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

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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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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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4answers
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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
29 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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51 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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30 views

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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40 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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2answers
36 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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2answers
38 views

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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1answer
37 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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1answer
25 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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0answers
13 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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2answers
53 views

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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1answer
38 views

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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40 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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1answer
29 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
27 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 ...
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2answers
37 views

Did I Inverse Laplace correctly?

$$L^{-1}\frac{4s}{(s-6)^{3}}$$ $$4L^{-1}\frac{s}{s^{3}}|s=s-6$$ $$4L^{-1}\frac{1}{s^{2}}|s=s-6$$ $$4L^{-1}\frac{1!}{s^{1+1}}|s=s-6$$ $$4te^{6t}$$ Is this correct? symbolab and Wolfram are giving me ...
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27 views

Laplace Transform method to solve a differential equation

I'm trying to solve this diffrential equation using Laplace Transform method but have trouble finishing it (x(0) = 0): $$\frac{dx}{dt}+2x=15e^{-2t}$$ $$L\left(\frac{dx}{dt}+2x=15e^{-2t}\right)$$ ...
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24 views

Finding Inverse Laplace Transform (Using the table of Laplace transform)

I want to take the inverse laplace transform of $$\frac{e^{-s}}{s(s^{2}+1)}$$ So I separate the equation into $$e^{-s}\times\frac{1}{s(s^{2}+1)}$$ Now, I take the partial fraction of ...
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14 views

Replicate Matlab integrator block in MS excel

I created a simple diagram to solve ordinary differential equation as shown below. Simple ODE I was trying to compute the result of xf_dot manually in Ms Excel but I did not get the same answer ...
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1answer
34 views

How to obtain this partial fraction decomposition?

I am studying Laplace transforms right now and got stuck at this step that involves a weird partial fraction decomposition. It looks like the instructor skipped a bunch of steps and assigned ...
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2answers
47 views

Solving a differential equation with the heaviside unit step function

I am having trouble figuring out what exactly to do for this question. Given the initial conditions y(0)=1 and the equation y'-2y=4-3u(t-2) where u is the heaviside unit step function. I took the ...
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1answer
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Restrictions on repeated use of initial conditions in ODE

It seems to be common practice when solving ODE's to keep a count of what conditions you have used. I was under the impression that once a condition has been used it cannot be used again. However, I ...
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49 views

Inverse Laplace transform seems to be always vanishing but it couldn't!

Let's consider $x\in (0,1)$ and the distribution $p(x)=\lambda x^\lambda$, $\lambda>0$. I would like to find the pdf of the sum. The characteristic function of the $N$ sum reads: \begin{equation} ...
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1answer
28 views

Inverse Laplace transform of convolution

$$\int_{0}^t \sin(2\pi(t-T)) \delta(t-5) \, dt$$ Wouldn't you just replace the $T$ in $\sin(2\pi(t-T))$ with a $5$ and that would be the answer?
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Laplace transformation of a piecewise function?

So I know in general how to do the laplace transformation of piecewise functions, but I ran into a different kind of piecewise than I have been doing so far. So I know for a function like: I just ...
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Find the Fourier transform for this function

Find the Fourier transform for this function $$f(x)=e^{x-e^x}$$ My Solution:- $T[f(x)]=\frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} e^{-ikx} f(x)dx$ $=\frac{1}{\sqrt{2\pi}} ...
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1answer
25 views

Solve this integral equation using Laplace transform

Solve this integral equation using Laplace transform $$f(x)=x^2 + \int_{0}^{x}f^{\prime}(x-t) e^{-at} dt ,f(0)=0 $$ Please Help see mu answer below Thank you for your participation
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Find the Laplace transform for this function

Find the Laplace transform for this function $$f(x)=(1+2ax)x^{-\frac{1}{2}}e^{ax}$$ Please, help me see my answer below Thank you for your participation
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1answer
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Laplace Transform Question

I was looking in my differential equations textbook and I found an interesting problem and I have no idea on how to approach it. I am supposed to let $F(s) = \mathcal{L} \{f(t) \} $ where $f(t)$ is ...
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23 views

Find the Laplace transform of the Gamma pdf

Per wikipedia the Laplace transform of the gamma distribution is $$L_X(s) = (1+\theta s)^{-k} = \frac{\beta^\alpha}{(s+\beta)^\alpha}$$ As an exercise I would like to show this.The definition I have ...
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32 views

laplace of piecewise (possibly dumb question but should have quick answer)

Find the Laplace Transform of $$f : (0, +\infty) \rightarrow \mathbb{R}, \quad f(t) = \left\{\begin{array}{lr} \sin(t), & \text{for } 0 < t < 2\\ 1+2t^3, & ...
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29 views

Sum of random variable and Laplace transform

Let $\tau$ be a r.v. $\in(0,\infty)$ with PDF $f_\tau(\tau)=\lambda e^{-\lambda \tau}$. How do I find the PDF of $f_{\sum_{i=1}^NX_i}(\sum_{i=1}^NX_i)$ where $X=e^{-\tau}$? I can easily find the PDF ...
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What is wrong with my Laplace Transform?

I need to find the Laplace transform of sine(t), and it is proving rather difficult. I integrate by parts, then integrate by parts again so that the original integral is on both sides of the ...
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1answer
24 views

A system of diferential equations solved by Laplace Transform

I have this system of diferential equations $$ \begin{cases} q'+q+i=50e^{-t}u_1(t) \\ i'+i-q=0 \end{cases}$$ $q(0)=i(0)=0$ ** $ u_1(t) $ is the Heaviside step function Solution Rewriting: $$ ...
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Laplace Transform of equation

I'm having trouble with the laplace transform: $\mathcal{L} \lbrace \sqrt{\frac{t}{\pi}}\cos(2 t) \rbrace$ The problem gives me the transform identity $\mathcal{L} \lbrace \frac{\cos(2 t)}{\sqrt{\pi ...
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1answer
32 views

Inverse Laplace Transform of $\frac{1}{\sqrt{s+a}+\sqrt{s+b}}$

I need to calculate the inverse laplace of: $$F(s)=[\frac{1}{\sqrt{s+a}+\sqrt{s+b}}] \qquad \qquad (s>-a\quad ;\quad s>-b;\quad a\neq b) $$
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1answer
48 views

Inverse Laplace Transform and error function

Express your answer in terms of the error function: $$L^{-1}\left[\frac{1}{\sqrt{s^3+as^2}}\right]$$ Clue: $\qquad L\left[\frac{1}{\sqrt{t}}\right]=\sqrt\frac{π}{s} \qquad , \qquad s>0$ Error ...
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3answers
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solve this partial integration with step function

I would like to know how to solve this partial integration. The equation I got is based on the following convolution: $$t^2e^{-2t} * te^t$$ The part I am having a hard time with is the (t-u) ...
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1answer
30 views

How to obtain Laplace transform of {f(t-a)U(t-b)}

$f(t)=g(t-10)U(t-15)-g(t-10)U(t-20)$ The above $f(t)$ contains terms of the form $g(t-a)U(t-b)$, where $a$ doesn't equal $b$. Describe the form that $L\{f(t-a)U(t-b)\}$ takes. [Hint: The formula for ...
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Inverse Laplace Transform in terms of the error function

Express your answer in terms of the error function: $\frac{2}{\sqrt{π}}\int_0^te^{-w^2}dw$ $$ $$ $$L^{-1}[\frac{1}{\sqrt{s^3+as^2}}]$$ $$ $$ Clue:$\qquad L[\frac{1}{\sqrt{t}}]=\sqrt\frac{π}{s} \qquad ...
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1answer
18 views

How to find the Laplace Transform of $\sin(2t) u(t-\pi)$

Using a translation theorem to determine that $ℒ{f(t-a)u(t-a)}$ is equivalent to $F(s) e^{-as}$, I determined that $ℒ\sin(2t)u(t-\pi)$ is equal to $ℒ\sin(2t) * e^{-(\pi)t}$ and found the Laplace ...
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On Laplace transform of periodic functions

I recently bumped into this theorem regarding the Laplace transform of periodic function: Given a periodic function $f(t)=f(t+p)$, where $p$ is its period, then its Laplace transform is given by: ...