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Questions tagged [inverselaplace]

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0
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1answer
20 views

Inverse Laplace Transform - Pulling out the constant

If you refer to my picture: https://i.stack.imgur.com/lVsU1.png I'm having a hard time understanding why in the 2nd step the fraction is split up in two terms when 2 is a constant. I get why you ...
0
votes
0answers
13 views

Inverse laplace transform using delay function

I have transfer function, G(s), that i made an partial fraction decomposition. From this i get the following equation: $\frac{2}{s+2}+\frac{2s+1}{s^2+s+1}$ Assume we have a unit impulse $u(t)=1$ and i ...
-1
votes
1answer
24 views

Inverse Laplace transformation of F(s) [closed]

Show that the laplace transformation of $F(s)= e^{-a (\sqrt s)}/s$, $a \gt 0$, is given by $$f(x)= 1 - \frac1{\pi} \int_{0}^{\infty} \frac{\sin(a \sqrt r)e^{-rx}}{r} dr$$
0
votes
0answers
17 views

Inverse Laplace of an exponential function $d \exp(ax-ax\sqrt{1+bs+c})/s$ [closed]

I want to find the solution of the inverse Laplace transform about this equation: $$f(s)=\frac{d\exp(ax-ax\sqrt{1+bs+c})}{s}?$$ Thank you very much for your time.
1
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2answers
72 views

Inverse Laplace Transform of $e^{-s^{\alpha}} $

I would like to compute the inverse Laplace of $$F(s) = e^{-s^{\alpha}}$$ For the case $\alpha = \frac{1}{2}$ I've found the following on this website: Compute the inverse Laplace transform of $e^{-...
0
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0answers
35 views

Inverse laplace of $ \ln(s+4-\frac{12}s)$

$$\ln\left(s+4-\frac{12}s\right)$$ I have done these steps but I'm stuck:
-1
votes
1answer
16 views

Prove that $\mathbb L^{-1}\{\mathbb p^\mathbb k\}=0$

There is a question in my book at the end of which it is written that $\mathbb L^{-1}\{\mathbb p^\mathbb k\}=0$ for $\mathbb k$= 0,1,2,..... But we know that $\mathbb L\{\mathbb 0\}$ = $\mathbb 0$ So ...
0
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0answers
14 views

Definition of H(0) when inversing Laplace transform results in heaviside step function

We have: $ L^{-1}\{e^{-cs}F(s)\} = H(t - c)f(t-c) $, with $ H $ is a heaviside function. In many documents, $ H(t - c) $ is defined as: $ H(t - c) = \left\{\begin{matrix} 0 &, t < c \\ 1 &...
2
votes
1answer
23 views

Is there an inverse Laplace Transform of $-a\cdot e^{-b\cdot x^c}$

Is there an inverse Laplace Transform of $-a\cdot e^{-b\cdot x^c}$, where $a,b$ and $c$ are constants, in my case its: $a=-0.9898; b=0.3511; c=0.2553$ The property, that the function tends to zero ...
1
vote
1answer
61 views

Calculate inverse Laplace transform of $\exp(\sqrt{s^2-r^2})$

I have some trouble with the calculation of the inverse Laplace transform $e^{-k\sqrt{s^2-r^2}}$ , $k\geq0$, $r$ is known. ## ## And I believe it has some relation with the inverse Laplace transform ...
0
votes
1answer
25 views

Inverse Laplace with a irreducible quadratic in the denominator and a 1 in the numerator

Please help me find the Inverse Laplace transform of: $$F(s) = \frac{1}{s(s^2+8s+4)}$$ After completing the square, I obtained $$F(s) = \frac{1}{s((s+4)^2-12)}$$ Thank you.
0
votes
0answers
31 views

Inverting a Laplace transform (for a Lévy process)

Let $\psi(\theta) = c\theta + \frac{\sigma^{2}}{2}\theta^{2} - \frac{\lambda\theta}{\alpha + \theta}.$ For those who are wondering where this function comes from, $\psi$ is the Laplace exponent for a ...
3
votes
0answers
26 views

Inverse Laplace Transform of $F(s)=\frac{1}{\sqrt{s} \coth(\sqrt{s})-1}$.

I try to find the inverse laplace transform of the function $\displaystyle F(s)=\frac{1}{\sqrt{s} \coth(\sqrt{s})-1}$. I check numerically that this function has no root in the right half complex ...
0
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0answers
27 views

Compute inverse Laplace transform

Compute the inverse Laplace transform of $$\frac{3s +1}{{s^{3}}+{4s^{2}}+{(k-3)}}$$ Already tried with partial fraction expansion, without success.
0
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0answers
35 views

Prove the inverse transform of unilateral Laplace transform

I'm reading this article and having a question. Consider a function $f$ and its Laplace transform $\hspace{3.0cm} F(s) = \int_0^\infty f(t) e^{-st} dt$, with $\{s|\text{Re}(s) = 0\} \in \text{ROC}[...
0
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0answers
21 views

Laplace inverse of functions involved with square roots

Let $ F(s) = \sqrt(s+ia)g(s) $, where $s$ is a complex number with $\Re(s) >0 $. I want Laplace inverse of $F(s) $. I have tried contour integration and convolution theorem but couldn't come up ...
1
vote
1answer
58 views

Inverse Laplace Tranform of a function involving Bessel functions

I need to evaluate (if it exists) the inverse Laplace transform of the following complex function $F(s)$: $$ F(s)=\sqrt{\frac{s}{a}} J_{1}(\sqrt{as}) $$ where $J_{1}(\cdot)$ is the Bessel function of ...
1
vote
0answers
37 views

Partial integration paradox, from odd to even integrand

I'm investigating the following integral: $$ g(y) = \int_{-\infty}^\infty \frac{f(x)}{x - i y} dx $$ where I know that $f(-x) = - f(x)$, and $\lim_{x \to \infty} f(x) = 0$. (In practice I have data ...
0
votes
1answer
33 views

A state-space representation of an integro-differential equation implies a false statement

I would like to convert the equation $\ddot{y}+\int_0^t y(\tau)d\tau=0$ to state-space representation. Below, I present my attempt, which seems to be contradicting, and then ask my question at the end....
1
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1answer
73 views

Limitations of Bromwich integral for inverting Laplace transform

Suppose: $$f(t)=e^{at}+e^{bt};\quad a>b>0$$ Its Laplace transform is: $$\mathbf{L}[e^{at}+e^{bt}]=\frac{1}{s-a}+\frac{1}{s-b}$$ for $Re(s)>a$ where $Re$ stands for the real part; for $Re(s)&...
-1
votes
1answer
62 views

Inverse Laplace transform of the Bessel function

I want to calculate the inverse Laplace transform of Bessel function: $$J_{as}(x)=\sum_{m=0}^\infty\frac{(-1)^{m}(\frac{x}{2})^{2m+as}}{m!Γ(m+as+1)}=\sum_{m=0}^\infty\frac{(-1)^{m}(\frac{x}{2})^{2m}(\...
0
votes
0answers
18 views

A general form of a solution for laplace inverse of $\frac{f^2(s)}{f^3(s)}$?

Is there any general form of inverse Laplace transform for a function like $$F(s)=\frac{f^2(s)}{f^3(s)}=\frac{a_2 s^2 + a_1 s+a_0}{b_3 s^3+b_2 s^2 + b_1 s+b_0}$$ when $f^3(s)$ cannot be simply ...
0
votes
1answer
48 views

Inverse Laplace transform of Gamma with branch cut

In solving a particular physical problem I have had to perform inverse Laplace transforms of sum and products of Gamma functions. Since my actual problem is complicated, I will state a simple example. ...
0
votes
1answer
51 views

How to find the inverse Laplace transform with complex shift

How can we take the inverse Laplace transform of $$f_1(s)X(s)+f_2(s)\left(e^{i\phi}X(s-i\alpha_1)+e^{-i\phi}X(s+i\alpha_1)\right)= f_0(s)$$ Where $ f_1(s)$ is in the form of $ \frac{f^2(s)}{f^3(s)}$...
0
votes
0answers
21 views

Laplace inverse using Bromwich integral

I have to recover $f(t)$ from its $F(S)=1/(S+\sqrt S)$. we should use Bromwich integral and using Cauchy's integral formula to do it, but we have a two valued function, how we should evaluate it? ...
0
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0answers
32 views

Series Approximation Before and After Inverse Laplace Transformation

We have the following series involving some Laplace transformed function $F(s)$, $$\sum^{\infty}_{n=0} \left( \frac{s}{a} \right)^{n} F(s).$$ Using the following formula, $$\left( \frac{1}{a} \...
5
votes
3answers
179 views

Using Laplace Transforms to solve $\int_{0}^{\infty}\frac{\sin(x)\sin(x/3)}{x(x/3)}\:dx$

So, I've come across the following integral (and it's expansion) many times and in my study so far, Complex Residues have been used to evaluate it. I was hoping to find an alternative approach using ...
0
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0answers
35 views

Uncommon Inverse Laplace Transforms

I'm interested in calculating inverse Laplace transforms of functions such as $\frac{1}{(1-s^{2})\log(s)}$, $\frac{1}{arccosh\left(1+\frac{s^{2}}{4}\right)}$ and more exotic examples. In general ...
0
votes
1answer
103 views

Using Laplace Transforms to solve $\int_{0}^{\infty} \sin\left(x^2\right)\:dx$

I recently came into the following definite integral $$ I = \int_{0}^{\infty} \sin\left(x^2\right)\:dx$$ To solve this, I used a composition of both the Feynman Trick with Laplace Transforms as ...
0
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0answers
26 views

inverse Laplace transform of exponential function and gamma function

I came to this final problem to be solved. I would like to understand a way to tackle this problem: Inverse Laplace transform of $$\frac{e^{-as}}{Γ(bs+c)}$$ I cannot find an easy way of finding an ...
0
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0answers
28 views

inverse laplace transform of $ \frac{s^2 +1}{s^2+s-2} {e^{-2s}}$

inverse laplace transform of $$ \frac{s^2 +1}{s^2+s-2} {e^{-2s}}$$ I tried solving it like that, but I'm not sure: 1- Apply partial fraction to the f(t) part: $$(\frac{2/3}{s-1} + \frac{-5/3}{s+2})...
1
vote
1answer
19 views

Inverse Laplace of $\frac{s}{(s+3)^{5}}$

Struggling to answer this transform. Anyone able to give me a walkthrough? I know that the inverse transform of $$\frac{1}{(s+a)^{n}}$$ is $$\frac{1}{(s+a){!}}{t}^{n-1} {e}^{-at}$$ But I am unsure ...
9
votes
1answer
214 views

Using Laplace transforms to evaluate$\int_{0}^{\infty}\frac{\sin^2(x)}{x^2(x^2 + 1)} dx$

Recently I've been playing around with Feynman's Trick to evaluate integrals. Obviously, one of it's many great features is that it allows derivatives to make expressions simpler. I was wondering ...
1
vote
0answers
21 views

Initial function and inverse laplace transformation are not the same [closed]

Hi I applied the Laplace transform to the following function so I had. Laplace transformation of the system but if I applied the inverse Laplace I will have T2 as the function. I know that it is ...
3
votes
1answer
337 views

Proof of inverse Laplace transform

Why is $$f(t) = \frac{1}{2πj}\int_{\sigma-j\infty}^{\sigma+j\infty} F(s) e^{st} \, ds,$$ provided that $$F(s) = \int_{0}^{\infty} f(t) e^{-st} \, dt \ ?$$ I tried to find out myself, or searched ...
1
vote
1answer
34 views

Problem with simple inverse Laplace transform

I have the function: $$F(s) = \frac{s^4+3s^3+2s^2+4s+4}{(s+3)(s^2+1)}$$ and I have to make inverse Laplace. I tried to collect $s^3$ from the first and second element of the numerator in order to ...
-1
votes
2answers
40 views

Inverse laplace transform of $\frac{1}{s^3(s-1)}$

I have a problem with this transform. I know that using this theorem (see the theroem) I can get the answer, and i have the resolution. But I don't understand why the put a $-1$ in the integral. IMG ...
0
votes
0answers
30 views

how to solve this Inverse Laplace Transform using Convolution Theorem

Given below is the problem :Click for the problem Here the difficulty I am having is in calculating the inverse Laplace of e^(s^2/2).So how should I solve this problem ?
0
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0answers
20 views

Inverse Laplace Transform in Thermal Problem with Newton Cooling Condition

I need to compute the inverse Laplace transform of the following function: $$\hat u(x,s)=\frac{\exp(-\sqrt{s}x)}{s(1+\Lambda^{-1}\sqrt{s})},$$ where $x\geq0$ and $\Lambda>0$. This function comes ...
1
vote
2answers
95 views

Inverse Laplace transform of $\frac{40.5}{s(s^2-9)}$ using convolution theorem

Find the inverse Laplace transform of $$\frac{40.5}{s(s^2-9)}$$ using the convolution theorem. I see how you can solve this using partial fractions, but apparently it's supposed to be easier if you ...
0
votes
2answers
43 views

Inverse Laplace transform of $\dfrac{s}{(s + 1)^2 - 4}$

I am trying to find the inverse Laplace transform $\mathcal{L}^{-1} \left\{ \dfrac{s}{(s + 1)^2 - 4} \right\}$. My textbook says that the solution is $e^{-t} \cosh(2t) - \dfrac{1}{2}e^{-t}\sinh(t)$. ...
0
votes
0answers
26 views

Why would inverse Laplace transform of a function of time yield a function of exponent ?

From what I understand of LT, it transform a function of time to a function of complex frequency $e^{a+ib}$, whose real part is an exponential. The inversion of LT should be the other way around. ...
0
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0answers
60 views

Is there a way to evaluate analytically or numerically the inverse Laplace transform of the following parameterized expression?

Everybody hello, I am trying to evaluate analytically or numerically the inverse Laplace transform of the following parameterized expression: \begin{equation} F(p) = \frac{1}{ p^2 \left[ 1 + \alpha \...
0
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2answers
56 views

Find $\mathcal{L}^{-1} \frac{9}{(s+3)^3} $

Find $$\mathcal{L}^{-1}\left[ \frac{9}{(s+3)^3}\right].$$ How do I go about with the fraction inside? There is no fixed formula for this expression. I did a partial fraction of the repeated linear ...
0
votes
2answers
47 views

Laplace inverse of $(s+2)U(s)=0$ and $(s+1)U(s)=0.$

I asked this question here and I was given an answer but with some steps unfolded Solve the following problem, $u'(t)+p(t)u(t)=0,\;\;u(0)=0,$ $p(t)=\begin{cases}2& 0\leq t< 1,\\1 &t\geq ...
0
votes
0answers
37 views

Inverse Laplace, residue, simple or essential pole from bivaluated hyperbolic trigonometric function?

I would like to compute the following function: $$f(t)=\mathcal{L}^{-1}\Big[\frac{1}{s(e^{a+\text{arcosh}(s+\cosh a)}-1)}\Big](t)$$ However, it seems that there is no other pole than the pole of ...
0
votes
0answers
70 views

Inverse Laplace transform related to modified Bessel function of the second kind

To solve a fluid diffusion problem, I need to calculate an inverse Laplace transform. The integral, according to the Inverse Laplace theorem, has the form: \begin{equation} \label{i_laplace_1} p(r,t)...
1
vote
1answer
43 views

Convolution of complementary Error functi0ns

I'm interested in the solution to the following integral, given that $a$ is a positive real: $$I=\int_0^1 \text{erfc}\!\left(\frac{a}{\sqrt{t}}\right) \text{erfc}\!\left(\frac{a}{\sqrt{1-t}}\right) dt=...
4
votes
1answer
249 views

Inverse Laplace transform of a product using convolution

I want to calculate $\mathcal{L}^{-1}\left\{\frac{1}{s^2(s^2+a^2)}\right\}$ using the convolution theorem $\mathcal{L}\{f*g\}=\mathcal{L}\{f\}\cdot\mathcal {L}\{g\}$. I have already calculated it ...
2
votes
1answer
70 views

solve $y''+y'+y=\sin(x)$ with $y(0)=0$ and $y'(0)=1$ with Laplace Transformation

I have this problem: $$y''+y'+y=\sin(x),$$ with $y(0)=0$ and $y'(0)=1$. I solve it using the Laplace Transform: $$\mathcal L(y''+y'+y)= \mathcal L(\sin(x))$$ $$s^2\mathcal L(y)-sy(0)-y'(0)+s\...