# Questions tagged [laplace-transform]

The laplace-transform tag has no usage guidance.

160
questions

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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 ...

0
votes

2
answers

129
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### 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):=\...

1
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0
answers

105
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### 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
$...

10
votes

3
answers

1k
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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 ...

0
votes

0
answers

119
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### 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 <...

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(...

1
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0
answers

151
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### 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 \\ ...

33
votes

8
answers

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### 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&...

2
votes

0
answers

142
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### 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:
$...

0
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0
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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(...

0
votes

1
answer

368
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### 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'...

5
votes

1
answer

372
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### 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 ...

3
votes

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}\...

0
votes

1
answer

112
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### 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 ...

0
votes

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) ...

3
votes

1
answer

121
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### 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, \...

1
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0
answers

60
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### 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-...

1
vote

1
answer

54
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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 ...

0
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1
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267
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### 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{...

0
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0
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### 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, \, \...

0
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0
answers

110
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### 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$...

1
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0
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111
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### 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 ...

1
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0
answers

118
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### 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 ...

5
votes

1
answer

317
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### 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 ...

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}{...

2
votes

1
answer

186
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### 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 ...

1
vote

1
answer

121
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### 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}...

1
vote

1
answer

253
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### 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 ...

0
votes

0
answers

92
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### 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}...

1
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0
answers

59
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### 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{...

2
votes

1
answer

123
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### 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) = \...

4
votes

1
answer

517
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### $\frac {f (0)}{2}+ \sum_{k=1}^{\infty}f (k)=\sum_{n=-\infty}^{\infty} \mathcal{L} \{ f \} (2 \pi i n)$

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 ...

0
votes

1
answer

123
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### 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, \...

1
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1
answer

74
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### 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 ...

3
votes

0
answers

142
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### 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)...

0
votes

1
answer

126
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### 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 ...

3
votes

1
answer

318
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### 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}-...

2
votes

0
answers

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### 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 ...

5
votes

1
answer

459
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### 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) \...

6
votes

2
answers

328
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### 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 \...

1
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0
answers

178
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### 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 ...

5
votes

2
answers

184
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### 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 ...

1
vote

1
answer

75
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### 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 ...

6
votes

1
answer

233
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### 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 ...

-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] = ...

0
votes

1
answer

114
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### 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}{...

0
votes

1
answer

210
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### 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 ...

1
vote

0
answers

36
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### 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 ...

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 ...

2
votes

1
answer

185
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### 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
...