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

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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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320 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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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 ...
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How can I find the RDC of bilateral Laplace Transformations of the signal: $\frac{1}{1 + t^2}$?

I'm a engineering student and I can't solve this question. How can I find the RDC of bilateral Laplace transformation of this specific signal? $$\dfrac{1}{1 + t^2}$$ graciously,
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Is an inverse Laplace Transform always solvable?

I just read on Wikipedia that if we got a certain Laplace Transform $$\mathcal{L}\{f(t)\}= \frac{A}{s-\alpha_1} + \frac{B}{s-\alpha_2} + ... $$ can be solved like this: $$f(t)= A e^{\alpha_1 t}+ ...
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178 views

Laplace transform of sin(at)

Given $f(t)= \sin (at)$ I want to calculate the Laplace transform of $f(t)$. I have determined by using integration by parts twice, that the answer should be $$F(s)= \frac{a}{s^2+a^2}$$ Now I want ...
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443 views

Continuity of integral function

How to show that the following function is right continuous at $0$ (that is, when $a\to0+$): $I(a) = \int_0^{\infty}\frac{\sin x}{x}e^{-ax}dx$? I know that Lebesgue integral $I(0) = \frac{\pi}{2}$. ...
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101 views

Integration with respect to a non-decreasing function on $\mathbb{R}$

Let $\alpha(t)$ be a non-decreasing function on $\mathbb{B}$ and consider the integral $$ \int_{-\infty}^{+\infty} e^{-xt}d\alpha(t) $$ absolutely convergent in $I$. Does exist a measure ...
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61 views

Inversion of a two-sided Laplace transform

I have the function $$ F(s)=\frac{1}{1+s^2}\frac{1}{1+4s^2} $$ and I would like to know if exists a non-decreasing function $f(t)$ such that $F(s)$ is the two-sided Laplace transform of $f(t)$. Of ...
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Compare Laplace, second order nonhomogenous differential equations

What other methods besides Laplace can be used to solve second order non homogenous diff eq? When would you choose to use or not use Laplace?
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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 ...
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Domain of the Laplace transform.

The unilateral Laplace transform of an $f:[0,\infty]\rightarrow \mathbb{C}$ is defined as $$F(s)=\int_{0}^{\infty}e^{-st}f(t)dt$$ My lecturer didn't go into detail on the domain of the transform, ...
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46 views

Interval of convergence of a Laplace-Stieltjes transform

I have a two-sided Laplace-Stieltjes transform, $$ \int_{-\infty}^{+\infty} e^{-xt}d\mu(t) $$ that converges absolutely in $(a,b)$. If the measure $\mu$ is finite,then $$ ...
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Analyticity of Laplace transform

Let $f(x)$ be a bilateral Laplace transform of a measure $\mu$: $$ f(x)=\int_{-\infty}^{+\infty} e^{-xt} d\mu(y). $$ Suppose that $f(x)$ converges absolutely in $(a,b)$, and $(a,b)$ do not contain the ...
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Find the inverse Laplace transformation of $\dfrac{s+1}{(s^2 + 1)(s^2 +4s+13)}$

My question is : find the function $f(t)$ that has the following Laplace transform $$ F(s) = \frac{s+1}{(s^2 + 1)(s^2 +4s+13)} $$ thanks
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Laplace transform of $|\sin(t)|$

There's already an answer to this, but I'm curious as to why my method of solving doesn't work. I take the integral where $\sin{t}$ is positive and the negative integral where it is negative: ...
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Laplace transform of $f(t)=10te ^{-5t}$

Find the Laplace transform of $$f(t)=10te ^{-5t}$$
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257 views

Laplace transform of a random variable

My professor says that the Laplace transform of a nonnegative RV uniquely determines the RV up to distributional equality among all nonnegative RVs. He says one can argue this by appealing to a fact ...
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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 $l(t)$ ...
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113 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 \\ ...
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Finding the inverse laplace transform of $s$ [closed]

How do I find the inverse laplace transform of $s$, i.e. $$L^{-1}\{s\}=\ ?$$
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101 views

Differential equations - show solution given by expression

Show that the solution of the problem of Cauchy $\ddot{x}(t)=-a^2x(t)+ b(t)$ with $ x(0)=x_0$ and $\dot{x}(0)=v_0$ Is given by $\displaystyle x(t)=x_0 \cos(at)+\frac{v_0}{a} ...
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A question arising from some misunderstanding involving the Laplace Transform of the Heaviside function

I am trying to compute the Laplace Transform of the Heaviside Step-function. I define the Heaviside step-function with the half-maximum convention: $H(t-t_0) = 0$ for $t < t_0$ ; $H(t-t_0) = 1/2$ ...
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solving the PDE of a beam under a moving load using Laplace transform

Solve this PDE using Laplace transform : $$ EI {\partial^4 y(x,t)\over\partial x^4}+\mu {\partial^2y(x,t)\over\partial t^2}= F(x,t) $$ $$F(x,t)= P\delta(x-u) / ...
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130 views

Calculate the next Inverse laplace transform

This question may be very basic, but I dont know how to get the next inverse laplace's transform: $${\scr L}^{-1}\left\{\frac{1}{(s-1)^2}\right\}$$ I can only use these two formulas: ${\scr ...
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165 views

Solve linear system of ODEs using Laplace transform

I need to solve the following initial value problem via Laplace transform \begin{align*} \dot{\mathbf{x}} = \begin{pmatrix} 2 & -5 \\ 1 & -2 \end{pmatrix} \mathbf{x} + \begin{pmatrix} \sin t ...
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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 ...
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307 views

Inverse Laplace transformation using reside method of transfer function that contains time delay

I'm having a problem trying to inverse laplace transform the following equation $$ h_0 = K_p * \frac{1 - T s}{1 + T s} e ^ { - \tau s} $$ I've tried to solve this equation using the residue method ...
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Laplace transform of $\int_t^{+\infty}\,\psi(\tau)\,d\tau$

Why this equality? $$\mathcal{L}\left(\int_t^{+\infty}\,\psi(\tau)\,d\tau\right)=\frac{1-\psi(s)}{s}$$ Can you help me with the steps?
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PDE initial value problem

Show that the solution of the initial value problem for $u_t+u_x=\cos ^2 u$ is given by $u(x,t)=\tan^{-1} \{ \tan [u_o(x-t)]+t\}$, where $u_0(x)$ is the initial condition. My attempts at a ...
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Inverse Laplace transform after derivative of transform.

I've been using the following theorem in my intro ODE course: If $F(s) = \mathscr{L}\left\{ f(t) \right\}$ and $n\in\mathbb{N}$, then $\mathscr{L}\left\{t^n f(t) \right\} = ...
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How do I apply partial fraction expansion on $\dfrac{K}{(a+bz^{-1})(x+yz)}$?

I want to apply partial fraction expansion on $\dfrac{K}{(a+bz^{-1})(x+yz)}$. I'm not able to do it in the standard way, because one term has $z^{-1}$ term and the other has $z$. What is the approach ...
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267 views

Finding the steady state error in the Laplace domain

I have the following block diagram: Now I like to find the steady state error for theta_ref being a step input and for several values of n, Td, K1 and K2. For the moment we can assume all gains ...
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inverse of laplace transform

How to compute this inverse Laplace transform ? $$\displaystyle{ \mathcal{L^{-1}} \left\{ \frac{1}{s(\exp(s)+1)} \right\} }$$ Thanks.
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Proof $L{\rm{[}}\frac{{x(t)}}{t}{\rm{] = }}\int_s^\infty {X(u)du} $

I see that we usually use the theorem to solve the Laplace transform, however i want to proof the theorem, who could give me some details!!! $L{\rm{[}}\frac{{x(t)}}{t}{\rm{] = }}\int_s^\infty ...
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Inverse Laplace transform of the function: $F(s)=e^{-a\sqrt{s(s+r)}}$

I would like to find inverse Laplace transform of the function: $$F(s)=e^{-a\sqrt{s(s+b)}}$$ which $a$ and $b$ are positive real numbers and $s$ is a complex variable. It would be appreciated if ...
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How to find the inverse Laplace Transforms?

How do you find the inverse Laplace Transform of the following, $$\frac{2 (s^2+4 s+5)^2+40}{(s^2+4 s+5)^2}$$ Separating them into complex coefficients is to long.
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Laplace transform $f(t) = f(t + \frac{2\pi }{a}) = \frac{\sin at}{|\sin at|}$

I had done the laplace transform with the period fucntion (T): $F(s) = \frac{1}{1 - e^{ - sT}}\int_o^T {e^{ - st}}f(t) \, dt $ Howerver, in this fuction: $$f(t) = f \left(t + \frac{2\pi}{a}\right) = ...
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Laplace transform:$\int_0^\infty \frac{\sin^4 x}{x^3} \, dx $

I have a trouble with a integral: Using this Laplace trasform equation: $$\begin{align} \int_0^\infty F(u)g(u) \, du & = \int_0^\infty f(u)G(u) \, du \\[6pt] L[f(t)] & = F(s) \\[6pt] ...
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Basic Partial Fractions

I feel super foolish asking this, but I've reached a mental block. I'm trying to find the inverse laplace transform of: $$\frac{s+3}{(s + 1)^2 (s-2)}$$ but, when I expand it into partial ...
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Proof the theorem $\int_0^\infty {{t^n}f(t)dt = {{( - 1)}^{n + 1}}\int_0^\infty {{F^{(n + 1)}}(u)du} } $

Knowing that: $$ L\left[ {\int_0^\infty {\frac{{f(t)}}{t}dt} } \right] = \frac{1}{s}\int_0^\infty {F(u)du}$$ with: $L[f(t)] = F(s) $ Show that: $$\int_0^\infty {\frac{{f(t)}}{t}dt ...
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101 views

Find the Laplace transform:$f(t) = \int_0^t {{e^{it}}\frac{{dt}}{{\sqrt {2\pi t} }}}$

Find the Laplace transform of the function: $$\begin{array}{l} f(t) = \int_0^t {{e^{it}}\frac{{dt}}{{\sqrt {2\pi t} }}} \\ {\rm{Us}}e:\Gamma \left( {\frac{1}{2}} \right) = \sqrt \pi \\ ...
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Find the inverse Laplace transform of $f(t) = \int_t^\infty \frac{e^{ - u}}{u}du$

Find the inverse Laplace transform of the integral:$$f(t) = \int_t^\infty {\frac{{{e^{ - u}}}}{u}du} $$ If the integral: $$f(t) = \int_0^\infty {\frac{{{e^{ - u}}}}{u}du} $$ I had done. However the ...
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100 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 - ...
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How do a transformation 'born'?

Well, there are several transformations in math. Like the laplace transformation. My question is about the utility and motivation of these transformations. Like, when we have an equation, and we ...
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51 views

Laplace transform of $\frac{f(t)}{1-e^{-at}}$

I would like to know if there is a way to calculate the Laplace transform for a given $f(t)$: $$\dfrac{f(t)}{1-e^{-at}}$$ Thanks!
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Prove that L[f' ' ](s)$ = $sL[f](s)

Can anyone prove this question ? Let $f$:$\mathbb{R}$$→$$\mathbb{C}$ be continuous function such that $f$$(0)$ $=$ $0$ and that $f'$ be a piecewise continuous function and absolutely integrable on ...
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16 views

Which value of $k$ in $e(k) = 5$ where $E(z)=5z^{-1}+z^{-2}+2z^{-4}$?

I'm new to Laplace transforms so here goes: From the lookup table we have that $$t^{k-1} = \frac{(k-1)!}{s^k}$$ using that: $$\begin{align} \scr L\{e(k)\}&=5z^{-1}+z^{-2}+2^{-4}\\ e(k) &= ...
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1answer
299 views

Using Convolution Theorem to find the Laplace transform

In previous questions I have used Laplace transform to find the inverse Laplace transform. I have worked through this work booklet ...