# Questions tagged [inverse-laplace]

This tag is for questions regarding to "Inverse Laplace Transform" which is the transformation of a Laplace transform into a function of time.

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### Finding the function $g(t)$ given its Laplace transform $F(1/s)$

I am trying to find a function g(t) given that its Laplace transform is F(1/s), where F(s) is the Laplace transform of another function f(t). I know that if f(t) has the Laplace transform F(s), then f(...
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### Inverse Laplace transform of $\frac{s}{2\sqrt{s^2-1}}e^{-\sqrt{s^2-1}|k|}-\frac{e^{-s|k|}}2$

To compute the Fourier transform described in this question, I need to perform the inverse Laplace transform of $$\tilde F_1(s)=\frac{s}{2\sqrt{s^2-1}}e^{-\sqrt{s^2-1}|k|}-\frac{e^{-s|k|}}2$$ with ...
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### Laplace Transform of a Second Order Systems Response to Triangular Pulse

I've been trying to derive the time domain response of a second order system to triangular pulse input using Laplace Transformation but even if I seem to be able to derive it Simulink simulations ...
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### Continuity of Inverse Laplace transform on suitable subspaces

I came across of the following relation. Taking a sum of exponentials (SOE) $E(x)=\sum_{j=1}^N b_j e^{\gamma_j x}$, with $b_j\in \mathbb C$ and $\gamma_j \in \mathbb C_-$ (left part of the complex ...
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### Laplace transforms coinciding on a vertical line

It is a well known result that if two Laplace transforms $L(f_1)$ and $L(f_2)$ coincide for a periodic set of values along an horizontal line, then it follows that $f_1=f_2$ a.e. I was wondering if I ...
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### Find the inverse Laplace transform of F(s) = 1/(s+exp(-sτ)), where τ is a positive real parameter.

I'm looking for the inverse Laplace transform of $$F(s) = \frac{1}{s + e^{-s\tau}}$$ where τ is a positive real parameter. I am trying to use general inverse formula of Laplace transformation to solve ...
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### The distribution of the first hitting time for the Constant Elasticity of Variance process.

The Constant Elasticity of Variance (CEV) process is a one dimensional diffusion process given by the following stochastic differential equation. d X_t = \mu X_t \cdot dt + \sigma X_t^...
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### Where did I go wrong in using the residue theorem to find the inverse Laplace transform of this function?

I used the residue theorem to solve the inverse Laplace transform of: $$f(t)=\mathcal L^{-1}\Bigg( {s e^{zs} \over (k-s)^2(k+s)^2}\Bigg) \tag 1$$ where $k$ and $z$ are non-negative. I have two poles ...
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### to find inverse laplace transform of $f(s)=s\ln\left|\frac s{\sqrt{s^2+1}}\right|$

I tried the differential property of the Laplace transform to the logarithm component, then I tried the “multiplying by s” property, but the answer would be infinity. What do you think is the solution?...
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### How to find the inverse Laplace transform of this expression?

While solving an ODE using the Laplace transform, I ran into the following problem: $$f(t) = \mathcal L^{-1}\Bigg( {se^{cs} \over (e^s+e^{-s})(k-s)^2(k+s)^2}\Bigg) \tag 1$$ where $c$ and $k$ are ...
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### Difficulty With An Inverse Laplace Transform

I am a physics student modeling a quantum mechanical system for my undergraduate research. I am currently solving for the time-dependent coefficients $c_1(t), c_2(t), c_3(t)$ on a super position of ...
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### Verifying the inverse Laplace transform for a production-inventory problem: total expected backlogs when demand is Poisson

I am entirely self-taught when it comes to Laplace transforms, and I am seeking an independent opinion on my attempt to work out how to arrive at the below expression (note: I am interested ...
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### Inverse Laplace transform of $\frac{\sinh(\sqrt z a)}{\sqrt z}$

I was trying solve this inverse Laplace transform, given by $\frac{\sinh(\sqrt z a)}{\sqrt z}$, with $a \in \mathbb{R}$. But i dont have any good idea to solve this. Please someone have a idea?
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### Stuck on Inverse Laplace transform, trying to convert convolution to ODE

I want to express a convolution of a known signal $b(t)=c(t)\ast h(t)$ as an ODE $b'(t)$ instead. The kernel $h(t)$ takes a double exponential form $Au(t)(1-e^{-T_{on} t})e^{-T_{off} t}$. After ...
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### Inverse Laplace transform of hyperbolic functions

can anyone tell me how can I seek the inverse Laplace transform of equation $F(s)=\frac{p}{s}-\frac{p}{s}\frac{sinh(a\sqrt{s})}{sinh(b\sqrt{s})}$, where $p$, $a$, and $b$ are positive constants ...
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### An inverse Laplace transform and its asymptotics

I am wondering if there is an analytic expression for the inverse Laplace transform $f(t):=\mathcal L^{-1}[F](t)$ for $F(s):=\frac1{s(\cosh{\sqrt{2s}}-1)}$. If there is no such analytic expression, ...
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### Defining a fractional derivative operator over Laplace transforms

$$\Large{ \text{Introduction} }$$ I'm interested in solving methods for fractional-differential-equations (FDEs) without specifying the nature of the fractional derivatives. In doing so, I keep coming ...
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### Inverse Laplace transform of $F(s) = \exp(-s)/s$ via its infinite series expansion

To invert $\frac{s}{{(s - 2)(s + 3)}}$ one might split it as: $$\frac{s}{{(s - 2)(s + 3)}}\; = \;\frac{A}{{(s - 2)}}\; + \;\frac{B}{{(s + 3)}}$$ solve for $A$ and $B$ and invert the fractions in the ...
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### Solve the boundary-value problem with Laplace transform

Solve $$\frac{d^2U}{dx^2} - sU = A, \qquad (0<x<1)$$ subject to the boundary conditions $$\frac{dU}{dx}(0) = 0, \qquad U(1) = 0.$$ The part I'm stuck on is that after taking Laplace transforms, ...
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### What is the inverse Laplace transform of $I_{0} (\sqrt {as+b} )$? [closed]
What is the inverse Laplace transform of $I_{0} (\sqrt {as+b} )$? Note: $I_{0}$ is the modified Bessel function of first kind with zero-order. $a$ and $b$ are constants. $s$ the Laplace parameter