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

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express as contour integral $ f(x) = \int_0^\infty dt \; e^{-t/g} \; \frac{1}{\sqrt{1 - 2 t x}} $

Let $0 < x < 1$, I have to compute this Laplace transform: $$ f(x) = \int_0^\infty dt \; e^{-t/g} \; \frac{1}{\sqrt{1 - 2 t x}} $$ I am not 100% this interal is defined. If $t > ...
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28 views

Why is causality important for laplace transformations? [closed]

Could someone please explain why causality is important for laplace transformations?
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3answers
30 views

Verify second order Cauchy Riemann equations

How do I differentiate the equations in 12? I understand the hint, but I'm not sure how to act on it.
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24 views

Where are the particular and homogeneous solution of the ODE when using Laplace?

When solving an ODE with Laplace, it seems as if there is no distinction between the homogeneous and particular solution. As if you calculated both at once. Is this correct? How does it come? Where ...
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172 views

What are disadvantages/limitations of Laplace?

I was curious about what limitations the famous Laplace theorem for solving ODE had and what drawbacks it may have. PS: I am NOT familiar with Fourier
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36 views

Comparison between Laplace, operator calculus and system of first order ODE

I am trying to understand those three methods to solve differential equations. I would like to know what actually are the differences between the three: Laplace calculus operator conversion to a ...
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1answer
33 views

Given integral equation, find $y(1)$

Let $y(t)$ be a continuous function on $[0,\infty)$ whose Laplace transforms exists. If $y(t)$ satisfies $$\int\limits_0^t(1-\cos(t-\tau))y(\tau)d\tau=t^4\to(1)$$ then $y(1)=$ I was able to find ...
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2answers
28 views

To solve given differential equation using laplace transform

I am solving following diff eqn using laplace transform: \begin{eqnarray} y''+y= \begin{cases} 0, & \text{if 0<t<2 $\pi$}\\ \sin t, & \text{t>$2\pi$} ...
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185 views

Book on applied mathematics/analysis

My Applied Mathematics course covers these subjects: -Calculus of Variations -Laplace Transform -Fourier Analysis -Special Functions -Integral Equations And as an introduction to the subject it has ...
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32 views

To find the value of a constant when the Laplace transform of a function is given

This question is regarding my previous post Find the value of a constant when the Laplace transform of a function is given where the hint was given by Moo to find the laplace transform of ...
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33 views

Find the value of a constant when the Laplace transform of a function is given

I am given that $F(s) = \tan^{-1}{s} + k$ is the laplace transform of some function $f(t)$ $t\geq 0$ . I have to find the value of $k$. What I get is: $F(s) = L(f(t))$ $\Rightarrow L(f(t)) = ...
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1answer
26 views

Laplace Transform of Shifted Function

Why do we need to multiply the shifted function $f(t - a)$ by the shifted step function $u(t - a)$ to obtain the Laplace transform? $$ \mathcal{L\{f(t - a)\}} = \int_0^\infty u(t - a)f(t - ...
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33 views

Convolution of complex functions (Laplace Domain)

Convolution of functions in the time domain is equivalent to multiplication in the frequency domain. However, I am interested in multiplication of functions in the time domain, which is convolution in ...
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1answer
23 views

Using Laplace Transform to solve this ODE

How to solve this ODE, with Laplace Transform: $$ \begin{cases} 20y'(x)+y(x)+4y''(x)=20\\ y(0)=10\\ 4y'(0)=-2 \end{cases} $$ Thanks in advance. My work: ...
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1answer
65 views

Solve the IVP $xy'' + y' + 4xy = 0, y(0) = 3, y'(0) = 0$

It has to be solved with Laplace transform and then converted to Bessel equation. $L(xy'') = -\frac{dL(y'')}{ds}$ $L(4xy) = -\frac{4dL(y)}{ds}$ $L(y'') = s²L(y) - sy(0) - y'(0) = s²L(y) -3s$ ...
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14 views

The need for two laplace transforms

So I have recently come across Laplace transforms, but I have seen one sided and two sided laplace transforms, my question is why do we need two kinds of transforms, when do we use which transform?
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35 views

2-sided Laplace transform of $\exp(-(t + e^{-t}))$

I'm having trouble finding an analytic solution to the 2-sided Laplace transform of; $$f(t) = \exp(-(t + e^{-t}))$$ Integration by parts doesn't seem to help. Any pointers appreciated. It seems like ...
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48 views

Laplace trasform

i am trying to do this exercise but i do not get it. The laplace trasform is: \begin{equation} T(f)(s)= \int_{0}^{\infty} f(t)e^{-st} dt \end{equation} The exercise is: a) If $f$ is the ...
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1answer
69 views

Solve $y''-xy'+y = 1 , y(0)=1, y'(0) = 2 $ with Laplace transform

What's making me get stuck is the Laplace transform of $xy'$. I'm aware of different methods of solving this, but it's asking specifically for Laplace transform.
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99 views

Inverse Laplace transform of $1/\sqrt{s^2-a^2}$ using complex integration

I want to find the inverse Laplace transform of $$F(s) = \frac{1}{\sqrt{s^2-a^2}}$$ preferably using the Bromwich integral: $$f(t) = \frac{1}{2\pi i}\int_{\beta -I \infty}^{\beta +i ...
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1answer
24 views

Calculating Laplace inverse

I'm having difficulties calculating a simple Laplace inverse : $$ \frac{S-4}{S^2-2S-11} $$ I'm new at this and couldn't find good examples for this case. could you please guide me ?
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Closed-loop transfer function in the time domain

In a simple linear system with feedback (figure 1), the closed-loop transfer function $H(s)$ can be written as $$ H(s)=\frac{X_o(s)}{X_i(s)} = \frac{G(s)}{1+G(s)F(s)} $$ by solving the equations $$ ...
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2answers
19 views

How to get the Laplace transform of $t \cdot f(t) \cdot e^t$

Is there a formula to get the Laplace transform of $t \cdot f(t) \cdot e^t$ ? I tried integration, but that got me nowhere, because I'm probably missing something. Any ideas?
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58 views

How can I find the Fourier transform of constant value like $1$.

The textbook told me that $\mathbb F[1] = \delta(f)$ and $\mathbb F[\delta(t)]=1$. It is easy to prove that $\mathbb F[\delta(t)] = 1$. $$ \mathbb F[\delta(t)] = \int_{-\infty}^\infty ...
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The inverse Laplace transform of $\Gamma\left(\zeta\right) \, W_{\zeta,\mu}(z) $

Someone has a reference that addresses an integral of the followns type $$I = \frac{1}{2i\pi} \int_{\sigma-i\infty}^{\sigma+i\infty} e^{t\zeta} \, \Gamma\left(\zeta\right) \, W_{\zeta,\mu}(z) \, ...
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31 views

Confusion in the usage/property of Laplace Transform.

While proving that $$\int^{\infty}_0 \frac{\sin x}xdx$$ I saw the Laplace Transform proof. It used that $$\cal L\left\{\frac{\sin t}{t}\right\}=\int^\infty_0 \cal L\left\{\sin(t)\right\}d\sigma$$ So ...
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92 views

What is the difference between an impulse response and a transferfunction?

An imupulse response, is the output you get when you apply an impulse, like a delta dirac function, to your system (only for LTI?). By knowing the impulse response you know the system. The ...
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zeros/poles of Laplace transforms of Dirac combs (Riemann zeta function)

let's define $p_\alpha(n) = \displaystyle\int_1^n x^\alpha dx$ so that $\left\{\begin{array}{lll} p_0(n) &=& n-1 \\ p_{-1}(n) &=& \ln n \\ p_\alpha(n) &=& \frac{\textstyle ...
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2answers
63 views

Inverse Laplace transform of $\tan^{−1}\left(\frac{1}{s}\right)$

I'm studying Laplace transformations, but I don't understand where $-\frac{1}{t}$ comes from. And what is the relationship between the corollary and the example?
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What is the Laplace transform of $\cos(4t+8)$?

Could someone please explain how to transform this to the Laplace domain? I've tried to use the definition of Laplace (not sure this is the easiest way). $$\int_{0}^{t}e^{-st}f(t)\,dt$$ ...
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Residue theorem with pole on integration path

I have to calculate the inverse Laplace transform of $\dfrac{1}{s^2+1}$ (which I know is sin(x)) by residue theorem: $\int^{i \infty}_{-i \infty}exp(t\cdot s)\cdot \dfrac{1}{s^2+1}\mathrm{d}s$. ...
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1answer
33 views

A Partial fraction expansion questions about Laplace transform

I am learning signals and systems. Our teacher give us the following answer, it's about Laplace transform . But I can't figure out the second line, the calculation of k1,k2,k3,k4. why they can be ...
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52 views

complex integration, residues, inverse Laplace transform, calculus

Dear Mathematicians, I kindly ask your expertise on complex integration. The problem is the last step in the solution to a differential equation, using an inverse Laplace transform. I know that the ...
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Application of Initial Value Theorem

Let $$F(s):=\frac{s}{2s-i}$$ be the Laplace transform of some $f(t)$. I have been asked to compute $f(0^+)$ assuming that this quantity, intended as limit, exists. I thought I could apply the IVT ...
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43 views

Inverse Laplace Transform of a polynomial

It seems to me that most distributions (positive, bounded, finite integral, continuous (to some degree)) must have a polynomial Bilateral Laplace transform. How is this inverted? Most inverse ...
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41 views

Inverse Laplace problem using partition fraction

Hello I am solving inverse Laplace transform using partial fraction. The question is: $$ X(s) = \frac{10(s+1)}{s(s^2+4s+8)} => \frac{10(s+1)}{s((s+2)^2+4)} $$ $$ \frac {C1} {s} + ...
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Separating differential equatons

The initial equation: $$ y''= g-((C*(y')^2)/m) $$ and I am trying to separate it into two differential equations. I also have that the aerodynamic force $F=C*(y')^2$. The initial equation describes ...
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Moment Generating Function of R.V.'s Y/X?

I want to calculate the MGF of $$ \left(\frac Y X \right)^\alpha, $$ where R.V.'s $Y \in Exp(1)$, $X$ has the Laplace transform $L_X(s)=e^{-s^\alpha}$ and $\alpha \in (0,1)$. $X$ and $Y$ are ...
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41 views

Eigenvalues/vectors of the Laplace transform?

I'm learning about eigenvalues and eigenvectors (finally starting to get them). This might be a silly question, but what is/are the eigenvector(s) of the Laplace transform? I mean, what ...
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26 views

Find the normal form of this function

A second order control theory function looks like: $$\text{H}_{(s)}=\frac{\text{K}_p}{\frac{1}{\omega_0^2}\cdot s^2+\frac{2\beta}{\omega_0}\cdot s+1}$$ Now I've got the function, with ...
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16 views

State space representation for fractional order transfer function

What is the state space representation for the following filters? $H(s)=\frac{Y(s)}{U(s)}=\frac1{s^\frac12}$ $H(s)=\frac{Y(s)}{U(s)}=\frac1{s^\frac12+1}$ Where $u(t)$ is the input and $y(t)$ is ...
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inverse Laplace transform by finding residues of essential singularities

I want to find the inverse Laplace transform of $$F(s)=\exp\Big(-\sqrt{2s}\tanh(\sqrt{2s})\Big).$$ Despite the square roots, $F$ doesn't have any branch points since ...
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Inverse Laplace transform of an exponential function

What is the inverse Laplace transform of $$\frac{e^{\frac{-2}{s}}}{s}$$ I have seen an answer using Maclaurin series expansion of this function. This function is not analytic at $0$, so, is such ...
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Inverse Laplace Transform (Natural Logarithm Case)

I have a problem about Inverse Laplace Transform, I would be appreciated to get your help for solving this problem (It took me about several hours to think but didn't come up with any solution). ...
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Inverse Laplace Transform of $F(s) = \frac{(3s^2+9s+14)*e^{-5s}}{(s^3+4s^2+7s)}$

Find the inverse Laplace Transform of $F(s) = \frac{(3s^2+9s+14)*e^{-5s}}{(s^3+4s^2+7s)}$ I have found the Simplified $F(s) = (\frac2s+\frac{s+1}{(s+2)^2+3})*e^{-5s}$ I am having trouble figuring ...
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43 views

Inverse Laplace transform of complicated function

I have a Laplace transformed function that I'd like to transform back. It's quite a complex function however, which is why I am stuck: $$C(x,s) = ...
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15 views

Time scaling in Laplace transformation

A Laplace transform property: $f(at)\leftrightarrow \frac{1}{a}F(\frac{s}{a})$ where $f\leftrightarrow F$. Question: Is $a>0$ necessary for this property?
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Laplace transform existence condition

Laplace transform existence conditions has the following phrase: "..piecewise continuous in every finite closed interval $0 \leq t \leq b$ " Can we replace it with "...piecewise continuous in ...
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31 views

How to inverse laplace the following

Been stuck on this for a while now and I have an exam tomorrow at 10am and I fear this might come up and it's only inverse laplace I can't do, this is my attempt at it but I know it's wrong..can ...
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1answer
28 views

Laplace Transform with unit step function

Here is the link to the question and the answer: https://gyazo.com/d98918d0e0eaabe88606c0314aff0aca Here is the link to what I have done: https://gyazo.com/20ddc3822c0c7bca888cd3334dd78281 What ...