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

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What is Fourier transform of space variable? on the similar grounds what is the Laplace transform of the same?

I understand that the transform of time domain is frequency domain and the transformation of time to frequency domain is done by Fourier/Laplace transforms. I am confused about the transformation of ...
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Laplace transform $\mathcal{L}\{ t^{-2}\}$ [on hold]

What are the steps for the Laplace Transformation for $t$ to some negative power ?
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riemann zeta function : entire and even Laplace transforms

$$\xi(s) = s(s-1)\pi^{-s/2}\Gamma(s/2) \zeta(s)$$ $$\xi(s) = \xi(1-s)$$ thus $\Xi(s) = \xi(1/2+s) = \Xi(-s)$ is even, and furthermore it is an "entire and even Laplace transform" : $$\Xi(s) = ...
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Laplace transform of the logarithmic integral function

What is the Laplace transform of the logarithmic integral function $\text{li}(t)$. Meaning, how to compute the integral : $$\int_{0}^{\infty}\text{li}(t)e^{-st}dt$$
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Laplace Transform for solve ODE (RLC circuit)

I have an RLC circuit and I want to know the charge on the capacitor $q(t)$ using Laplace transform: The diferential equation is: $$ Lq'' + Rq' + \frac{1}{C}q = E(t),$$ where $L = 1H , R = 20 ...
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Discrete PID controller Laplace formula

I saw the following formula: the transfer function is: $$Gr(s) = K_p \bigg(1 + \frac{1}{T_n s}+ \frac{T_v s}{1 + T_d s}\bigg) $$ From my understanding: $K_p$ is the proportional gain $T_n$ is the ...
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Relation Fourier/Laplace Transform

I have a question about the relation between Fourier and Laplace transforms. I have seen in some places that the transfer functions in the Laplace space are represented as $G(s)$ where $s$ is the ...
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Find the Laplace inverse of the following.

$$ \frac{2s+5}{s^2+6s+34} $$ I am stuck on this part: Wolfram has the step by step showing that you simply split up the original fraction into $$ \frac{2s}{s^2+6s+34} + \frac{5}{s^2+6s+34} $$ and ...
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Laplace transform of sine cubed [closed]

I don't know how to get the Laplace transform of $$f(x)= \sin^3 x.$$ I have difficulties with this kind of problems, please help me step by step.
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help with laplace [closed]

I have to get to the second part of the ecuation with laplace and I don't know how to do it step by step, help please! thanks!! $$\int\limits_{-\infty }^{+\infty }{\left( \frac{\sin x}{x} ...
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Laplace Transform conundrum

consider, x1(t) + constant = x2(t) => w/ laplace, X1(s) + c/s = X2(s) but, take the time derivative of the first equation, x1dot = x2dot => sX1(s) = sX2(s) => X1(s) = X2(s). Which is correct?
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Why can't Wolfram calculate the Laplace transform of $\sinh(t)\sin(t)$ correctly?

Question Show that the Laplace transform of $\sinh(t)\sin(t) = \frac{2s}{s^4+4}$. Wolfram can't calculate this as is, so I tried to simplify it a bit. I defined $\sinh(t)$ as $e^t-e^{-t}$ and split ...
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Use the Laplace transform to solve the initial value problem.

$$ y''-3y'+2y=e^{-t}; \quad\text{where}~ ~ y(2)=1, y'(2)=0 $$ Hint given: consider a translation of $y(x)$. I am stuck on this problem on our homework. I don't understand what they mean by a ...
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Numerical Laplace Transform

I want to compute the Laplace transform of data vectors. I have tried the usual numerical software and I'm surprised to see that does not have this operation available. I wonder if there is a straight ...
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Convergence of Laplace transform and its inverse

There is a sequence of functions $F^{\epsilon}(\lambda)$ which converges to 0 as $\epsilon \rightarrow 0$. Assume that each $F^{\epsilon}(\lambda)$ has a inverse Laplace transform f(s) such that ...
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Computing the Laplace transform of $\tan(pt)$

I've been thinking of using complex number approach , what's your view guys ?.
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Is it possible to say that $L(f^n)=s^nL(f)$ when the differential equation is not in the rest condition?

Question Use the Laplace transform to solve the following equation: $y'+2y=\cos(3t)$ ; where $y(0)=1$ In class our teacher wrote that "When in rest condition: $L(f^n)=s^nL(f)$", but I want to use ...
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Find the inverse Laplace transform of: $\frac{1}{(s^2+a^2)(s^2+b^2)}$

I'm having trouble doing this homework problem because I'm not sure how to deal with the $a$ and $b$. I did it the usual way we were taught - use partial fraction decomposition and then try to solve ...
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Solve the following initial value problem: $2y''+y'-y=e^{3t}$

$$ 2y''+y'-y=e^{3t}; \text{ with } y(0)=2,\ y'(0)=0 $$ I got to this point: $$ L(y)=\frac{1}{(s-3)^2}\cdot\frac{1}{(2s-1)(s+1)} $$ but now I'm not sure what to do with these polynomials. I know ...
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Inverse laplace transform with square completion

I need to find the inverse laplace of this : $$\frac{s+2}{s^2+2s+5}$$ I know that completing the square should help me to solve this so I get $$\frac{s+2}{(s+1)^2+4}$$ Then separating this equation ...
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About the causality of the signal whose frequency spectrum is not continuous as follows

Consider the signal in frequency domain: $$ \alpha(\omega) = \begin{cases} 1, & |\omega|<\omega_c \\ 0, & |\omega|\ge\omega_c \end{cases} $$ $$ =A(-j\omega)A(j\omega) $$ $$ =|A(j\omega)|^2 ...
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Laplace-Stieltjes :Functions of independent random variables

I am reading a book about stochastic modelling and I came across something and I couldn't really figure it out. First question would be are Probabilty Generating Functions (PGF) only for discrete ...
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Laplace Transform of Kelvin functions

What is the value of the Laplace transform, in terms of the G-function, \begin{align} \int_{0}^{\infty} e^{-st} \, t^{m} \, \left(ber_{\nu}^{2}(t) + bei_{\nu}^{2}(t)\right) \, dt \hspace{5mm} ? ...
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What does s=jω actually mean in terms of the complex plane and Laplace transforms?

I was trying to solve a problem on RC circuits. The current source was of the form $\cos (\omega t)$ which transforms in the manner of Laplace to $\frac{s}{s^2+\omega^2}$. I thought I’d use the ...
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Prove that $\mathcal{L}\left( \int_{0}^t f(u)du \right)=\frac{1}{s}\mathcal{L}(f)$

Prove that $$\mathcal{L}\left( \int_{0}^t f(u)du \right)=\frac{1}{s}\mathcal{L}(f)$$ I started out with the following identity: $$ \frac{1}{s}\mathcal{L}(f)=\frac{1}{s}\int_{0}^\infty e^{-st}f(t)dt ...
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Laplace Inverse

I want to find the laplace inverse of $$s^{-3/2}$$ the steps given in the solution manual are as follows: $$\frac{2}{\sqrt\pi}\frac{\sqrt\pi}{2s^{3/2}}=2\sqrt{\frac t\pi}$$ I know the first part ...
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Inverse Laplace Transform using Hetnarski's Algorithm

I'm trying to find the velocity component of an MHD flow using Laplace transforms. R.B. Hetnarski's algorithm for inverting the laplace transforms of some exponential functions was recommended to me ...
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How do I find the Laplace Transform of $ \delta(t-2\pi)\cos(t) $?

How do I find the Laplace Transform of $$ \delta(t-2\pi)\cos(t) $$ where $\delta(t) $ is the Dirac Delta Function. I know that it boils down to the following integral $$ \int_{0}^\infty ...
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Inverse Laplace transform of $\operatorname{arccot}(s)$, $\arctan(s)$

How would one find inverse Laplace transforms of $\operatorname{arccot}(s)$ or of $\arctan(s)$ without knowing in advance that this is related to $\dfrac{\sin x}{x}$?
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Inverse Laplace Transform and the Unit Step Function

I need to find the inverse Laplace transform of the following function: $$ F(s) = \frac{(s-2)e^{-s}}{s^2-4s+3} $$ I completed the square on the bottom and got the following: $$ F(s) = (e^{-s}) ...
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Laplace transform and IVP at $\infty$

Solving the following differential equation $$ty^{''}\left ( t \right )+\left ( t-1 \right )y^{'}\left ( t \right )-y\left ( t \right )=0$$ with initial values $$y\left ( 0 \right )=5, y\left ( ...
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Laplace transform of convolution integral

If $f(t)$ an $g(t)$ are piecewise continuous functions on $[\ 0, \infty)$ then the convolution integral of $f(t)$ and $g(t)$ is, $$(f*g)(t) = \int_{0}^{t}f(t-\tau)g(\tau) \text{d} \tau.$$ The text ...
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Laplace Transforms of Step Functions

The problem asks to find the Laplace transform of the given function: $$ f(t) = \begin{cases} 0, & t<2 \\ (t-2)^2, & t \ge 2 \end{cases} $$ Here's how I worked out the solution: ...
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Finding the eigenvalues and eigenfunction (tricky)

I'm given $$X"- vX' +X \lambda=0$$ (v is a constant) I have worked x' to be: X'(x) = $$\frac{1}{2} B v e^{\frac{v x}{2}} \sin \left(\frac{1}{2} x \sqrt{v^2-4 \beta ^2}\right)+\frac{1}{2} B ...
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Bessel equation of half-order (asymptotic)

Not really optimistic about getting a reply for a question tagged under "Bessel function" but here goes, I have $$J_{\frac{1}{2}} = (a_1 \cos(z) + a_2 \sin(z))Z^{-\frac{1}{2}} $$ and ...
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Solving a differential equation by using Laplace transform

I need to solve this equations by using laplace-transform. I tried to solve it but when I reach to the point that it's needed to use partial fraction expansion in order to transform the laplace ...
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Do we have a inverse Laplace transform of $\frac{1}{\arctan s}$

Do we have a closed form of this seemingly very simple inverse transformation? If no closed form, what about its asymptotic form? Does this satisfies the criterion to have its inverse ...
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$F^{(n)} (p)$ do you first differentiate and afterward apply the Laplace?

If you have a Laplace transform: $F^{(n)} (p)$, do you first differentiate and afterwards apply the Laplace? $F(p)$ meaning $L[f(t)](p)$
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Fourier transform of PDE on finite and infinite bound simultaneously.

Consider $$u_{xx} + u_{yy} = 0 $$ on the bounds: $$o < x < L$$ and $$-\infty<y<\infty$$ The initial condition is: $$u(0,y) = f(y)$$ and $$u(L,y)=g(y)$$ I've tried performing fourier ...
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The inverse Laplace transformation of $e^s$

I am solving the differential equation: $$y'' + 3xy' -6y = 1, \ y(0) = y'(0) = 0$$ Using Laplace transformations. I arrived at: $$L(y)(s) = \frac{c}{s^3} e^{s^2 / 6} + \frac1{s^3}$$ Where $c$ is ...
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Applying Fourier transform to heat equation with source

I haven't had any experience with applying of FT to heat equation with source. But this popped up in an exercise. Any help in the right direction would be great. Consider: $$\frac{\partial ...
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Fourier transform of a piecewise

How should I go about seeking the Fourier transform for the piecewise function: $$f(x) = \left\{\begin{matrix} 0 ,&|x|>a \\ 1 ,&|x|<a \end{matrix}\right.$$ Is this the correct ...
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Reference for an identity from Abramowiz and Stegun

I am curious as to where this identity was originally obtained. Any suggestions? $$ \frac{1}{\mathop{\Gamma}\nolimits\!\left(1+2\mu\right)2\pi i}\int_{-\infty}^% ...
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Integral of Absolute Value of $\sin(x)$

For the Integral: $\int |\sin (ax)|$, it is fairly simple to take the Laplace transform of the absolute value of sine, treating it as a periodic function. $$\mathcal L(|\sin (ax)|) = ...
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The Dirichlet problem for the Laplace equation: classical solutions versus weak solution

Let $B_R$ a ball in $\mathbb{R}^n$. Consider $u^{\star} \in H^{1}(B_R) $ and $f \in H^{1}(B_R) \cap C(\overline{B_R})$. Suppose that $u^{\star}$ minimizes $$\int_{B_R} |\nabla u|^2, u \in \{ v \in ...
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Why does the imaginary part of $s$ have no effect in analyzing region of convergence for Laplace Transform?

The tutorial that brought this assertion to me was: http://fourier.eng.hmc.edu/e102/lectures/Laplace_Transform/node2.html "As the imaginary part $\omega=Im[s]$ of the complex variable ...
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Laplace Transform of derivative squared

I'm trying to solve a problem while I'm studying Control Theory and I came up with a difficult question. $ \mathcal{L}\left[y'(t)^2 \right] $ Basically I need to find the Laplace Transform of this ...
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“eigenfunction” of a transformation

Fourier transform of a gaussian is another gaussian. Fourier/Laplace transforms of $\frac{1}{\sqrt t}$ is something like $\frac{1}{\sqrt \omega}$. I realize that we can't call these eigenfunctions ...
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What is the inverse laplace transform of $\large{\frac{ s^3 - a^2s }{(s^2 + a^2)^2}}$

I tried convolution and partial fractions but both turned out to be too much work. Is there any easy work around??
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Laplace transfer function and quasi-sinusoidal input

Let's suppose we have an LTI system whose Laplace domain transfer function is: $$ F(s)=\frac{1}{s^2 + \frac{\omega_y}{Q_y}s + \omega_y^2} $$ Its input is the Coriolis force. Such force is experienced ...