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

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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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laplace transform and infinitely differentiation

This fact appears in my statistics textbook (Pg 543, statistical decision theory and bayesian analysis). it says : for normal distribution the generalized bayes estimator becomes \begin{align*} ...
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Is impulse response always differentiation of unit step response of a system?

I was trying to solve a question in which the transfer function of a system was asked, its unit step response being given: c(t) = 1-10exp(-t) The method that the book followed was to first find out ...
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64 views

Computing the inverse Laplace transform of this?

What's the correct way to go about computing the Inverse Laplace transform of this? $$\frac{-2s + 1}{(s^2+2s+5)}$$ I Completed the square on the bottom but what do you do now? $$\frac{-2s + ...
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How do I go about performing the following Laplace transform?

I'm unable to compute the following Laplace transform. How do I deal with cases such as $$f(t) = \sin(t-3)\theta(t) \quad \text{or} \quad f(t) = \sin(t-3)\theta(t-3),$$ where $\theta(t)$ is the ...
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Complex Integral with exponential

I've been struggling with this: $$\int_{0}^{\infty }\frac{e^{-px}}{x^{2}+1}\mathrm{d}x, \; \; p\ge 0.$$
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Laplace transform of product of $\sinh(t)$ and $\cos(t)$

My question is this: If i have a function $f(t)=\sinh(t)\cos(t)$ how would I go about finding the Laplace transform? I tried putting it into the integral defining Laplace transformation: $$ F(s)= ...
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214 views

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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How do I perform Inverse Laplace on this function?

$$ F(S) = \frac{-S+11}{S^2-2S-3} $$ Howo do I find $f(t)$? What is a good strategy for attacking these types of problems? Thanks a bunch in advance for your help!
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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 ...
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74 views

Laplace transformation problem

There is a timely unchanged continuous function : $$H(s)=\frac{s-1}{s+1}$$ At the entry of the system exists a $x(t)$ which Laplace's transformation is: $$X(s)=\frac{(5s^2 - 15s + ...
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Use Laplace transform to solve the following initial–value problems.

Use Laplace transform to solve the following initial–value problem. $y′′′′ + 2y′′ + y = 0, y(0) = 1, y′(0) = −1, y′′(0) = 0, y′′′(0) = 2$ Answer $s^4 L(s) - s^3y(0) -s^2 y'(0) - s y''(0) - y'''(0) ...
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Solving initial value problem using Laplace Transform

Use Laplace transform to solve the following initial–value problems. a). $y'' + y = e^{−t}\cos 2t, \\ y(0) = 2, y′(0) = 1$ After using the concept of partial fraction and using Elementary Laplace ...
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666 views

Power series for the Bessel function using Laplace transforms?

The Bessel's function of the first kind of order zero, $J_0$ is the solution to $$ty''+y'+ty=0$$ which satisfies $J_0(0)=1$ The Laplace transform of this equation gives ...
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49 views

Laplace transform restriction and differentiation

every one.I have just started learning Laplace transform.However, there are two main conceptual problems I can't convince myself. The first problem is about the restriction of this integral, I ...
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52 views

help with laplace transform

Can you please help me with this Laplace transform? I used wolfram alpha to get the answer but I need some hints about the procedure to get to that answer. $$ \mathcal{L}\left(\frac ...
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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 ...
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392 views

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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46 views

inverse laplace transform - with symbolic variables

Transform: $$ F(s) = \frac{2s^2 + (a-6b)s + a^2 - 4ab}{(s^2-a^2)(s-2b)} $$ My steps: $$ F(s) = \frac{2s^2 + (a-6b)s + a^2 - 4ab}{(s+a)(s-a)(s-2b)} $$ $$ = \frac{A}{s+a} + \frac{B}{s-a} + ...
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236 views

Laplace Transforms and third-order derivatives

The question is to calculate the Laplace transform of $(1 + t.e^{-t})^3$. I know that this can be done using a property where the problem is of the form of $t.f(t)$. However, I seem to be messing up ...
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Stuck on laplace transform question

I have to solve the following initial value problem using the laplace transformation: $$y'' + 4y = 0$$$$y_0 = c_1, y'(0) = c_2$$ I have taken the laplace transform of both sides, then rearranged ...
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87 views

Partial fractions for inverse laplace transform

I have the following function for which I need to find the inverse laplace transform: $$\frac1{s(s^2+1)^2}$$ Am I correct in saying the partial fraction is: ...
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59 views

Evaluating Laplace Transform

I have a Laplace transform function of the following form and I'm trying to evaluate it. From my research I think I need to take the Inverse Laplace Transform and then integrate, but I'm having ...
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53 views

Evaluating integration with Laplace transform

I am taking a differential equation class and for Laplace transformations and I have to find $$\displaystyle \int_0^\infty \dfrac{\sin t}{t}dt.$$ How can I do that?
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Laplace transform and integration together

The question, given in the textbook, is somewhat different. However, I am rephrasing it as follows: $$ \frac{2}{\pi}\int_{0}^{\frac{\pi}{2}} \left[ \mathcal L \lbrace \cos(t\cos\theta) \rbrace ...
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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} ...
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More Laplace! - help needed

Here is the exam question that I am practicing: I have completed the first two parts to this question (thankfully to stackexchange) Laplace question - help needed Laplace question continued ...
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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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96 views

Find the inverse Laplace transform.

What is Laplace inverse of $$\dfrac{1}{\left({s}^{2}+{1}\right)^{2}}$$
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70 views

Laplace question - help needed

I am currently studying the Laplace transformation and came across this question: I have no idea of how to start this and am completely lost. If anyone could help I would be really grateful. ...
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98 views

inverse Radon transform

I'm having a trouble understanding a proof in regards to the inversion of the Radon transform in $\mathbb{R}^3$. The statement is as follows: if $f \in \mathcal{S}(\mathbb{R^3})$, then ...
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129 views

Laplace Transformation Applications

In one of our Mathematics lecture our Prof told us that similar to Logarithmic Transformations we can use Laplace Transformations to solve difficult equations. What kind of equations do Laplace ...
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63 views

Integration by Parts confusion

I am using this video to learn Laplace Transform. The example used is a fairly basic one: $$ \int_{0}^{\infty}t.e^{-st}dt $$ Simple enough, you need to integrate by ...
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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, ...
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Is the Laplace transform a functor?

I may be oversimplifying, as I know very little about category theory, but: Does the Laplace transform, which—to my limited recollection—is a morphism between differential equations and algebraic ...
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What is the Laplace transform of $\dfrac{y'(t)}{t}$?

I know that ${\mathcal{L}}\left\{ {\dfrac{1}{t}y\left( t \right)} \right\} = \int\limits_s^\infty {Y\left( u \right){\text{ d}}} u$ and that ${\mathcal{L}}\left\{ y'(t) \right\}=sY(s)-y(0)$ How ...
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Is it true that the Laplace Transform of a real function with compact support is always entire?

Is it true that the Laplace Transform of a real function with compact support is always entire (entire = complex derivative exists on the entire complex plane)?
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What $f(t)$ satisfies the inverse Laplace transform $\mathcal{L}^{-1}\left\{\frac{p'(s)}{p(s)}\right\}=f(t),$ where the polynomial $p$ is given.

What is the general form of $f(t)$ in the inverse Laplace transform $$\mathcal{L}^{-1}\left\{\frac{p'(s)}{p(s)}\right\}=f(t),$$ for some given polynomial $p(s)$ ? Put another way, what $f(t)$ ...
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Finding Transfer functions for linearised systems

I'm using Nise for my control systems class. Finding a linearised system is all gravy baby, but when it comes to finding the transfer function Nise does some stuff which confounds me: See page 6/7, ...
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151 views

A finite difference method for robust convergence despite large time steps in first order ODE

Suppose we're looking at a first order ODE of the form $$ \frac{dx}{dt}=-\lambda x+ b u $$ where $\lambda$ and b are functions of $x$ and $u$ is an 'energy generating' term which is a function of $x$ ...
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Fourier, Laplace, … and other Integral-transformations

I know Laplace, Fourier and Mellin-Transformation. Is there a general theory of transformations? My main interest is about classification of transformations satisfying specified properties like ...
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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 ...
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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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1answer
114 views

1st order linear DE with step function input

the 1st order linear equation is: $y'(t) + \frac D M y(t) = f(t)$ with constants: $D = 100kg/s$ $M = 1000kg$ $f(t) = Fu(t)$ <-- that's Force x the unit step function an initial condition: ...
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Inverse Laplace Transform of Discontinuous Function

I'm currently studying transform of discontinuous and periodic functions (Differential Equations.) I was presented with the following question. $$\dfrac{se^{-3s}}{s^2+4s+5}$$ (Sorry, I couldn't get ...
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43 views

Analyticity of a two-sided Laplace-Stieltjes transform

Consider $$ g(y)=\int_{-\infty}^{+\infty} e^{-yt}d\mu(t) $$ convergent for $y\in(a,b)$ for some $a,b>0$; and with $\mu(t)$ a $\sigma$-finite and non-negative Borel measure on $\mathbb{R}$. I'm ...
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Analytic Function vs Exponential Order function

We say that a function $f$ is of exponential order $\alpha$ if there exist constants: $M$, $\alpha$, $T$ such that for $x>T$ $$f(x)<M\cdot e^{\alpha x}$$ Polynomials are of exponential order. ...
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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, ...
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346 views

Simplifying trig expression for Laplace transform

I'm working on the following Laplace transform problem at the moment, and I'm a little stuck. $$\mathcal{L} \{\sin(2x)\cos(5x) \}$$ I don't recall any trig identity that would apply here. I know ...
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161 views

discretize a function using $z$-transform

I would like to discretize the following continuous function using $z$-transform: $$G(s)=\frac{s+1}{s^2+s+1}$$ The process I am using is to take the inverse Laplace transform of $\frac{G(s)}{s}$ and ...