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

learn more… | top users | synonyms

0
votes
1answer
17 views

Reason for $0^-$ bound on unilateral Laplace transform

I have the definition of the unilateral $\mathcal{L}$-transform, valid for causal signals, to be: $$\mathcal{L}\left[f(t)\right](s)=\int_{0^-}^{+\infty}f(t)e^{-st}dt$$ My question is regarding the ...
1
vote
0answers
35 views

Inverse of Mellin transform with lower bound at $1$

I've seen two definitions of the Mellin transform: more commonly, $g$ is the Mellin transform of $f$ if $$g(s)=\int_0^\infty x^{s-1}f(x)\; dx,$$ or secondly, and more rarely, defined by the same ...
0
votes
1answer
36 views

Why is Laplace transform so useful

I recently encountered the term Laplace transform. Since My background is mathematical, the definition itself is very easy for me. The thing which is less clear for me is the context. I assume, that ...
2
votes
1answer
94 views

Numerical or analytical or exisistence: Inverse Laplace Transform

Edit 1: With the hint of Ron, we can simplify the question to : $$\bar{f}(s)=\frac{1}{(s^2+1)\arctan s }$$ So what about this function's inverse Laplace Transform? Or can anyone tell me that the ...
3
votes
1answer
148 views

Laplace transform,Fourier transform and Z transform mathematical equations

Fourier transform $x(w)$ of signal x(t) is given by $$ x(w) = \int\limits_{t=-\infty}^{+\infty} x(t) e^{-j w t} dt -(1)$$ Laplace transform $x(s)$ of signal x(t) is given by $$ x(s) = \int\...
1
vote
1answer
109 views

Proof of a Bromwich integral formula

I am trying to prove that: $$\frac{1}{2\pi i }\int_{\alpha-i\infty}^{\alpha+i\infty}\frac{(\log s)^{n}}{s}e^{sx}ds=(-1)^{n}\frac{d^{n}}{dz^{n}}\frac{x^{z}}{\Gamma(1+z)}\left.\begin{matrix} \\ \\ \...
0
votes
1answer
213 views

Conditions for existence of inverse Laplace transform.

Given a function $F(s)$, how to check if inverse Laplace transform of $F(s)$ exists? In other words, I want to know conditions for existence $f(t)$ such that $$ \int_0^\infty e^{-st}f(t)\,ds = F(s) $$...
2
votes
1answer
34 views

Proving an equation involving integrals and limits

I have to show the following equation: $\large\int_0^\infty \! e^{-st}\cos(\beta t) \, \mathrm{d}t=\frac{s}{s^2+\beta ^2}$ with $s>0$ I've come so far: $\large\int_0^\infty \! e^{-st}\cos(\beta ...
2
votes
1answer
71 views

Interchanging Inverse Laplace Transform

I have a function $f(|\boldsymbol{k}|,s,\theta)$ for which I am interested in its inverse Laplace transform. I am also interested in the function's mean value for constant $|\boldsymbol{k}|$, but ...
1
vote
1answer
65 views

Laplace transform of function

Assume that $f(u)=(\frac{b}{πu^3})^{1/2} e^{2b} e^{-bu} e^{-b/u}$, where $b>0.$ I am trying to calculate the Laplace transform $L\{f(u)\}(s)$ and then the $n_{th}$ derivative of this transform, $L^{...
0
votes
1answer
31 views

Inverse Laplace of a function

I am really searching for hours now for the inverse laplace transformation of the following function: $$\frac{75s + 12739.726}{s( 0.0365s^2 + 81.2s + 12739.726)}$$ If I put this in WolframAlpha the ...
1
vote
1answer
270 views

Laplace transform of inverse gaussian distribution [closed]

Can someone write in details how i can derive the Laplace transform of the Inverse Gaussian distribution? I think i am missing something during the calculation of the interval which gives the Laplace ...
1
vote
1answer
120 views

What does it mean “Laplace transformable functions”

I am reading about the The convolution operation, and the notion Laplace transformable functions is mentioned there. Doe anyone know what is the definition of Laplace transformable functions? Thank ...
0
votes
1answer
34 views

Inverse Laplace transform with minus $\Delta$ in denominator

Please help me find this inverse Laplace transform. $$ F(s)=\dfrac{2s-3}{s^{2}-2s+2} $$ I couldn't resolve the denominator, because the quadratic has discriminant $\Delta=-4$.
2
votes
1answer
87 views

Use of Laplace transform to solve initial value problem.

--Short Explanation: I have to say I am going crazy with this problem as it does not give me the same as the suggested solution in the book: Problem: $y''-7y'+10y=9\cos{t}+7\sin{t}$ $y(0)=5$, $y'(...
1
vote
4answers
73 views

Laplace transform of the wave equation

I started of with the wave equation $$\frac{\partial^2 u}{\partial x^2}=\frac{\partial^2 u}{\partial t^2}$$ with boundary conditions $u=0$ at $x=0$ and $x=1$ and initial condition $u=sin(\pi x)$ and $\...
3
votes
1answer
64 views

Laplace transforms to solve heat equation

I have the heat equation $$\frac{\partial u}{\partial t}=\frac{\partial^2 u}{\partial x^2}$$ Boundary conditions are $u=0$ at $x=0$ and $x=1$ The initial condition is $u=\sin(\pi x)$ I know that $$L ...
2
votes
0answers
106 views

Laplace transform of inverse error function

I want to calculate the convolution of a function with the inverse error function. Therefore I chose to try to first find an integral transform of the inverse error function like the laplace transform:...
1
vote
1answer
30 views

Question in regard to solving for inverse laplace transform

I am having some confusion when it comes to solving for the inverse laplace transform. ( We are allowed the tables with the common values by the way). Il give an example. Take, $$y''+4y'+8y=-6e^{-...
0
votes
0answers
267 views

For Laplace Transforms; What is the interpretation of $s$ compared to $t$? Why is each Laplace transform only defined for some values of $s$?

What is the interpretation of $s$ compared to $t$? Why is each Laplace transform only defined for some values of $s$?
0
votes
1answer
81 views

use laplace transform to solve the given integral equation

use Laplace transform to solve the given integral equation I don't know how start because it differences on other Laplace question I see before
0
votes
0answers
47 views

Can this definite integral of an inverse Laplace transform by simplified?

Can either of the below expressions involving an unknown analytic function $h(s,t)$ and the inverse Laplace transform $\mathcal{L}^{-1}$ be simplified? $$ \int\limits_{0}^1 \mathcal{L}^{-1} \left\{ \...
0
votes
1answer
45 views

Laplace diffrential equation

$$\frac{dx}{dt}=2x +3y$$ $$\frac{dy}{dt}=3x +2y$$ Find general solution. I know there is a solution through eigenvalues. But I want to solve it with Laplace transformation. I almost get the right ...
12
votes
1answer
175 views

The Laplace transform of $\frac{\ln(1+at)}{1+t}$

By expressing the square of the exponential integral as a double integral and then making a change of variables, one can show $$ \int_{0}^{\infty} e^{-2zt} \ \frac{\ln(1+2t)}{1+t} \, dt = \frac{e^{2z} ...
-1
votes
1answer
69 views

Laplace Transforms

Solve the initial value problem for y(t) using Laplace Transforms. $$L\{y''+3y'\}=L\{f(t)\}$$ $$s^2Y-sy(0)-sy'(0)+3(sY-sy(0))=L\{t\}+L\{1\}-L\{u4(t)(t-4)\}-5L{u8(t)}$$ $$Y(s^2+3s)=(1/s^2)+(1/s)-(...
0
votes
0answers
16 views

Unilateral Laplace transform calculation

I'd like to verify that $\mathcal{L}[e^{-at}]=\frac{1}{s+a}$, $t\ge 0$. So I calculate: $$\int_{0^+}^{+\infty} e^{-at} e^{-st} dt=\int_{0^+}^{+\infty} e^{-t(a+s)} dt = \frac{1}{-(a+s)} e^{-t(a+s)}\...
5
votes
2answers
180 views

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 ...
2
votes
2answers
195 views

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....
1
vote
0answers
53 views

The Laplace transform of $\exp(t^2)$

A naive attempt to calculate the Laplace transform of the function $f(t)=e^{t^2}$ results in integrals of the form $$\int_0^\infty e^{t^2-st}dt,$$ which obviously don't exist as the integrand grows ...
2
votes
2answers
53 views

Unilateral Laplace transform

I tried to do the same unilateral Laplace transform in two ways, but I got different results. I have to calculate: $\mathcal{L}[r(t-1)]$, where $r(t)$ is the ramp function, that is $r(t)=t, t\ge0$. $...
2
votes
2answers
52 views

Find the Laplace Transform

Could anyone enlighten me on how to find the Laplace Transform of $$\frac{1-\cos (t)}{t}$$
1
vote
1answer
49 views

Inverse Laplace Transform of $1/(s+1)$ without table

The pole is on the left half plane, so $\gamma =0$ $$\frac{1}{2i\pi}\int ^{i\infty}_{-i\infty}\frac {e^{st}}{s+1}ds$$ substituting $iu=s$ $$\frac{1}{2i\pi}\int ^{\infty}_{-\infty}\frac {e^{iut}}{iu+...
5
votes
1answer
61 views

How do you find the inverse Laplace transform of $\frac{1}{\sqrt{s}(s-a)} $

When I use the convolution method, I can't avoid getting a divergent integral.
0
votes
4answers
83 views

Laplace of $\int_0^t \frac{sinx}{x}dx$

What is the Laplace transform of $\int_0^t \frac{\sin x}{x}dx$ I'm thinking about approaching it as a convolution but I am not sure how. Could I define it as the convolution of $1$ and $\frac{\sin ...
1
vote
1answer
79 views

System of differential equations using Laplace transform

Using Laplace transform, solve the system: $w'+y=\sin(x)$ $y'-z=e^x$ $z'+w+y=1$ where $w(0)=0$ and $z(0)=y(0)=1$.
4
votes
0answers
121 views

Integration Around Part of a Branch Cut

I am studying the integral, given by a Laplace transform, $$\int_0^\infty\!e^{-\alpha x}\sinh^{-2/3}x\left(1+\frac 12\sinh^2x\right)^{-1/6}\left(1-\beta\sinh^{4/3}x\right)^{1/2}\,\mathrm dx$$ From ...
0
votes
1answer
115 views

Relationship between Inverse Fourier and Inverse Laplace Transform?

Suppose we are given a fourier transform $$ F(\omega) = \frac{1}{\omega^2+4} $$ Can we use inverse laplace tranform by taking $i\omega = p$ to find the inverse fourier transform? I did this and got ...
0
votes
1answer
49 views

Laplace transform, Bochner integral

I have a quesition about linear operators on a Banach space. Let $B$ be a real Banach space. $(T_{t})_{t>0}$ is called strongly continuous contraction semigroup on $B$ if For all $t>0$, $D(T_{...
1
vote
2answers
92 views

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 ...
2
votes
1answer
134 views

A proof which results in Gamma (or Erlang) distribution-From Karlin & Taylor's “A First Course in Stochastic Processes”

The random variables X and Y have the following properties: X is positive, i.e., $P\{X > 0\} = 1$, with continuous density function $f_X(x)$, and $Y\mid X$ has a uniform distribution on $\{0,X\}$. ...
0
votes
1answer
40 views

laplace step function $H(π-t)(\sin(t))^2$

How to calculate the laplace transformation of $H(π-t)(\sin(t))^2$ ? I know that I have to use $\sin^2(t)= 1/2(1-2\cos(2t))$ but i am stuck of how to proceed``
1
vote
2answers
28 views

Find the Laplace Transformation of $H(\pi-t)$.

I know how to find the Laplace Transformation of $H(t-\pi)$, but what about if the $t$ is negative. Any help is much appreciated.
1
vote
1answer
297 views

When does Fourier Transform be the same as Laplace's?

I have the TI nspire CX CAS... it can perform Laplace Transform but can't perform Fourier Transform. They are equal in some problems, but not all the time! So, when does both of them be equal so that ...
0
votes
1answer
62 views

Laplace transform (Simple factorization)

The question require me to find the inverse of Laplace transform. In the first line of solution, how does it go from LHS to RHS? Does it simply apply partial fractions?
1
vote
0answers
46 views

Solving initial value problem using Laplace transforms, one other method, and comparing results

So for my solution using characteristic equations I get (fixed a typo for first coefficient) $$\frac{11}{30} e^{-3t} - \frac{21}{20} e^{-2t} + \frac{21}{20} e^{2t} - \frac{11}{30} e^{3t}$$ For the ...
1
vote
0answers
37 views

Evaluating the inverse Laplace transform of $1/(s^2-\sum_{n=1}^\infty{n!s^{3-n}x^n})$

I want to evaluate at $t=1$ the inverse Laplace Transform $\mathcal{L}^{-1}\{F(s)\}\vert_{t=1}$ of $$ F(s) = \frac{1}{s^2-\sum\limits_{n=1}^\infty{n!s^{3-n}x^n}} $$ and find out the $x^n$ ...
0
votes
1answer
207 views

Are there functions that are not of exponential order for which you can define a Laplace transform?

I'am in a course of Introduction to Linear Differential Equations and teacher made us this question in class. we work in $\mathbb{R}$, and any help to answer this is welcome
-3
votes
1answer
47 views

Find the roots of the corresponding characteristic equation

The equation is $${Y_s\over F_s}={1\over s^2+2s+2}$$ I have got to $$r^2+2r+2=0$$ what do i need to do next?
1
vote
1answer
80 views

Relation between Laplace and Fourier transform

I have a function that has the property $\tilde f(s) = \tilde{f}(abs(s))$. For this function, I need the inverse Fourier transform. I actually know the inverse Laplace transform of $\tilde f$ and I ...
1
vote
1answer
42 views

solving second order linear differential equation

Can somebody please show me how to solve the following differential equation: $$ a\ddot{x} + b\dot{x} = c $$ given these initial conditions $x(0) = 2$, $\dot{x}(0) = 0.5$ and $a = 4, b = 1.5$ First ...