The Laplace transform is a widely used integral transform (transformation of functions by integrals), similar to the Fourier transform.

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Inverse Laplace transform of fraction $F(s) = \large\frac{2s+1}{s^2+9}$

Is there a general method used to find the inverse Laplace transform. Are there any computational engines that will calculate the inverse directly? For example, can a procedure be followed to find ...
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340 views

Convolution Laplace transform

Find the inverse Laplace transform of the giveb function by using the convolution theorem. $$F(x) = \frac{s}{(s+1)(s^2+4)}$$ If I use partial fractions I get: $$\frac{s+4}{5(s^2+4)} - ...
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Compute the inverse Laplace transform of $e^{-\sqrt{z}}$

I want to compute the inverse Laplace transform of a function $$ F(z) = e^{-\sqrt{z}}. $$ This problem seems very nontrivial to me. Here one can find the answer: the inverse Laplace transform of ...
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Laplace transform:$\int_0^\infty \frac{\sin^4 x}{x^3} \, dx $

I have a trouble with a integral: Using this Laplace trasform equation: $$\begin{align} \int_0^\infty F(u)g(u) \, du & = \int_0^\infty f(u)G(u) \, du \\[6pt] L[f(t)] & = F(s) \\[6pt] ...
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$\mathcal{B}^{-1}_{s\to x}\{e^{as^2+bs}\}$ and $\mathcal{L}^{-1}_{s\to x}\{e^{as^2+bs}\}$ , where $a\neq0$

http://en.wikipedia.org/wiki/Integral_transform#Table_of_transforms claims than the integral form of inverse bilateral Laplace transform and inverse Laplace transform are both the same. But are they ...
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What exactly is Laplace transform?

I've been working on Laplace transform for a while. I can carry it out on calculation and it's amazingly helpful. But I don't understand what exactly is it and how it works. I google and found out ...
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Inverse Laplace of $ \frac{1}{\sqrt{s} - 1} $?

please help with this. I found this in textbook. Not derived from any differential equation. Also found the answer $$ \frac{1}{\sqrt{\pi}\sqrt{t}} + e^t * erf(\sqrt{t}) $$ (but don't know how)
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Evaluate Integral with $e^{ut}\ \Gamma (u)^{2}$

I am trying to integrate this integral: $$f(x)=\frac{1}{2\pi j}\int_{c-j\infty}^{c+j\infty}x^{-s}\sigma ^{ms-m}\left [ \frac{\Gamma \left ( \frac{s}{\beta} \right )}{\Gamma \left ( \frac{1}{\beta} ...
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471 views

find the inverse Laplace transform of complex function

It would be appreciate if someone help me to obtain the inverse Laplace transformation of the complex function $F(s)=\frac{e^{-\frac lc\sqrt {s(s+r_0)}}}{\frac lc\sqrt {s(s+r_0)}}$ where $r_0,l,c$ are ...
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About evaluating $\mathcal{L}^{-1}_{s\to x}\bigl\{\frac{F(s)}{s}\bigr\}$ by considering contour integration with different entire functions $F(s)$

Detailedly compare the difficulties of different entire functions $F(s)$ where $F(0)\neq0$ when evaluating $\mathcal{L}^{-1}_{s\to x}\bigl\{\frac{F(s)}{s}\bigr\}$ by considering contour integration, ...
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Laplace transform of $ t^{1/2}$ and $ t^{-1/2}$

Prove the following Laplace transforms: (a) $ \displaystyle{\mathcal{L} \{ t^{-1/2} \} = \sqrt{\frac{ \pi}{s}}} ,s>0 $ (b) $ \displaystyle{\mathcal{L} \{ t^{1/2} \} =\frac{1}{2s} \sqrt{\frac{ ...
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About the inverse laplace transform of sinc function

How to calculate $\mathcal{L}^{-1}_{s\to x}\{\text{sinc}(s)\}$ ? Note: $\text{sinc}(s)=\dfrac{\sin s}{s}$ when $s\neq0$ . Also note that $\lim\limits_{s\to\pm\infty}\dfrac{\sin s}{s}=0$ .
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Differential equations and Fourier and Laplace transforms

Why do both the Fourier transform and the Laplace transform appear in the study of differential equations? I've never understood why there are some situations where the Fourier transform is used and ...
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How can I evaluate $\int_{-\infty}^{\infty}\frac{e^{-x^2}(2x^2-1)}{1+x^2}dx$?

How can I solve this integral: $$\int_{-\infty}^{\infty}\frac{e^{-x^2}(2x^2-1)}{1+x^2}dx.$$ Can I solve this problem using the Laplace transform? How can I do this?
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Inverse Laplace Transform of $\bar p_D = \frac{K_0(\sqrt[]s r_D)}{sK_0(\sqrt[]s)}$

I solved the following partial differential equation using Laplace Transform: $\LARGE \frac{1}{r_D}\frac{\partial}{\partial r_D}(r_D\frac{\partial p_D}{\partial r_D})=\frac{\partial p_D}{\partial ...
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Laplace transform of integrated geometric Brownian motion

Is there any closed form of the Laplace transform of an integrated geometric Brownian motion ? A geometric Brownian motion $X=(X_t)_{t \geq 0}$ satisifies $dX_t = \sigma X_t \, dW_t$ where ...
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How to figure of the Laplace transform for $\log x$?

I was looking at a table of common Laplace transforms of functions when I came across the transform for $\log x$. Apparently, the transform is as follows: $$\mathcal{L} \left\{ \log ...
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inverse laplace transform by using complex integral

given function $$f(s)=\frac{1}{s}\frac{\sqrt{s}-1}{\sqrt{s}+1}$$ and $$\int_{0}^{\infty}{\frac{e^{-xt}}{\sqrt{x}(x+1)}dx=\pi e^t {erfc}(\sqrt{t})}$$ my steps: ...
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Inverse Laplace Transform of e$^{-c \sqrt{s}}/(\sqrt{s}(a - s))$

I am trying to find the Inverse Laplace of the following function: $$ F(s) = \frac{\mathrm{e}^{-x b \sqrt{s}}}{ b (a - s)\sqrt{s}} $$ I really don't know where to start on this one as I have only ...
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1answer
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What is the laplace transform and how is it performed? (detailed explanation)

I am a high school student and I became interested after someone mentioned it. Although I am not quite at the level where I am taught this it just captured my attention. Could someone give me an ...
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Using laplace transforms to solve a piecewise defined function initial value problem

I want to use laplace transforms to solve the following: $$\frac{d^2 y}{dt^2}+16 y = f(t) = \left\{\begin{array} 1 1&t\lt\pi\\0&t\geq \pi\end{array}\right.\text{ with } y(0)=0 \text{ and } ...
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Laplace transform of $\sin(\sqrt{t})$

I'm tired and completely blank on how to find a solution of Laplace transformation of $\sin(\sqrt{t})$.Please specify the method used and if possible something other than using series method. thank ...
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Laplace From Fourier transform?

This video (no need to actually watch it) makes a great point. If we interpret the $f$ in $f(x)$ as a function of time, then the fourier transform of $f$ takes the representation of this function in ...
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Calculate Inverse Laplace transform

Can anybody help me with the answer of this question? Find the inverse Laplace transform of $$\frac {1}{(s-3)^4}$$
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Laplace Trouble to find solution

Trying to figure out how to use Laplace Transform to find $y(t)$: The problem is $$y''+4y'+4y=f(t)$$ where $f(t) = \cos(\omega t)$ if $0 < t < \pi$ and $f(t)=0$ if $t > \pi$? Initial ...
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Proof $L{\rm{[}}\frac{{x(t)}}{t}{\rm{] = }}\int_s^\infty {X(u)du} $

I see that we usually use the theorem to solve the Laplace transform, however i want to proof the theorem, who could give me some details!!! $L{\rm{[}}\frac{{x(t)}}{t}{\rm{] = }}\int_s^\infty ...
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Initial values are lost (diff eq to Transfer function)?

I read eternal Julius O. Smith III and he says that $$x_{n-m} = z^{-m}X(z)$$ Particularly, difference relation $$y_{n} = y_{n-1} + x_{n}$$ is solved by by $$Y = z^{-1}Y + X = {X \over ...
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Relationship Between The Z-Transform And The Laplace Transform

Below I've quoted Wikipedia's entry that relates the Z-Transform to the Laplace Transform. The part I don't understand is $z \ \stackrel{\mathrm{def}}{=}\ e^{s T}$; I thought $z$ was actually an ...
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Calculating Inverse Laplace Transform of stretched exponential

I am trying to solve a Laplace transform problem that has gotten way over my head in terms of complex analysis knowledge. I would like to solve the Inverse Laplace Transform $(s\rightarrow t)$ of ...
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Partial fraction expansion for non-rational functions

With regard to this answer to an inverse Laplace transform question, I derived the following result: $$\frac1{i 2 \pi} \int_{c-i \infty}^{c+i \infty} ds \, e^{s t} \Gamma(s)^2 = 2 K_0 \left ( 2 ...
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ODE Laplace Transforms: what impulse brings an oscillating system to rest?

$2y''+y'+2y=\delta(t-5)$ $y(0)=0, y'(0)=0$ Consider the system given by ODE above in which an oscillation is excited by a unit impulse at $t=5$. Suppose that it is desired to bring the system to ...
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inverse Laplace Transform: $ L^{-1} \{\log \frac{s^2 - a^2}{s^2} \}$.

I am styding Laplace transforms and for some reason I have stuck in the followning exercise. Find the inverse Laplace Transform $ L^{-1} \{\log \frac{s^2 - a^2}{s^2} \}$. Any help? Thank's in ...
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transforming ordinary generating function into exponential generating function

I have seen a post here that says that you can convert an exponential generating function into an ordinary one with the aid of the Laplace transform. Is it possible to do the reverse transformation? ...
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Does the Laplace transform biject?

Someone wrote on the Wikipedia article for the Laplace trasform that 'this transformation is essentially bijective for the majority of practical uses.' Can someone provide a proof or counterexample ...
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Differential Equations with Discontinuous Forcing Functions

$$ y''+y'+1.25y = g(t), \quad t > 0, $$ $$y(0) = 0, \quad y'(0) = 0 $$ $$g(t) = \left\{ \begin{array}{ll} \sin{t} & 0 \le t < \pi \\ 0 & t \ge \pi \end{array}\right.$$ ...
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Prove that the Laplace transform of $I_0\left(2\sqrt t\right)$ is $\exp\left(1/s\right)/s$

Wolfram Alpha gave me the answer to this, but unfortunately Wolfram Alpha doesn't show its work, I can't find a proof anywhere else, and my feeble attempts to show it myself went nowhere. How can it ...
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$\int_0^\infty x e^{-\mathrm i x\cos(\varphi)}\mathrm dx=-\frac{1}{\cos (\varphi )^2}$ is that correct?

Good day. This integral looks very simple, yet I don't know how to start. $$\int_0^\infty x e^{-\mathrm i x\cos(\varphi)}\mathrm dx$$ I know that if the lower integration limit was $-\infty$ it would ...
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1answer
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Functional form of a series of a product of Bessels

This question arises from my answer to an inverse Laplace transform question. The result I got took the form $$ f(t)= e^{-r_0 t/2} H(t-a) \left [ J_0\left(\frac{1}{2} a r_0\right) ...
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Evaluating improper integrals using laplace transform

I want to calculate the following improper integral using Laplace and transforms (and laplace transforms only). $$\int_0^{\infty} x e^{-3x} \sin{x}\, dx$$ I propose the following method. I plan to ...
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Special Laplace Inversion

Use complex analysis to show $$\frac1{2\pi i}\int_{a- i\infty}^{a+i\infty} e^{st}/s^{1/2} ds = \frac1{\sqrt{\pi t}}\ ,\quad a >0, t> 0 .$$ This is a special case of Bromwich's integral for the ...
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Existence of Inverse Laplace Transform

Let $F(s)$ the Laplace transform of a function $f(t)$ . Under which conditions on $f(t)$ there exist a unique $g(t)$ such that $g(t) = \mathcal{L}^{-1}\{e^{- F(s)}\}(t) \quad $ ? ...
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inverse Laplace transfor by using maple or matlab

I want to use inverse Laplace transform to F function by using maple or matlab. However I cannot get any answer. I know the answer from table but I want to use one of softwares. from table: ...
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Laplace transform of unit step function

Im given a graph of $f(t)$ and i need to find the Laplace transform of $f(t)$. From looking at the graph i have $$f(t) = \begin{cases} t, & \text{$0 \le t \le 1 $} \\ 0, & \text{$1 \lt t \lt ...
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Laplace question continued (partial fractions)

Last night I attempted and successfully finished (with the help of stackexchange) the first part to this question on laplace transformations: Laplace question - help needed The second part to this ...
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Using Convolution Theorem to find the Laplace transform

In previous questions I have used Laplace transform to find the inverse Laplace transform. I have worked through this work booklet ...
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Finding the inverse Laplace transform of $\frac{s^2-4s-4}{s^4+8s^2+16}$

$$F(s) = \frac{s^2-4s-4}{s^4+8s^2+16}$$ My work is as follows, $$\frac{s^2-4s-4}{(s^2+4)^2}=\frac{s^2+4}{(s^2+4)^2}-\frac{8}{(s^2+4)^2}-\frac{4s}{(s^2+4)^2}$$ The inverse laplace of the first term ...
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152 views

Show that $\int_{0}^{\infty}\frac{\cos(at)-\cos(bt)}{t} =\ln\frac{b}{a}$ [closed]

It should be using Laplace transform. I found similar problems already solved but I need this to be shown using Laplace transforms: $$\int_{0}^{\infty}\frac{\cos(at)-\cos(bt)}{t} = \ln\frac{b}{a}$$
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Inversion of Laplace transform $F(s)=\log(\frac{s+1}{s})$ (Bromwich integral)

I am looking for the inversion of Laplace transform $F(s)=\log(\frac{s+1}{s})$. I started by using the general formula of the Bromwich integral: $\displaystyle \lim_{R\to\infty} \int_{a-iR}^{a+iR} ...
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189 views

Contradiction in inverse Laplace transform problem with Mellin's inverse formula?

Let say we have to solve a given differential equation $$ty''+y'+ty=0$$ $$y(0)=1,\ y'(0)=0$$ (which is Bessel equation with the solution $y=J_0 (t)$, of course) with the Laplace transform. Then we ...
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161 views

Finding Inverse Laplace Transform

Can somebody help me to find the inverse Laplace transform of $$F(s)\exp\left(-\sqrt{\frac{s}{a}}\right)$$ or at least $$\exp\left(-\sqrt{\frac{s}{a}}\right)?$$ I'll be so grateful.