The Laplace transform is a widely used integral transform, similar to the Fourier transform.

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Inverse of Mellin transform

I would like to invert the following Mellin transform $M(s)$ of a function $f(x)$ defined on $[0,a]$ with $a>0$ (or get the $x\rightarrow 0$ asymptotics): $$ M(s) = \frac{2a^s}{s-2(1-a^s)} $$ We ...
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asymptotics from Laplace transform

Suppose I know that a non-negative random variable with density $f$ has the following Laplace transform: $$\hat{f}(s)=\int_0^{\infty}e^{-st}f(t)dt=\frac{1}{\cosh(\sqrt{2s}x)}$$ where $s>0$ and ...
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The existence of the laplace transform

I don't understand why the laplace transform of some function, say f(t), has to be "piecewise continuous" and not "continuous". Is "piecewise continuous" sort of like the minimum requirement? This ...
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inverse Laplace transform by integral

I've seen this formula for the inverse Laplace transform in several books: $$f(t)=\mathcal{L}^{-1}\{F\}(t)=\frac{1}{2\pi i}\lim_{T\to\infty}\int_{\alpha -iT}^{\alpha +iT}e^{st}F(s)ds$$ where $f$ is ...
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24 views

Find the Laplace transform of integral(from 0 to x) sin(2t) dt

Find the Laplace transform of $\int_0^x\,\sin\,(2t)\,dt$ So basically, $$\int_0^x\,\sin\,(2t)\,dt = -\frac{1}{2}(\cos\,(2x) - 1)$$ So $$\mathcal{L}\{\cos\,(2x)\} = \dfrac{s}{s^2 + 4}$$ ...
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An identity derived from the Laplace transform

It seems that $$\int_0^t \int_0^l f(\tau) ~d \tau ~d l = \int_0^t z f(t-z) dz $$ since the Laplace transform of both sides is $F(s)/s^2$, where $F(s)$ is the Laplace transform of $f(t)$: the left-hand ...
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26 views

Fourier series: term-by-term Laplace transform.

Quick question: If a Fourier series is uniformly convergent should the term-by-term Laplace transform of the series equal the result of the periodic function theorem for the Laplace transform?
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50 views

Laplace transform of the autocorrelation of a wss random process

Consider a wide-sense-stationary random process $x(t)$. The autocorrelation function is $r(t-\tau):=E[x(t)x(\tau)]$. Let $S(s)$ be the Laplace-transform of $r(t)$. Can I compute $S(s)$ as ...
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33 views

Inverse Laplace Transform of exponential

Is it possible to compute the inverse Laplace transform of: $$\frac{1}{1-e^{-sa}}$$ where $a>0$ ?
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Are FT and LT both isomorphic?

As the following diagram:(from a textbook) Note: 1. L2: L2 space, H2: H2 space 2. The upper one is in t-domain; the lower one, f-domain 3. : the Laplas transform operator : the fourier tansform ...
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Does an inverse Laplace transform for $\hat{F}(s)=e^{-is}$ exist? If not, why?

Does an inverse Laplace transform for $\hat{F}(s)=e^{-is}$ exist? If not, why? The Bromwich integral is not covered in my course so I can't use it. I'm hoping and guessing that the answer is simple! ...
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18 views

Any closed formula for $\mathcal{L}\big(u_c(t)\cdot f(t)\big) $?

As in the title, is there any closed form formula for such Laplace transform, with denoting $\mathcal{L} \ f(t)=F(s)$?
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31 views

Find $\mathcal{L}^{-1}\frac{s}{s^2-6s+9}$

It is easy to see that $\frac{s}{s^2-6s+9}=\frac{s}{(s-3)^2}$ and now I want to use use the convolution integral for $s\cdot \frac{1}{(s-3)^2}$. So I get this integral: $$\int_0^t \delta '(\tau)\cdot ...
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The Laplace transform of $\mathcal{L}(te^t \cos t)$

How do I find it? I know that $\mathcal{L}(e^t \cos t) =\frac{s-1}{(s-1)^2+1^2}$ but what is it when multiplied by $t$, as written in the title?
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45 views

Laplace Transform's phase delay

I have read this example about time shift of Laplace Transform somewhere. It used a unit step function that has been shifted along $x$ axis for $a$ unit. So, to find the Laplace Transform of it, ...
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27 views

Laplace transform of $\frac{\sin at}{t}$

Laplace transform of $\displaystyle \frac{\sin at}{t}$ My Attempt: Rule used: $\displaystyle L[\frac{1}{t}f(t)]=\int_{s}^{\infty}\bar f(s)ds$ So, $\displaystyle L[\frac{1}{t}\sin ...
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34 views

Laplace transform, Inverse Laplace transform

Let $(\mathcal{L}f)(s)$ be the Laplace transform of a piecewise continuous function $f(t)$ defined for $t\geq 0$. If $(\mathcal{L}f)(s)\geq 0$ for all $s\in\mathbb{R^+}$ does this imply that $f(t)\geq ...
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Laplace transform on a finite interval $f(t)= \int_0^1 e^{-xt} f(x) \, dx$

What is the name of this transform? It's basically the Laplace transform where we integrate over a finite interval. $$ F(t)= \int_0^1 e^{-xt} f(x) \, dx$$ Is it still just the Laplace transform? ...
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inverse laplace transform of $s/(s^2+6s+13)$

Hi can anyone help with this inverse Laplace transform $$s/(s^2+6s+13) $$ I tried to do partial fraction $s+3/(s+3)^2+4 - 2/(s+3)^2+4$, but then I don't know what to do next...
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17 views

hybrid function into one-line form

I came across a non-homogeneous ODE with the non-homogeneous term $g(t)$ defined by a few functions like this one below: $$g(t)=\left\{\begin{matrix} f_1(t), & 0\leq t<a\\ f_2(t), & a\leq ...
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Laplace transform of $\dfrac{\sin2t} t$

So I'm taking a look at my notes and the professor wrote this: ${\scr L}(\frac {\sin2t}{t}) = \arctan \frac 2s$ But I can't see this anywhere in the tables. So, where does this come from? Thanks in ...
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274 views

Contour integration with branch points inside the contour.

In my scientific research I ran into an unpleasant situation with specific type of contour integrals. Being more specific I have problems not with integrals themselves (I can use various numeric ...
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Are there “formal” versions of the Laplace transform?

I am reading on formal power series theory, which among other things appears to give autonomous existence to the recurrence solving techniques otherwise based on z-transform. Is there such a purely ...
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30 views

Laplace transform quick answer check :) using second shift theorem

I want to get $L((t-4)^2u(t-4))$ I say this is a second shift with $g(t)=(t^2-4t)$ and my friend says "NO you are wrong, you are dumb!!!!!! $g(t)$ is MOST CERTAINLY equal to $t^2$" Mine gives me ...
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26 views

Laplace transform convolution attempt

I can't seem to get this Laplace working using the convolution method. $H(s) = \frac{1}{s^2(s+2)}$ Which I can't get to work using convolution. So I am separating it into $\frac{1}{s^2} * ...
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24 views

Laplace transform on a non-standard sort of problem

I don't know where a laplace comes into play here: $\ddot{a}+2a=0,a(0)=b_1,\dot{a}(0)=b_2$ I am meant to solve the above using a Laplace transform, but I don't see how I would use it here? I ...
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11 views

Inverse Laplace Transform Table, Absolution of Form

Do I need to ensure I don't stray from the transform in the table? $\frac{-2}{s-1}$ this looks like $-2*\frac{a}{s^2-a^2},$ for $a=1$ Does this yield $-2\sinh(t)$, or should it fit perfectly to ...
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44 views

Laplace Transform assistance

Find the inverse laplace transform of: $\frac{25}{(s-1)^2(s^2+4)}$ $\frac{25}{(s-1)^2(s^2+4)}=\frac{A}{s-1}+\frac{B}{(s-1)^2}+\frac{C}{s^2 + 4}$ $$25=A(s^2+4)(s-1)+B(s^2+4)+C(s-1)^2$$ ...
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What is the Laplace Transform for the next equation?

I have some doubts in the correct way to solve this part of a mathematical model using the Laplace transform: $8 y'(t) + 3 y'(t) - 6 y(t)$ = $4x'(t - 2) + 5x (t - 2)$ This is my solution: $Y(s) [8 ...
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138 views

Taylor series expansion and Laplace transform final value theorem

I cant figure out how some transformations are made in one article on physics. Here is expression in s-domain and they want to find its asymptotic value. $$ \xi(s) = \nu_1(s+1)=\frac{1}{(s+1)} ...
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55 views

Laplace transformation using second shifting theorem

$$f(t)=\begin{cases}t^2,&t\lt4\\t,&t\geqslant4\end{cases}.$$ can anyone tell me how to evaluate the solution? I really get stuck.
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Inverting weird Laplace transform

Solving a PDE gave me this expression: $U(x,s) = 2/((s+1)^2)+1) e^{-\sqrt{s}x} + sin x/(s+1)$ I suppose there's a trick involved since I can't find it in my table. How do I invert this thing?
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help in Laplace and partial fractions

Can any one teach me how to solve C2.(a) and (b) step by step? C2. (a) Resolve $\frac{1}{s^2(s^2+s+1)}$ into partial fractions of the form $\frac{A}{s}+\frac{B}{s^2}+\frac{Cs+D}{s^2+s+1}$. Hence, ...
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Using partial fraction for inverse Laplace transform of $1/[s(s+5)^2]$

my question is the last part $1/5(s+5)^2$, how is it become $-5te^{-5t}$ I thought is should be -$1/5 te^{-5t}$
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Fourier Transform, Laplace Transform, but what about…

I have a question regarding the fourier and laplace transform. First, the Fourier transform essentially takes a function, divides it by a frequency (imaginary exponential), and then sees how much of ...
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the jump in $\ddot y$, Laplace transform

Given the following IVP: $$\ddot y+4y=\cos t-\cos t \cdot \theta(t-2\pi), y(0)=0, \dot y(0)=1$$ Check that $y(t)$ is continuous at $t=2\pi$. Find the jump in $\ddot y(t)$ at $t=2\pi$ i.e find $\lim ...
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Laplace transform of 1/(1+t)

In a table and also on WolframAlpha, I stumbled upon this http://www.wolframalpha.com/input/?i=laplace+transform+1%2F%281%2Bt%29 So the Laplace transform of $1/(1+t)$ is apparently $-e^s Ei(-s)$. ...
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40 views

linear time-constant causal system

I have a linear time-constant causal system with the transfer function: And I have the insignal How do I get the output signal? I thought of Laplace transform the insignal and then get Y and ...
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65 views

Laplace transform of $f'(t)/t$

A question regarding the computation of $\mathcal{L}_s[f'(t)/t]$, where $f(t)$ is a differentiable function, was asked few hours ago. Unfortunately, this question was voluntarily deleted by the OP. I ...
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Fourier transform of a Laplace transform

Is there an easy way to find the Fourier transform of a Laplace transform of function? $$ F[L[f(t)]_{s}] $$ Where my $f(t)$ is $\sqrt{t}$. However, Before finding the Fourier transform I do the ...
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Deriving Laplace Transform of Laguerre polynomial

I'm given this definition for the Laguerre polynomials: $$L_n(t)=\frac{e^t}{n!}\frac{d^n}{dt^n}\left[t^ne^{-t}\right],~\text{for }n=0,1,2...$$ and I have to show that the Laplace transform is ...
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Laplace transform of a product of two functions

I have read questions and answers about this topic and i am still confused, using this formula we can calculate the Laplace transform of a product of two functions: $$ L[a_{(t)} b_{(t)}]={{1}\over{2 ...
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'Deriving' the Laplace Transform from the $z$ Transform: Missing a $\Delta t$

Textbooks normally give the following 'derivation' (or justification, if you prefer) of the z-Transform from the Laplace Transform. Let $x(t)$ be a signal defined on $t\geq 0$, and write a discretized ...
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65 views

Convolution and Total Response Differential Equations

Convolution with differential equations is extremely confusing to me. The two following questions were asked in class and we were asked to think about them. I want to work them out but I don't know ...
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ODE with Laplace transform: the jump of $\dot y$

I solved this eq. using the Laplace Transform: $\ddot y+4\dot y+13 y=\delta(t-2\pi)-\delta(t-7\pi)$ The sol. is: $y(t)=\frac{1}{3} e^{2 t} (-e^{14 \pi} \theta(t-7\pi) sin(3 t)+e^{4 \pi} \theta(t-2 ...
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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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58 views

Inverse Laplace transform (using table) when denominator cannot be factored

Usually when performing inverse Laplace transforms, I decompose the function into partial fractions, and then look up standard transforms in a table. For example: $$Y(s) = ...
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36 views

Finding an inverse laplace transform for $\displaystyle\frac{a}{\left(s^2 + a^2\right)^2}$

I am asked to show that $x'' + w^2x = f\sin(wt)$ has a solution given by $x = \frac{f}{2w^2}(\sin(wt) - wt\cos(wt))$ where $w$ and $f$ are constants, by means of Laplace transforms. By taking a ...
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60 views

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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Elliptical Coordinates PDE, wave equation and separation of variables

I need some help with this problem. I know how to use the method of separation of variables and that the constant lambda should give you trig functions with solutions at some interval of pi, which ...