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

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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 ...
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Inverse Laplace Transform,

I have been stuck on this problem for quite a bit, have tried to look at similar answers on website but no help... The original questions is, Solve the IVP $\ y''+y=\sin(t);y(0)=1;y'(0)=0$ I ...
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321 views

Tandem queue - response time distribution

In tandem queue with two queuing system, each server has exp(mu0) and exp(mu1) service time distribution and arrival rate is poisson(lambda). Scheduling policy is FCFS. What would be the response time ...
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40 views

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 ...
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2answers
74 views

Intuition behind convolution identity for Laplace transforms

Convolutions, relatively speaking, are fairly straightforward for simple systems (from an applied perspective), but I cannot, at all, find the intuition behind the Laplace identity for convolutions. ...
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32 views

Why does s = z+1?

What exactly is Laplace transform? motivated me to ask why unit function is 1/s by Laplace transform and 1/(1-z) by Z-transform? Both seem to be integrals of delta-pulse and secondary integration ...
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30 views

The Laplace transform of the delta function

$f(t) = 1$ must equal to delta function in the Laplace domain since "constant in one domain is delta in the other domain". On the other hand, table says that it must be $1/s$ in the Laplace domain. ...
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32 views

Laplace transform of product of two function

Consider the following integral of product of two function $$ \int_0 ^t f(s)g(s-t)ds $$ i want to know the laplace transform of above term w.r.t t. if $g(s-t)$ is replaced by $g(t-s)$, there is a ...
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Laplace transform of noncentral chi-square distribution

I'm interested in non central chi-square distribution. More specifically, i want to derive the laplace transform of noncentral chi-sqruae disribution or density function. Let me know whether it ...
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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 ...
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26 views

Determine the Laplace transform using the heaviside fuction

Kindly assist with the question below $$ \mathcal{L}\{ \mbox{Sinh}(3t) \, H(t-3t) \} $$ I tried using the Heaviside property to determine the laplace transform, but i got stock on the way. i used the ...
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19 views

Solving Differential equations with Laplace transform

$\displaystyle y''+4y'+3y=e^{-t}$, given $\displaystyle y(0)=y'(0)=1$ My Attempt: Taking Laplace transforms on both sides $\displaystyle $ $\displaystyle [s^2\bar y-sy(0)-y'(0)]+4[s\bar ...
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Solving simultaneous equations using Laplace transforms

$\displaystyle \frac{dx}{dt}+y=\sin t$ $\displaystyle \frac{dy}{dt}+x=\cos t$, given $\displaystyle x(0)=2, y(0)=0$ My Attempt: Taking Laplace transforms on both sides $\displaystyle $ ...
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31 views

Laplace transform - Heaviside algebra

I'm strugling with some algebra around a laplace transform with heaviside. The start function is $L(2tH(1-t)) + L(2H(t-1))$ so from this, I'm supposed to convert it to $L(2t) + L(2(1-t)H(t-1))$ ...
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Find the inverse laplace transform of $\displaystyle \frac{s}{a^2s^2+b^2}$.

Find the inverse laplace transform of $\displaystyle \frac{s}{a^2s^2+b^2}$ My Thoughts: Take $\displaystyle s^*=\frac{s}{a}$ and $b^*=\frac{b}{a}$ and divide numerator and denominator by $a^2$. ...
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1answer
38 views

How to find the Direct Discrete Laplace Transform of ${2n \choose n}$

Some time ago I developed a discrete version of the Laplace transform for the purpose of calculating sums and solve finite difference equations with constant coefficients. The notes below are a ...
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Heaviside Expansion Theorem with matrices

Is the Heaviside Expansion Theorem (HE) for the determination of inverse Laplace Transforms true for matrix expressions such that $\mathscr{L}^{-1}[\mathbf{P}(s)\mathbf{Q}^{-1}(s)] = \sum_i^n ...
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Determine tha Laplace transform using Heaviside fucntion

I want to determine the Laplace transform of the following function: $$f(t):t \mapsto \begin{cases}0, \quad t< 0 \\t, \quad 0\leq t \leq2\\2,\quad t>2\end{cases}$$ I have done it using standard ...
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Confused by a Laplace transform of $f(t)=t^ne^{at}$

Having looked at my lecture notes I was confused by the following part of a derivation of a Laplace transform for the function $\;f(t)=t^ne^{at} ,\quad n\ge0,\; a \in \mathbb{C}, \; f(t)=0 \;\forall ...
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1answer
13 views

Inverse Laplace transform, none factorable denominator

I am really stumpted on this problem and can't seem to figure out where to go from where I am. Can anyone give me some advice or hint where I should do next? Here is the problem: ...
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1answer
28 views

Using Laplace transforms to solve a convolution of two functions

Hi I have this problem where I need to take the convolution of functions and I am not sure if I got the right answer or something close so any advice or help would be very appreciated. So here is the ...
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Laplace transform of Differential Equation with a piecewise function

Hi I have this question and I am horribly stuck at one part and I cant seem to figure out if i did something wrong so any advice or help would be greatly apprecaited. Here is the question: ...
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1answer
28 views

Dirac Delta Function, Initial Value Problem

Hi I finished this IVP but I cant seem to get the right answer can someone give me some advice as to where I went wrong and point me in the right direction as to how to fix it. Here is the problem and ...
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54 views

Contour integral: different answers with different contours

Good day to everyone. I have a following contour integral problem. I have to find a solution for the integral $$\underset{\gamma_r }{\oint }\frac{e^{\lambda s} }{(1-s) s^{a-b} \left(s-\theta ...
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$\lim_{s\to 0^+}\int_0^\infty a(t) e^{-st} dt $

$$\int_0^\infty a(t) e^{-st} dt = f(s)$$ What is the meaning of the limit of this integral as $s\to 0^+.$
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Inverse laplace Transform of

How to calculate the inverse laplace transform of $\frac{\omega }{\left ( s^{2}+\omega ^{2} \right )( s^{2}+\omega ^{2} )} $ ?
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Origin of Laplace Transform

Is the Laplace transform the continuous version of the infinite power series? $$ \sum_{n=0}^\infty a_nx^n$$ becomes $$\int_0^\infty f(t)e^{-st}dt$$ I learned this by watching this video lecture: ...
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23 views

Inverse Laplace Transform partial fraction

While solving a second order differential equation, I have reached at a stage where I have to calculate the inverse laplace transform of $\frac{\omega ^{2}}{\left ( s^{2}+\omega ^{2} \right )( ...
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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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1answer
75 views

Laplace transform of : $t^{\gamma-1} F(\alpha,\beta,\delta,\frac{t}{d})$, where $F$ is the Gauss' hypergeometric function

What is the Laplace transform of : $t^{\gamma-1} F(\alpha,\beta,\delta,\frac{t}{d})$, where $\gamma >0 $ and $F$ is the Gauss' hypergeometric function. Note that I have the Laplace transform of : ...
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Inverse Laplace Transform of a complicated function

Help with finding the inverse Laplace Transform of $$F(s) = \frac{1}{s\sqrt{M^{2}-s^{2}}}e^{\frac{\sqrt{N^{2}-s^{2}}}{s}}$$
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Maximum output value

I have the following transfer function: $\frac{sX_{2,r}(s)}{X_{2,r}^0} = \frac{s\omega_n^2R}{s^2 + 2\zeta \omega_n s + \omega_n^2}$ If I am correct, the signal $\dot{x}_{2,r}(t)$ is limited to the ...
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How did this function turn to the other through the laplace transform?

I was studying for my exam doing exercises, and in one of the questions where you had to use the laplace transform, they transformed this: $$X'(t)=AX(t)$$ into this:$$sX(s)-X(t=0)=AX(s)$$ where ...
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What is the $s$ in the Laplace transform?

I know the formula, which is $$F(s) = \int_0^\infty f(t) e^{-st}\,dt$$ but I don't understand what the $s$ is, and I've been searching everywhere and can't seem to find an answer, or maybe I'm just ...
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Laplace transform and Power series

As it can be read here, Discrete to Continuous Representations of Functions via Laplace Transforms? the Laplace transform is a continuous analog of a power series in which the discrete parameter n is ...
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Inverse Laplace Transformation of an exponential function

How one could find the inverse Laplace transformation of $\exp(-(b/(b+s))^k)$? Where both $b$ and $k$ are positive.
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Prove that some function is the solution of some equation

Show that $$x(t)=\sum_{n=0}^{\infty}\frac{(-1)^n(t/2)^{2n}}{(n!)^2}$$ is the solution of $$x*x=\int_{0}^t x(u)x(t-u)du=\sin t$$ My approach: I suppose that I have to use the Laplace transform. I ...
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Inverse Laplace transform of $\frac{s}{\sqrt{(s+a)^3}}$

Trying to find the inverse Laplace transform of $\frac{s}{\sqrt{(s+a)^3}}$. So solving $\oint_B dz \: \frac{z}{\sqrt{(z+a)^3}} e^{z t}$ (Bromwich contour). I tried doing a u-substitution with $u=z+a$ ...
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338 views

inverse Laplace transform of $e^\sqrt{as}$

I am trying to find the inverse Laplace transform of $e^\sqrt{as}$ for $a>0$. So we need to solve $\oint_B dz \: e^\sqrt{az} e^{z t}$ (Bromwich contour), but not sure how to start. How do we even ...
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52 views

Laplace Transform of $\frac{a}{2\sqrt{\pi}}t^{-3/2}\exp(-a^2/(4t))$

I'm trying to prove that the Laplace Transform of $$\frac{a} {2\sqrt{\pi}}t^{-3/2}\exp(-a^2/(4t)) $$ is $$\exp(-a\sqrt{s}); $$ from the definition of Laplace Transform we should compute ...
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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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Laplace Transform and differential equation

Question is d^2y/dt^2+3dy/dt+2y=u(t-1), where y(0)=0 and y'(0)=1 This is my working, is it correct? (For my 1st part of the answer)
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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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Inverse Mellin transform of simple function

How should the following inverse Mellin transform integral be evaluated: $ f(x) = \displaystyle \frac{\alpha}{2\pi i}\int_{c-i\infty}^{c+i\infty}(\alpha ...
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1answer
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Laplace transformation $y''+2y'+2y=3\sin x+\cos x$

Given$$y''+2y'+2y=3\sin x+\cos x$$ Transform to image region $$Y(s)(s^2+2s+2)=\frac{3}{s^2+1}+\frac{s}{s^2+1}-s-2$$ $$Y(s)((s^2+2s+1)+1)=\frac{3}{s^2+1}+\frac{s}{s^2+1}-s-2$$ ...
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Laplace transform non exponential order.

There's an existance/uniqueness theorem which states: if function f is of exponential order, then the Laplace transform of f exsists and is unique. Since there's no theorem in my book which states ...
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Transfer Function non-zero initial conditions Laplace Transform

okay I know how to get transfer functions using the Laplace Transform assuming zero initial conditions but I would like to know how to deal with non-zero initial conditions. One of my maths books ...
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Explanation on how is simplified expression $\frac{s^2+3s+3}{2s^2+7s+7}$

This is done in the solution of exercise in order to make it possible to do inverse Laplace transform. Though I am not sure how is that done, so here it is: ...
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Confusion regarding bilinear transform.

I was reading my book where the z-transform of a signal is derived to be ${1-e^{-2bT}z^{-1}}$ . Then it goes on to say that by applying the bilinear transform we can get ...
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What is Wolfram Alpha returning when I ask it to calculate $\mathcal{L}[e^{x^2}](s)$?

As my instructor mentioned more times than could ever seem necessary, arguments of the Laplace transform have to have growth of exponential or less. I typed this "nonexample" into W|A and noticed ...