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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22 views

How to realize the ILT of $\frac{1}{(-1+e^{2\sqrt{s}})s^{3/2}+(1+e^{2\sqrt{s}})s\lambda}$?

Please explain how to realize the ILT from s to t? $$\frac{e^{2\sqrt{s}x}(-F+u_0\lambda)}{(-1+e^{2\sqrt{s}})s^{3/2} +(1+e^{2\sqrt{s}})s\lambda} -$$ $$\frac{e^{2\sqrt{s}-\sqrt{s}x}(-F+u_0\lambda)}{(-1+...
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51 views

Inverse Laplace transformation involving exponential function

I am working on a project related to SH type of waves in layered media. and i got stuck at finding the inverse laplace $\displaystyle\exp{\left(-z\sqrt{\frac{s^2}{\beta^2}+k^2}\right)}$ where $k\; ,...
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1answer
39 views

Using Inverse Laplace to find the frequency response of a transfer function - Help needed!

The frequency response is the inverse Laplace transform of a transfer function. I am tasked to apply the inverse Laplace on the transfer function below in order to convert it into the time domain. $$...
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12 views

Bromwich contour of inverse transform

I have a given equation $$F(s)=b/(s-a)^2+b^2$$ where a and b are real numbers. While I was able to find two roots which are $$sqrt(a)$$ and $$-sqrt(a)$$. However I am stuck on algebra part, where I ...
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35 views

Find the inverse laplace transform of $\mathcal{L^{-1}}$

If $\mathcal{L^{-1}}=\frac{2-2e^{-5s}}{s(s+1)^2}+\frac{e^{-6s}}{(s+1)^2}$, how do I find the inverse Laplace transform? I see that there are fractions for this equation, so I have to use partial ...
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18 views

How to obtain the pole and the order of a complex function with a quotient of Bessel functions

I am trying to determine the Inverse Laplace Transformation of the following function. \begin{equation} p(\zeta) = \ \frac{ K\sqrt{{I_{2}{(\sqrt{\zeta})}}}} { \sinh\left[\sqrt{\frac{I_{0}{(\sqrt{\...
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1answer
49 views

Find the inverse laplace transforms of $\frac{\ln(s)}{s^2}$

I have the equation $\mathcal{L}^{-1}=\frac{\ln(s)}{s^2}$. How do I find the Laplace inverse transform? I know that $\mathcal{L}[\ln(t)]=-\frac{1}{s}(\gamma+\ln(s))$, so I need to probably transform ...
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1answer
54 views

Inverse Laplace transform (double integral)

Is it possible to find a function $F:\mathbb R^+\times \mathbb R^+\to\mathbb R$ such that \begin{align} \int_0^\infty\int_0^\infty F(x,y)e^{-xq-yq}dxdy=\frac{\alpha}{q^4}+\frac{\beta}{q^3}+\frac{\...
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15 views

Convergence speed of the tail of distribution using Tauberian remainder theorem

This question is related to this. Now I try to make some statistical inference using Laplace transform, but I face the following problem. Let $f$ be some one-sided probability distribution defined on $...
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30 views

Inverse Laplace transformation of a rational function [closed]

I just don't seem to be able to crack this inverse Laplace: $$ \mathcal{L}^{-1} \left\{ \frac{20000s}{s^2 + 20000s + 5\cdot 10^8} \right\} (t). $$ Could someone help me out? I'm totally lost. I don't ...
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34 views

How to take inverse Laplace transform of this expression?

I am trying to take inverse Laplace transform of this equation: $$ Y(s)=\frac{\omega_{n}^{2}}{s^{2}+2 \zeta \omega_{n} s+\omega_{n}^{2}}\times\frac{s}{s^2 + \omega^2} $$ Can I obtain partial fraction ...
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1answer
112 views

An exercise in "Mathematical Statistics Jun Shao" about the completeness of a 'modified' exponetial family

It is not the first time meeting this problem in StackExchange and I have read the answer to it(the original solution is copied at the bottom, also available in Show a statistic is complete but not ...
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18 views

How was this partial fraction solved with 2 variables?

How was this partial fraction decomposition done? partial fraction image
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23 views

Usage of Laplace transform in showing equality of two function.

I want to solve the following problem: given $$ f(x)=\int_{0}^{\infty} \delta\left(\sum_{i=1}^{n} \theta_{i}-x\right) \prod_{i=1}^n \theta_{i}^{u_{i}-1} d \theta_{1} \ldots d \theta_{n}, \; u_i > 0 ...
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22 views

How do I change the output of Matlab when using inverse Laplace?

Here is my input: syms s F = (2+2*s*(exp(-s))+4*(exp(-2*s)))/(s^2)*(s^2+2*s+10); ilaplace(F) An the output is : ...
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77 views

How to calculate ILT of an expression with terms $e^{\sqrt{s}}$ by residue theorem?

In order to realize the inverse Laplace transform of $$H(s)=\frac{1}{\sqrt{s}(-m+b\ s)}\frac{ \exp{[(r+r_0)\sqrt{s}/\sqrt{D}]} + \exp{[(-r+r_0+2 r_x)\sqrt{s}/\sqrt{D}}]}{\exp{(2r_0\sqrt{s}/\sqrt{D})}-\...
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26 views

DIfferential equation, transforming $\sigma(t-1)$

$y''(t)+5y'(t)+6y(t)=\sigma(t-1), y(0)=1, y'(0)=-1$ I'm confused how I am supposed to handle $\sigma(t-1)$?
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50 views

Trouble 'reversing' $\frac{dv(t)}{dt}=L\,\frac{d^2i(t)}{dt^2}+R\,\frac{di(t)}{dt}+\frac{1}{C}\,i(t)$

I'm having trouble with this differential equation. I would like to reverse it, and, in particular, express it in terms of $i(t)$ with on the other side some linear (integro-)differential operator on $...
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292 views

Is the reciprocal of the inverse tangent function, $\frac1{\arctan x}$, a (logarithmically) completely monotonic function on $(0,\infty)$?

A non-negative function $f$ is said to be completely monotonic on an interval $I$ if $f$ has derivatives of all orders on $I$ and \begin{equation*} 0\le(-1)^{n-1}f^{(n-1)}(x)<\infty \end{equation*} ...
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43 views

Using the theorem of convolution find the inverse Laplace transform

Using the theorem of convolution find the inverse Laplace transform: \begin{align} F(s)=\frac{s}{(s-1)(s+2)}\end{align} This is what I have so far: \begin{align} Let \space F_1(s)=\frac{s}{s-1} and\...
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36 views

Is the inverse Laplacian bounded in $\mathbb{R}^{2}$?

I'm searching for an inequality in the form $$\forall s>2,\quad\forall u\in H^{s-2}(\mathbb{R}^{2}),\quad\|\Delta^{-1}u\|_{H^{s}(\mathbb{R}^{2})}\lesssim\|u\|_{H^{s-2}(\mathbb{R}^{2})}$$ where $H^s$...
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15 views

Inverse Laplace transform of inverse function containing logarithm

I am trying to find the inverse Laplace transform of the following function but am currently stumped. My understanding is that I can't formally solve the integral because of the singularity at $s=0$, ...
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20 views

Inverse Laplace transform of infinite series

I am trying to find the inverse Laplace transform of $$ f(s)=\frac{1}{{1 - \frac{1}{{\left( {\left( {s + a} \right)\left( {s + b} \right)\left( {s + c} \right)} \right)}}}}, \left| {\frac{1}{{\left( {...
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1answer
57 views

Whats the partial fraction for $f(x) = (-2s^2 -1)/(s^2(s^2+2))$ [closed]

Im trying to find the answer for the following inverse laplace, but when doing that $A/s$ + $B/s^2$ + $(Cs+D)/(s^2+2)$ , Im finding that b = -1/2 ; d= -3/2 but a and c are undefinied.
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1answer
46 views

How to find general form of the inverse laplace transform?

I've been having trouble with understanding how to properly do partial-fraction decomposition on the inverse Laplace transform of $$ F(s) = \frac{s+1}{(s^2+1)^2} $$ I tried using the complex factors $$...
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2answers
78 views

Finding the inverse Laplace transform of $Y(s)=\frac{1-e^{-{\pi}s}}{(s^2+1)(s^2+4)}+\frac{s+1}{s^2+4}$

I've encountered an IVP, and I need to solve it by applying the Laplace transform. The problem is: $$y^{''}+4y=g(t), y(0)=y^{'}(0)=1$$ where $g(t)=\sin(t)$ for $0\leq{t}\leq\pi$ and $g(t)=0$ for $\pi\...
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1answer
27 views

How do you find the Inverse Laplace transformation for a product of two general functions?

If a function for s-domain is defined as, $$\frac{F(s)}{(s^2+\nu^2)}$$ how can I perform inverse laplace transform for above function? I think it can be transformed into time domain function, but I ...
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1answer
19 views

Verifying inverse Laplace transformation of an expression

I tried solving the next inverse Laplace transformation myself: $$f(t)=L^{-1}\Bigl({{s}\over {s-C}}X(s)\Bigl)={dx(t)\over{dt}}*e^{Ct}$$ but I am not sure if it is correct. I can not find a similar ...
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3answers
68 views

Laplace Inverse of $\frac{s+1}{s^2 + 2s}$ [closed]

Here I have $$F(s) = \frac{s+1}{s^2 + 2s}$$ Taking Laplace inverse on both sides, $$\mathcal{L}^\prime \{F(s)\} = \mathcal{L}^\prime \left(\frac{s}{s^2+2s}\right) + \mathcal{L}^\prime \left(\frac{1}{s^...
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0answers
74 views

Inverse Laplace transform of the Riemann zeta function

I wonder whether the inverse Laplace transform of the Riemann zeta function be obtained on the domain $s \in \mathbb{R}_{>1}$. Are any results on this known? In order to solve the problem, I tried ...
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1answer
127 views

How "practical" is the Laplace transform method for constant coefficient ODE?

I just finished teaching a chapter on using Laplace transform to solve constant coefficient second order linear differential equations. I touted how amazing the method was because it incorporates the ...
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1answer
47 views

Inverse Laplace transform of this transfer function

$$F(s)=\frac{125(s+8)}{(s^2+12s+136)(s+0.5)}$$ which becomes $$Y(s) = U(s)F(s)=\frac{125(s+8)}{s(s^2+12s+136)(s+0.5)}$$ I used partial fractions to obtain the following $$\frac{125(s+8)}{s(s^2+12s+136)...
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1answer
33 views

Inverse laplace transform of the expression

I am struggling to find the inverse Laplace transform of this expression $$X(s) = \frac{e^{-st_0}}{\omega^2 + s^2}$$ The inverse Laplace transform of the numerator is $\delta(t-t_0)$. However, I don't ...
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20 views

Inverse Laplace transform of Incomplete Gamma function

I have a Laplace transform of a function $f(x)$ I would like to obtain, i.e. $$ \mathcal{L}_f(s)=\Gamma\left(1-N,\frac{1}{s}\right)\frac{1}{s^N}e^{\frac{1}{s}} $$ with $N$ positive integer. ...
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57 views

Inverse Laplace transform through contour integration

$$ F(s)=\frac{\sin(s)}{\sqrt{s}} $$ Does the inverse laplace transform of this function exist? But how do we find this through contour integration? Any suggestions please. Because as this is special ...
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0answers
46 views

Proving $\mathcal{L}^{-1}(0) =0$ by definition of the Laplace transform.

I have seen a proof of $\mathcal{L}^{-1}(0) =0$ here: Inverse Laplace Transform of zero, but I would like to prove it by definition. We have $\int_0^\infty e^{-st}f(t)dt=0$. Here, I would like to say ...
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53 views

Inverse Laplace Transform of $\dfrac{1}{s^a+b}$

I could not find on tables, but trying some values I think that is something with $ ERFC$ functions. I am seeking for a general formula to the inverse Laplace transform of $\dfrac{1}{s^a+b},$ where $a$...
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102 views

Inverse Laplace transform of $\frac{\sqrt{s^2-2c^2s}}{2sc} e^{-(a/c)\sqrt{s^2-2c^2s}}$

Question: What is the ILT of $$H(s)=\frac{\sqrt{s^2-2c^2s}}{2sc} e^{-(a/c)\sqrt{s^2-2c^2s}},$$ where $a \geq 0$ and $c>0$. That is, how can we compute $$\frac{1}{2\pi i} \int_{\gamma -i\infty}^{\...
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2answers
41 views

Find the analytical formula of the function (picture included)

So we're practicing Laplace transforms and I think I came across a rather unorthodox question. The question is: Find the analytical formula of the function: NOTE $h(t)$ is the Heaviside function. a) $...
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1answer
34 views

Partial Fraction Decomposition for the Inverse Laplace Transform

I need to do the inverse Laplace transform of such fraction: $$\frac{\frac{3}{9 + (-1 + p)^2}+p-3}{p^{2} -2p+10}$$. I've thought of rewriting it as $$\frac{\frac{3}{9 + (-1 + p)^2}+p-3}{p^{2} -2p+10}=\...
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0answers
41 views

Homogeneous semi-infinite wave equation with inhomogeneous boundary and initial conditions

On page 115 of the third edition of Logan's Applied Partial Differential Equations, the reader is asked to solve the following: $$u_{tt} = u_{xx}, \,\,\: x,t>0$$ $$ u(0,t) = \sin t ,\,\,\: t\ge 0 $$...
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1answer
59 views

Inverse Laplace Transform of exponential fraction

I need to inverse Laplace transform this function: $$\frac{1-e^{-2s}}{s(s+a)}$$ (where a is any real, positive constant) So far I only have that it is equivalent to: $$\frac{1}{s(s+a)} - \frac{e^{-2s}}...
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2answers
124 views

Inverse Laplace transform calculation with steps

I need to calculate with steps this inverse laplace transform $\mathcal{L} [(3s+2)/(4s^2+12s+29)*e^{-2s}]$ Unfortunately,I cant factor the dividor and I tried different ways to spit it up (creating ($...
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1answer
66 views

What is the inverse Laplace transform of $\frac{\sqrt{1+s}}{\sqrt{s}+a}$?

I am trying to inverse Laplace transform of $\frac{\sqrt{1+s}}{\sqrt{s}+a}$ where a is a complex constant. Here are the three facts I get: $$ \mathcal{L}_{s}^{-1}\left[\frac{1}{a+k s^{\alpha}}\right](...
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30 views

How can I find the laplace of $F(s) = {1\over \sqrt{s^2+4} }$?

$$F(s) = {1\over \sqrt{s^2+4} }$$ I know that if it didn't have the srqt, the inverse would be f(t) = $sin (2t)\over 2$. But can't find the inverse above. Could you help me, please?
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24 views

Inverse Laplace Transfrom of $F(s^{1/n})$

Let $F:s \mapsto \int_0^\infty e^{-st} f(t) dt$ be the laplace transform of a function $f\geq 0$. Is there a formula for the inverse laplace transform of $F(s^{1/n})$ for a natural number $n$ ?
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16 views

How can I take the Inverse Laplace transform of the following expression?

I try to take the inverse Laplace transform of this equation; $$\frac{{a\sqrt s \left( {b - c\sqrt s } \right)}}{{s\left[ {ds + e\sqrt s } \right]\left[ {f - \sqrt {{f^2} + gs} } \right]}} + \frac{{\...
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1answer
162 views

Find the solution $x(t)$ of the differential equation $x''+3x+2x' = 0$ for the initial conditions $𝑥(0) =a$ and $𝑥'(0)= b$.

I don't know how I could solve this problem. $[s^2 X(s)-sX(0)-X'(0)]+3X(s)+2[sX(s)-X(0)]=0$ $[ s^2 X(s)-as-b ]+3X(s)+2[sX(s)-a]=0$ $X(s)(s^2+3+2s)-as-b-2a=0$ And I finally found this: $X(s)=(2a+as+b)...
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1answer
56 views

What is the inverse Laplace transform of the equation? $F(s)=\frac{1}{s\left( s^2 +s+1\right)}$?

$F(s)=\frac{1}{s\left( s^2 +s+1\right)}$ Can you help me? Is there someone to explain step by step?
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1answer
84 views

Inverse Laplace Using Heaviside Function

I have a function for which I need to both find the inverse Laplace transformation and sketch a graph. The function is $$ F(s)=\frac{2}{s^3}-\frac{4}{s^2}e^{-s}-\frac{2}{s^3}e^{-2s} $$ I've gotten as ...

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