sum with numerator

I've been struggling a bit with a double sum that arose as the trace of an operator: $$\sum_{(j,k)\in Z^2 \setminus (0,0)} \frac{(j+k)^n}{(j^2+k^2)^n},$$ where $n$ is an even natural number. Is there ...
R Grady's user avatar
  • 31
1 vote
1 answer
121 views

Which kind of convergence can we get from Laplace transform convergence?

This question is a related question see this post Vague convergence VS Laplace transform convergence. But now we assume that \begin{equation} \int_0^\infty e^{-sx}\mu_n(dx)\to \int_0^\infty e^{-sx}...
Fractional analysics's user avatar
1 vote
1 answer
253 views

Vague convergence VS Laplace transform convergence?

If we assume that $\int_0^\infty e^{-sx}\mu_n(dx)\to \int_0^\infty e^{-sx}\mu(dx), \forall s\geq0$, it is possible to show that $\mu_n\to\mu$ vaguely. Where $\mu_n$ is a measure. Please check here for ...
Fractional analysics's user avatar
0 votes
0 answers
92 views

Modeling decay of a linear system with a mixing term

I'm trying to analyze convergence of $x\in \mathbb{R}^{+d}$ which follows the following recurrence $$\mathbf{x}\leftarrow (\mathbf{1}-\mathbf{h})^2 \mathbf{x} + \mathbf{h}\langle \mathbf{x}, \mathbf{h}...
Yaroslav Bulatov's user avatar
1 vote
0 answers
59 views

Upper bounds on Inverse Laplace transform of a rational function

I need the inverse Laplace transform $\mathscr{L}^{-1}$ or a nice upper bound on $\mathscr{L}^{-1}$ for the following function: $$f(y)=\frac{\left(\sum_i \frac{u_i}{y-a_i}\right)^2}{1-\sum_i u_i \frac{...
Yaroslav Bulatov's user avatar
2 votes
1 answer
123 views

Injectivity of two sided Laplace transform

Let $\mu,\nu$ be finite Borel measures on $\mathbb R$. Assume that there is an open interval $(a,b)$ on which the Laplace transforms exist and coincide: $$ \int_{-\infty}^\infty e^{-tx}\,d\mu(x) = \...
Lau's user avatar
  • 749
4 votes
1 answer
517 views

$\frac {f (0)}{2}+ \sum_{k=1}^{\infty}f (k)=\sum_{n=-\infty}^{\infty} \mathcal{L} \{ f \} (2 \pi i n)$

I obtained the very strange formula above and at begining I was just wanted know how to interpretate it. But now when I know what is this with help of @Carlo Beenakker, I am leaving it as a proof. BTW ...
Wreior's user avatar
  • 161
0 votes
1 answer
123 views

What kind of functions can be represented as infinite linear combinations of exponential functions?

Let $f(x)$ be a real-valued function defined in $(0, \infty)$. I am curious what kind of $f(x)$ has the following representations: $$ f(x) = \sum_{j=0}^\infty a_j e^{-jx}, \quad \forall x \in (0, \...
fs l's user avatar
  • 81
1 vote
1 answer
74 views

Discrete uniqueness sets for the two-sided Laplace transform?

Let $f : \mathbb R_+ \to \mathbb C$ be a measurable and integrable function where $\mathbb R_+ = [0,\infty)$. The Laplace transform of $f$ is given by $$ Lf(s) = \int_0^\infty f(x)e^{-sx} \, dx. $$ A ...
r_l's user avatar
  • 190
3 votes
0 answers
142 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
0 votes
1 answer
126 views

Approximate inverse Laplace transform in terms of the moments of a function

If $F(s)$ is the Laplace transform of $f(t)$ and \begin{equation} F(s)=\frac{1}{1-aG(s)} \end{equation} where $G(s)$ is the Laplace transform of a known probability density distribution $g(t)$ whose ...
papad's user avatar
  • 272
3 votes
1 answer
318 views

Integral of a function changing sign

By some numerical tests, we can see that the following function is negative on $(0,1)$: $$\small f(x)=\int_0^\infty\frac{s^{x-1} e^{-2 s} (\pi \cos(\pi x) (s^{2 x}+(0.1)^2)-\sin(\pi x) \ln(s) (s^{2 x}-...
Migalobe's user avatar
  • 395
2 votes
0 answers
102 views

Evaluate action of $f(\frac{d}{dx})$ using the Fourier/Laplace transform

Consider a function $f(x)$ that is numerically defined in $-1 \leq x \leq 1$ interval (assume $N$ samples). I am trying to compute the action of $f(d/dx)$ on a function $g(x)$ using the Fourier ...
Mirar's user avatar
  • 350
5 votes
1 answer
459 views

Hardy–Littlewood Tauberian theorem for Laplace transform

The Hardy–Littlewood Tauberian theorem for Laplace transform in Chapter XIII in "An Introduction to Probability Theory and Its Applications" by Feller reads as follows Let $F : [0,\infty) \...
mnmn1993's user avatar
6 votes
2 answers
328 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
  • 4,103
1 vote
0 answers
178 views

Discretizing a differential operator which is a function of the derivative operator

Assume that $p(x)$ and $f(x)$ are sufficiently smooth functions and $D\equiv \frac{d}{dx}$. My question is concerned with the discretization of $p(x+D)f(x)$. As an example, let $p(x)=x^{2}+2x$. Then ...
Mirar's user avatar
  • 350
5 votes
2 answers
184 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
  • 695
1 vote
1 answer
75 views

Initial and final Theorem for upper and lower limits?

Let define $F(s)=\int_0^\infty f(u)e^{-su}du$. If $f$ is bounded and $\lim_{t\to 0}f(t)$ exists. Then we can get $\lim_{t\to 0}f(t)=\lim_{s\to\infty}sF(s)$. Can we use upper limits or lower limits to ...
Fractional analysics's user avatar
6 votes
1 answer
233 views

Convergence speed of the tail of distribution using Tauberian remainder theorem

This question may be related to this one. Now I try to make some statistical estimator using Laplace transform, but I face the following serious problem. Let $f$ be some one-sided probability ...
Seung Hyeon Yu's user avatar
-1 votes
1 answer
61 views

Finding the nth value of a dual series [closed]

I have a problem which i remember solving using Z transform in my uni. time, but i don't recall the EXACT way. I have 2 series of numbers: X[n] and Y[n] X[0] = Y[0] = 1 X[n+1] = aX[n] + bY[n] Y[n+1] = ...
Guy Barash's user avatar
0 votes
1 answer
114 views

Unique zero solution to a difference equation via Laplace transform

We want to prove that the unique solution to the following difference equation is the null one: $$ au(x)+b\mathbf{1}_{(0,\frac{1}{2})}(x)u(x+\frac{1}{2})+c\mathbf{1}_{(\frac{1% }{2},1)}(x)u(x-\frac{1}{...
Gustave's user avatar
  • 545
0 votes
1 answer
210 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
1 vote
0 answers
36 views

Solving an equation containing Laplace transform

Consider the equation \begin{equation} \frac{f(p)}{f(s_{1})}\mathcal{L}(y)(s_{1})+\frac{g(p)}{g(s_{2})}\mathcal{L}% (y)(s_{2})=\mathcal{L(}y)\mathbf{(}p), \end{equation} where $\mathcal{L}$ is the ...
Goga's user avatar
  • 47
2 votes
1 answer
533 views

What is the integral representation of the exponential function $e^{1/t}$ on $(0,\infty)$?

A function $q(x)$ is said to be completely monotonic on an interval $I$ if $q(x)$ has derivatives of all orders on $I$ and $(-1)^{n}q^{(n)}(x)\ge0$ for $x\in I$ and $n\ge0$. See Chapter 1 in the ...
qifeng618's user avatar
  • 942
2 votes
1 answer
185 views

Inverse Laplace transform of $\frac{1}{s^a + 1}$ with $0 < a \leq 1$

Problem I am looking for the following inverse Laplace transform, $$ f(t) = \mathcal{L}^{-1}\left[\frac{1}{s^a + 1}\right] \;\;\;\;\; \text{with} \;\;\;\;\; 0 < a \leq 1. $$ What I understand ...
Keisuke FUJII's user avatar