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

learn more… | top users | synonyms

9
votes
0answers
624 views

Physical interpretation of Laplace transforms

One may define the derivative of $f$ at $x$ as $\lim\limits_{h\to0}\cdots\cdots\cdots$ etc., and show that that has certain properties, but it also has a "physical" interpretation: it is an ...
5
votes
0answers
231 views

ODE Laplace Transform an impulse bring 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 ...
4
votes
0answers
39 views

Inverse Laplace Transform of …

I am trying to find the inverse Laplace transform of $$\frac{1}{pe^{p}}\int_2^p\frac{e^{q}q}{q-1}dq.$$ This function decays as $p$ goes to infinity, so it's reasonable trying to find it's Laplace ...
4
votes
0answers
5k views

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 ...
3
votes
0answers
63 views

Laplace Transform of the Wave Equation

I am given a damped wave equation $u_{tt}(t,x)+2u_t(t,x)=u_{xx}(t,x); \forall t>0$ Now I know the laplace transform of this given the initial conditions, $u(0,x)=\sin x, u_t(0,x)=0;$ is ...
3
votes
0answers
108 views

Interpretation of the Laplace transform

Here's my intuitive understanding of the Fourier transform of $f:{\mathbb R}\rightarrow{\mathbb C}$, defined by $$\mathcal{F}(f)(\omega) = \int_{-\infty}^{\infty}e^{-2 \pi i \, \omega \,x}f(x)dx$$ I ...
2
votes
0answers
32 views

Laplace transform to describe a bounded function

It is easy to show that if a real function $f:\mathbb{R}\rightarrow\mathbb{R}$ is contained in a strip $[a,b]$, that is if $\forall_{x}\, a\le f(x)\le b$, then its Laplace transform is bouned by ...
2
votes
0answers
73 views

Existence of the Laplace Transform

I'm seeing Laplace transforms for the first time, and I'm having trouble understanding the criteria for deciding when they exist. I've read a few websites and books that seem to say that we only ...
2
votes
0answers
221 views

How Heaviside step function changes limits of integration

This question involves the Laplace transform of the convolution of two functions. The derivation in my textbook has a step that really confuses me. First I'll lay out their argument. $$ f(t) = f_1(t) ...
2
votes
0answers
423 views

Inverse Laplace Transform as Bromwich Integral

I am seeking a references that provide a rigorous treatment of the inverse Laplace transform (Bromwich integrals), and how to compute them (beyond using tabled solutions - they don't cover my needs, ...
2
votes
0answers
100 views

Periodic Laplace transform

Here's the questions and the graph I've been struggling with this since Thursday and this is due today. I need help on problems a and b. For a, the question is: "Write $f(t) = \sum_{n=0}^\infty ...
2
votes
0answers
59 views

Criteria for $L^1$ convergence looking at Laplace transforms

Let $(X_n)_{n \geq 0}$ be a sequence of integrable ($\mathbb{E} |X_n| < \infty$) random variables and denote by $l_n(t)$ the Laplace transforms of $X_n$. Similarly, let $X$ be a r.v. and $l(t)$ ...
2
votes
0answers
79 views

Find the inverse laplace transform: $\frac{1}{{{{({s^2} + 1)}^3}}}$

Find the inverse Laplace transform: $$x(t) = {L^{ - 1}}\left[ {\frac{1}{{{{({s^2} + 1)}^3}}}} \right]$$ with $x(t=0)=0$. I did: $${\left[ {{\mathop{\rm R}\nolimits} {\rm{es}}\frac{{{e^{st}}{{(s - ...
2
votes
0answers
77 views

Continuity of the inverse Laplace Transform

If I know $Y(s)$, can I predict when $\mathscr{L}^{-1}[Y(s)]=y(t)$ will be continuous or continuously differentiable or even stronger conditions? For example; I'm solving an ODE with the Laplace ...
2
votes
0answers
105 views

A rational integral with exponential denominator

Prove that: $$\int_{-\infty }^{+\infty }{\frac{{{x}^{4}}\text{d}x}{\left( \beta +{{\text{e}}^{x}} \right)\left( 1-{{\text{e}}^{-x}} \right)}}=\frac{\left( {{\pi }^{2}}+{{\ln }^{2}}\beta ...
2
votes
0answers
1k views

Laplace transform of the square root of a generic function

Let $f(t)$ be a function (for example of time $t$). Is there a general expression of the laplace transform of $\sqrt{f(t)}$ ? Same question for the inverse Laplace transform : Let $f(s)$ be the ...
1
vote
0answers
33 views

Zeros/poles at Laplace and at Fourier Transform

I recently started "relearning" the Laplace transform, and I noticed something. It seems to me that the intuitive idea of poles and zeros is different between these two transforms! For example, in ...
1
vote
0answers
31 views

Help with an improper integral

Can someone please help me evaluate this improper integral? $$\int_{0}^{\infty}\exp\{-au^{-a}-u\}du$$ for $a>0.$
1
vote
0answers
29 views

Figuring out impulse response

I need a little help with figuring out this problem. I understand most of it but the main part I don't understand is: The signal $h''(t)+2*h'(t)+2*h(t)$ is of finite duration. In the problem we are ...
1
vote
0answers
30 views

The Laplace transform - does it have an associated differential operator, if the kernel is to be viewed as a Green's function?

I've begun learning about Green's functions, and if I understand correctly, the Green's function for a linear differential operator $L$ with appropriate boundary conditions is the kernel for the ...
1
vote
0answers
37 views

Transfer Function

Vehicle dynamic system (Bicycle model) is given by the following state space model (which also includes Road Bank angle): $\begin{Bmatrix}\dot{x_1}\\\dot{x_2}\end{Bmatrix}=\begin{bmatrix}a_{11}& ...
1
vote
0answers
38 views

Particular integral equation

Let $a,\sigma, n>0$ be some parameters and define the conditional probability density function $$ p(x,y):= \frac{1}{\sigma\sqrt{2\pi}}\exp\left(-\frac{(y-n-x)^2}{2\sigma^2}\right). $$ Is it ...
1
vote
0answers
50 views

Calculating convolutions of probability density functions

I have a PDE: $$\frac{\partial N (x,u)}{\partial x}=\int _0^uN(x,u)f(u-u')du'$$ $$N(0,u) = \delta (u)$$ Here $f(u)$ is a probability density function for $0 \le u \le u_{max}$, $\int _0 ^ {u_{max}} ...
1
vote
0answers
72 views

Laplace transform - frequency differentiation property (generalization)

Let $\mathcal{L(f(t);s)}$ be the Laplace transform of a function $f$. It is known that the Laplace transform of $\mathcal{L}{(t^nf(t);s)}$ is given as (frequency differentiation property) ...
1
vote
0answers
65 views

Laplace transformation of $\frac{\cos x}{x}$

How would you find the laplace transformation of $\frac{\cos x}{x}$? Like I know you need to add it to another transformation to solve for it. And I know that the transformation of $\cos x$ is ...
1
vote
0answers
125 views

Heat equation via Laplace transform, final inversion…

Working on a PDE that may have solution out there somewhere but I have yet to stumble upon it. I prefer to use the Laplace Transform approach when I can, as I find it more intuitive than some other ...
1
vote
0answers
55 views

Transformed Laplace “solution space”

From my own knowledge I can tell that when we take the Laplace transformation of a function we are in essence transforming our f(t) into a F(s). I've looked at several Q/A here asking for the ...
1
vote
0answers
24 views

Mellin transform and equality of two functions

If say, $\mathcal{M}f(s) = \mathcal{M}g(s)$ in a strip say $0 < a < \Re(s) < b$, then can we say that $f$ and $g$ are identical? Does the Mellin inversion theorem imply this?
1
vote
0answers
20 views

What polynomial transformation is this and how is it related to the original polynomial?

Let $f(X) = a_n X^n + \dots+a_0$. Then the Laplace transform of $f$ is $g(s) = \mathscr{L}\{f\}(s) = \frac{n! a_n}{s^{n+1}} + \dots + \frac{a_0}{s}$. If you now define the polynomial transform of ...
1
vote
0answers
73 views

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 ...
1
vote
0answers
57 views

Laplace transform of a sum of stochastic variables

I have a problem with interpretation of one transformation performed on equation consisting of continuous random variables. Here is the source equation describing recurent relationship between the ...
1
vote
0answers
74 views

Inverse Laplace transform of functions with jump discontinuities

Given a function $F(s)$, suppose we define its inverse Laplace transform as: \begin{equation} f(t) = \lim_{k \to \infty} \frac{(-1)^{k}}{k!}\left(\frac{k}{t}\right)^{k+1}F^{(k)}\left( \frac{k}{t} ...
1
vote
0answers
217 views

Creating intuition about Laplace & Fourier transforms

I've been reading up a bit on control systems theory, and needed to brush up a bit on my Laplace transforms. I know how to transform and invert the transform for pretty much every reasonable function, ...
1
vote
0answers
102 views

Final value theorem on coupled differential equations

Good day, I have two linear and coupled differential equations: $J_{11}\ddot{\theta_1}-n(J_{11}-J_{22}+J_{33})\dot{\theta_3}+n^2(J_{22}-J_{33})\theta_1=T_{c_1}+T_d \\ ...
1
vote
0answers
49 views

Laplace transform of $\sqrt{f(t)}$

I have a question about the Laplace transform. Suppose the Laplace transform of $f(t)$ is known. Is there any relation between the Laplace of transform $f(t)$ and that of $\sqrt{f(t)}$? Thanks in ...
1
vote
0answers
28 views

The Square of the Laplace Transform

I have been looking at the Laplace transform $$\mathcal{L}f(s)=\int_0^{\infty}f(t)e^{-st}dt$$ and I'm trying to find The norm of $\mathcal{L}^2$ The nullspace of The norm of $\mathcal{L}^2$ So ...
1
vote
0answers
39 views

Find $N$ independent solutions through Laplace Transform

The Laplace Transform method gives us one solution of an Ordinary Differential Equation. How can we use the same procedure to get $N$ independent solutions, being $N$ the order of the ODE? Where can ...
1
vote
0answers
211 views

Laplace transform of a product of functions

While trying to compute the Laplace transform of a certain product, part of the calculation leaves me with a Bromwich integral which has the form: ...
1
vote
0answers
64 views

Inverse Laplace transform is required

I shall be very very thankful if some one can find the inverse Laplace of the function given below. I really need it as early as possible. $$ \frac{1}{s-a}\exp ...
1
vote
0answers
68 views

Laplace inverse transform of a complex function

I have a complex expression and I need to find the laplace inverse transform of this complex term. The expression is $$\Large{ e^{\Large{-\frac{x\sqrt{A^2+4Bs}}{2B}}}}$$ In the above expression ...
1
vote
0answers
107 views

Identifiability of a state space system

I'm trying to solve assignment 4E.5 from this sheet (ship steering dynamics). My question are: Do I need to perform the Laplace Transform in order to check for identifiability? The state space model ...
1
vote
0answers
71 views

Laplace transform curiosity

Experimenting in Mathematica, I see that taking the Laplace transform of certain functions $f(t)\neq 0$ actually gives me a non-zero function $F(s)$. However, for these certain functions, taking the ...
1
vote
0answers
188 views

Inverse Laplace Transform by contour integration

In question 1) we get Laplace transform of $$ g(t) = t^a $$ is: $$ \hat g(t)= {1/s^{a+1}}\int_0^\infty e^{-t}x^a $$ then I was stuck at question 2) which asks me to evaluate the inverse laplace ...
1
vote
0answers
158 views

How to check solution of Laplace Transform ODE problem

$$\frac{\mathrm dw(t)}{\mathrm dt}+2w(t)=y(t)$$ $$\frac{\mathrm dy(t)}{\mathrm dt}+3y(t)=2w(t)+f(t)$$ The input to the system is $~f(t)$ and the output is $~y(t).$ The initial conditions are ...
1
vote
0answers
227 views

square root of s domain transfer function

I have a question about calculating the square root of the magnitude of a transfer function. When you take the square root, what is happening? My initial idea of a magnitude is a single value, but in ...
0
votes
0answers
9 views

Laplace transform of a majorated function

I have the following problem. I have an analytic function and I want to show that it is majorated by a convenient function. To do that, it is very helpful to solve the transformed equation. I have a ...
0
votes
0answers
22 views

Which method to use when— higher order differential equations

Over the last unit in my class we have learned various methods of handling higher order diffeqs. However, I want to know how to decide which method to use to solve given diffeqs most efficiently. So ...
0
votes
0answers
17 views

Existence of inverse Laplace tranform

I have two questions about inverse Laplace transform. Given a function $F(s)$, does its inverse Laplace tranform always exists ? If it's not, assume $F(s)$ has an inverse Laplace tranform, does the ...
0
votes
0answers
11 views

A question about Parseval's formula.

In operational calculus there is Parseval's theorem, which states that if $ f(t) \doteqdot F(p), \varphi(t) \doteqdot \Phi(p) $ and both $ F(p) $ and $ \Phi(p) $ are analytical in $ Re p \geq 0 $, ...
0
votes
0answers
33 views

Product of two Whittaker functions

According to 6.669.3 of Gradshteyn and Ryzhik the following identity $$ W_{a,b}(z_1)\,W_{a,b}(z_2) = \frac{2\sqrt{z_1z_2}}{\Gamma(1/2+b-a)\,\Gamma(1/2-b-a)}\int_0^\infty ...