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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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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501 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)} - \frac{1}...
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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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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] L[g(t)...
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Laplace transform of the Bessel function of the first kind

I want to show that $$ \int_{0}^{\infty} J_{n}(bx) e^{-ax} \, dx = \frac{(\sqrt{a^{2}+b^{2}}-a)^{n}}{b^{n}\sqrt{a^{2}+b^{2}}}\ , \quad \ (n \in \mathbb{Z}_{\ge 0} \, , \text{Re}(a) >0 , \, b >0 ...
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672 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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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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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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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, e....
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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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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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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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Moving a limit inside an integral

Under what conditions does $$ \lim_{a \to 0^{+}} \int_{0}^{\infty} f(x) e^{-ax} \, dx = \int_{0}^{\infty} f(x) \, dx \ ?$$ For example, for $a>0$, $$ \int_{0}^{\infty} J_{0}(x) e^{-ax} \, dx = \...
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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 t_D}...
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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 $W=(W_t)_{...
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Inverse Laplace transform $\mathcal{L}^{-1}\left \{ \ln \left ( 1+\frac{w^{2}}{s^{2}}\right ) \right \}$

Where $s\in \mathbb{C}$. I assume that this would be pretty easily handled by solving it by definition, but I haven't taken courses in complex analysis yet. Also, I can't think of any nice property of ...
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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? i....
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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 x\right\}=-\frac{...
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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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580 views

Laplace inverse of the sine function

I was wondering if there is a closed-form Laplace inverse of the sine function. I have tried the following: $$ \sin(as)=\sum_{n=0}^{\infty}\frac{(-1)^{n}(as)^{2n+1}}{(2n+1)!} $$ an $n$-th power of $...
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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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Inverse Laplace transform of s/s-1

Finding the inverse laplace transform: $$L^{-1}\left\{\frac{s}{s-1}\right\}$$ I wrote: $$L^{-1}\left\{\frac{s}{s-1}\right\}=L^{-1}\left\{\frac{1}{s-1}\right\} + L^{-1}\{1\}=L^{-1}\{1\} + e^{t}$$ And ...
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Converse of the Watson's lemma

Watson's lemma basically says $$ f(t) \sim t^{\alpha} \,\,\,(\text{for small } t) \implies \int_0^{\infty} f(t) e^{-st} dt \sim \frac{\Gamma(\alpha + 1)}{s^{\alpha + 1}} \,\,\,(\text{for large } s). $...
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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 {X(u)...
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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 (1-z^{-1})...
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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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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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If $\int_{0}^{\infty} f(x) \, dx $ converges, will $\int_{0}^{\infty}e^{-sx} f(x) \, dx$ always converge uniformly on $[0, \infty)$?

I previously asked about sufficient conditions to conclude that $$\lim_{s \to 0^{+}}\int_{0}^{\infty} e^{-sx} f(x) \, dx = \int_{0}^{\infty} f(x) \, dx$$ when $\int_{0}^{\infty} f(x) \, dx$ does not ...
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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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Integrating using Laplace Transforms

$$\int_{0}^\infty {\cos(xt)\over 1+t^2}dt $$ I'm supposed to solve this using Laplace Transformations. I've been trying this since this morning but I haven't figured it out. Any pointers to push me ...
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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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1answer
111 views

Inverse Laplace transform of one complicated function

I want to ask the inverse Laplace transform of the following function: $$F(s) = \frac{1}{s \cdot (1 + a \cdot s)^{m} \cdot (1 + b \cdot s)^{m-k}} \cdot \Bigl[\exp{(\frac{- c \cdot s}{ 1 + b \cdot s } ...
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Moment Generating Function and Inverse Laplace transform

I need to compute the inverse Laplace transform of the function $$ M(t)=e^{\frac{t^2}{2}} $$ Now, I know that this is a normal distribution with mean zero and variance 1, but how the computations are ...
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Laplace transform of the derivative of the Dirac delta function

$$\int_{0}^{\infty} \delta'(t) e^{-st} \ dt = \delta(t)e^{-st} \Big|_{0}^{\infty} + s \int_{0}^{\infty} \delta(t) e^{-st} \ dt $$ $$= 0 - \lim_{t \to 0} \delta(t)e^{-st} + s e^{-st}\Big|_{t=0} = - \...
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1answer
289 views

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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Generalized Fresnel integral $\int_0^\infty \sin x^p \, {\rm d}x$

I am stuck at this question. Find a closed form (that may actually contain the Gamma function) of the integral $$\int_0^\infty \sin (x^p)\, {\rm d}x$$ I am interested in a Laplace approach, double ...
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Using the Laplace transform to evaluate the steady-state of a function

My understanding is that the Laplace transform evaluated at $s = i \omega t$ can be used to evaluate the steady-state of a function. How is this done? I can't find any information on this in my ...
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Finding the inverse laplace transform of $s$ [closed]

How do I find the inverse laplace transform of $s$, i.e. $$L^{-1}\{s\}=\ ?$$
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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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$\int_0^\infty x e^{-\mathrm i x\cos(\varphi)}\mathrm dx=-\frac{1}{\cos (\varphi )^2}$ 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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387 views

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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176 views

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) I_0\left(\frac{1}{...
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1answer
49 views

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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193 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.
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Discrete to Continuous Representations of Functions via Laplace Transforms?

The Laplace transform can be thought of as the continuous analogue of a power series, as in this video. From this perspective, think of the function $ a : \mathbb{N} \rightarrow \mathbb{R}$ as a ...