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

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Integral parameter equation

How to solve the following equation, $$\int_{-i\infty}^{+i\infty} e^{st}\frac{\Sigma_{k=0}^{m}b_ks^k}{\Sigma_{k=0}^{n}a_ks^k} ds=0$$ to obtain $t$ where, $m\le n$ and $a_i$, $b_i$ are the given ...
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Finding the maximum of step response for a given transfer funtion

Assuming that the following transfer function is given: $$F(s)=\frac{\Sigma_{k=0}^m b_k s^k}{\Sigma_{k=0}^n a_k s^k}$$ $$m\le n$$ Lets say $g(t)$ is the step response to this transfer function. ...
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1answer
16 views

Use Laplace transform to solve the system

$\frac{\partial x}{\partial t} = -y$ , $\frac{\partial y}{\partial t} = x.$ For $t\geq 0$ , with $x(0) = 1$, $y(0) = 0$. I have no clue on how to start the question. Could some please give me a ...
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Laplace transform and distributions

I was studying for my course in Fourier Analysis and was going through some old exams, when this question came up: Let $s^{-1}_+$ and $s^{-1}_-$ denote the analytic distributions given by the ...
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1answer
30 views

Solving IVPs with Heaviside function where switch depends on function value

I am looking to solve an IVP of the format $y'(t) = \begin{cases} a, & y(t) < y_0 \\ b, & y_0 \le y(t) < y_1 \\ c, & y(t) \ge y_1 \\ \end{cases}$ That ...
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36 views

Can the z-transform/laplace transform be generalized?

The fourier transform is a special case of z/laplace where the countour being projected on is the unit circle. The ztransform/laplace transform then generalizes this to a circle of any radius. Is ...
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25 views

Laplace Transform of $\int_{0}^{t}u^2e^u\,du$

Good day, I am having some problems with the Laplace transform of $f(t) = \int_{0}^t{u^2e^u\textrm{d}u}$. $$\mathcal{L}\left [\int_{0}^t{u^2e^u\textrm{d}u} \right ]$$ using the property : $$\...
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Inverse Laplace transform of the form $F(s) = \frac{s^{m}}{(1+a \cdot s)^{n}(1+b \cdot s)^{h}}$

I am trying to solve the inverse Laplace transform of the form \begin{equation} F(s) = \frac{s^{m}}{(1+a \cdot s)^{n}(1+b \cdot s)^{h}} \end{equation} where, $a$ and $b$ are known constants, $m$, $n$, ...
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understanding basic Laplace transformation

I am trying to understand Laplace transformations. Could someone tell me from where the fraction (1/-s), in red on the first line is originated? http://imgur.com/bTqjLGl
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60 views

Inverse Laplace transform of a hypergeometric function

I managed to solve an initial value problem in the Laplace domain in terms of a special function $ F(s) = c_2 \frac{1}{{{\left( {{s}^{1 +\beta}}-1\right) }^{\frac{1}{\beta+1}}}}+ c_1 \frac{s}{{{\left(...
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26 views

How do you differentiate a Laplace transform?

Consider the Laplace transform of $\color{green}{t\cfrac{\mathrm{d^2}f}{\mathrm{d}t^2}}$: $$\mathcal{L}\left[{t\cfrac{\mathrm{d^2}f}{\mathrm{d}t^2}}\right]=\int_{t=0}^{\infty}e^{-st}{t\cfrac{\mathrm{...
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15 views

Find maximum $p$ so random variable is in $L^p$ using Laplace transform

For a (non-negative) random variable $X,$ define its Laplace transform as $$\mathbb{E}(e^{-\lambda X}) =: L(\lambda),$$ for $\lambda > 0.$ We know that this uniquely determines the law of $X$ as ...
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existence of laplace transfrom to 1

I'm asked to find existence of f(t) given equation in exam 1 * f(t) = 1 (* means convolution) I tried to solve it by transforming (1/s) x F(s) = (1/s) so I got F(s) = 1 and then I said f(t) is ...
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1answer
13 views

Solving $\frac{dV}{dt} = \frac{I}{C}$ using the Laplace transform

I have the following equation for the evolution of the membrane potential ($V$) of a neuron: $$ \frac{dV}{dt} = [-g_L(V-V_{rest}) + I_{syn}(t) + I_0] / C. $$ According to Equation 2.13 of this paper,...
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1answer
67 views

Laplace transform of the zeroth-order Bessel function [duplicate]

Consider $J_0$ the zeroth order Bessel function. I'm trying to compute the Laplace transform $$\mathcal{L}[J_0](s) = \int_0^\infty J_0(t) e^{-st}dt,$$ but until now I couldn't find a good way to do ...
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57 views

Inverse Laplace transform of $\frac{\exp(\frac{\lambda s}{1 - 2s})}{(1 - 2s)^{k/2}}$ (MGF of noncentral chi-squared distribution)

I am trying to use the countour integral to calculate the inverse Laplace transform of the function $$F(s) = \frac{\exp(\frac{\lambda s}{1 - 2s})}{(1 - 2s)^{k/2}} \hspace{1cm}\mathrm{for} \hspace{1cm} ...
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111 views

Inverse Laplace transform of one complicated function

I want to ask the inverse Laplace transform of the following function: $$F(s) = \frac{1}{s \cdot (1 + a \cdot s)^{m} \cdot (1 + b \cdot s)^{m-k}} \cdot \Bigl[\exp{(\frac{- c \cdot s}{ 1 + b \cdot s } ...
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Is there a mistake in my calculations?

I am trying solve the differential equation $y''+3y'+2y=u(t-1)-u(t-2), y(0)=y'(0)=0$, by calculating the convolution of $f(t)=1$ and $g(t)=e^{-t}-e^{-2t}$. The problem is that I get two different ...
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72 views

Solve the integral equation (convolution integral)

I have this problem: Solve the integral equation $$e^{-t}=y(t) +2 \int_0^t \cos(t-u)y(u) \, du$$ So I'm thinking that Laplace transform will be the way to go here, so I get $$\frac{1}{s+1}=Y(s)+2\...
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73 views

Laplace Transform

The question I had was Find the Laplace transform of $$f(t)=10e^{-200t}u(t).$$ Would it be correct to take out the 10 because it is a constant, find the Laplace transform of $e^{-200t}$ and then ...
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63 views

How to find the inverse Laplace transform of this?

I am looking for the inverse Laplace transform of $$\frac{1}{s-1}e^{-\sqrt{s}x}$$ This is for an introductory partial differential equations class so I am thinking that it should not be too hard. If ...
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34 views

Analytical Solution of diffusion equation(with Laplace Beltrami operator)

The analytical solution of diffusion equation is an exponential function. How can i find the analytical solution of the diffusion equation, if we have Laplace Beltrami operator(on a sphere), instead ...
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29 views

Laplace transform for convolution integral

I'm having some trouble with this problem: Solve the integral equation with Laplace transform $$e^{-t}=y(t)+\int_0^t(t-u)y(u)du$$ I know how to use the Laplace transform for more "normal" equations ...
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Laplace Transformation with initial value problem

Consider the the initial-boundary value problem $u_t=u_{xx}$ where $u(x,0)=f(x)$ and $u_x(0,t)-u(0,t)=0$ for $x>0$ and $t>0$ and $u$ remains bounded. Solve this problem by observing that the ...
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86 views

Inverse Laplace Transfrom of $s^{-1}e^{-a\sqrt{s} + b/s}$

I am trying to find the inverse Laplace transform for following function and it seems almost impossible for me to find the answer. Can anyone help me please with final answer and also the way to get ...
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1answer
34 views

Unilateral and bilateral Laplace transorm

My friend and I had an argument upon the Laplace transform of $\sin \omega t$. He's saying that its Laplace transform does not exist and only the Laplace transform of $u(t) \sin \omega t$ exist. But ...
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Inversion of Laplace transforms - simplifying the Bromwich integral

I have trouble following the derivation of equation $(2)$ in this paper. The authors define the Laplace transform of a real-valued function $f(t)$ of a positive real variable $t$ as \begin{equation} ...
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21 views

laplace transform of smirnov density, i.e. how to calculate this integral?

I am trying to figure out how to perform the following computation. The objective is to compute the laplace transform of the smirnov density. The lecture notes I've seen online state that $\...
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Application of Laplace transform to stopping times and expectations

Let $X_k$ be i.i.d. random variables such that $E[X_1]=m<\infty$. Consider $S_n = \sum_{k=1}^{n} X_k$. Let $\tau$ be a stopping time independent of $X_k$ with respect to the filtration $\{F_n\}_{n \...
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38 views

Solving a pair of kinetic equations using the Laplace transform

I have the following set of kinetic equations: \begin{align} \frac{dx}{dt}&=r_1\delta(t-t_0)-(r_2+r_3)x(t)\\ \frac{dy}{dt}&=r_3x(t)-r_5y(t). \end{align} How can I solve for $y(t-t_0)$ using ...
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1answer
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Inverse Laplace transform of $\frac{r_1e^{-t_0s}}{s + r_2 + r_3}$

I have the Laplace transform of $x$ as follows: $$ x_L=\frac{r_1e^{-t_0s}}{s + r_2 + r_3}, $$ where $x$ is a function of $t$, and $x_L$ is a function of $s$. I know the inverse Laplace transform of ...
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1answer
59 views

Minimum number of zeros of this Laplace transform

I've come across this question in my Signals and Systems class but I can't seem to understand what the answer might be. Here is the entire question: Consider a signal x(t) which has its Laplace ...
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1answer
56 views

Breaking a biexponential function in two

I have the following two equations: $$ I(t) = \sum_iw_i\alpha(t-t_i) $$ $$ \alpha(t) = \beta\frac{\tau_2}{\tau_2-\tau_1}(e^{-t/\tau_1}-e^{-t/\tau_2}). $$ When implemented in a particular software ...
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37 views

Taking inverse laplace transform of $\dfrac{1}{(s^2+1)^2}$ using residues, something's wrong

I am trying to compute the inverse laplace transform of $\dfrac{1}{(s^2+1)^2}$ using residues So $$\mathcal{L}^{-1}\dfrac{1}{(s^2+1)^2} = res(\dfrac{e^{st}}{(s^2+1)^2}, i) + res(\dfrac{e^{st}}{(s^2+...
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How to quickly compute the inverse laplace transform of $\dfrac{1}{(s^2+1)^2}$

I wish to find the inverse laplace transform of $\dfrac{1}{(s^2+1)^2}$ I tried using partial fraction but things do not seem to work out i.e. $\dfrac{1}{(s^2+1)^2} = \dfrac{A}{s^2+1} + \dfrac{B}{s^2+...
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Laplce transform solution to this system of Nonlinear ODEs

I want to solve this system of advective-diffusive-reactive equations analytically: $$\left(\alpha - k_0c_B\right)c_A+v\frac{dc_A}{dx}-D\frac{d^2c_A}{dx^2} = f_A $$ $$\left(\alpha - k_0c_A\right)c_B+...
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Inverse Laplace Transform of $(3s)/((s^2+2s+3)(s+1))$ [closed]

Use the partial-fraction expansion to calculate the time-response to a ramp input of $u(t)=t$ for a system with the following transfer function. $$G(s)=\frac{3s}{(s^2 + 3)(s + 1)}$$ enter image ...
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Laplace transform of $\frac{1}{\sqrt{t^2+1}}$

What is the Laplace transform of this function? $$\frac{1}{\sqrt{t^2+1}}$$
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Laplace Transform of uniformly convergent series

Let $\sum_{n=1}^{\infty} f_n(x)$ be a uniformly convergent series of functions each of which has a laplace transform defined for $s \geq \alpha$. Show that $f(x)=\sum_{n=1}^{\infty} f_n(x)$ has a ...
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45 views

Explain what the teacher did - system of ode, control theory.

There are a few things I'm not clear about in her solution and would appreciate a short explanation. We are given the system $\dot{x}=-ax+bu$. with an initial value $x(0)=x_0$. We want to find a ...
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44 views

Turn this integral into a Laplace transformation by Change of Variables

Question from Advanced Engineering Mathematics - Greenberg. Page 268, section 5.4 question 6. $C(T)$ = $\int_0^{\infty} e^{-0.0744v^2/T^2}p(v)dv$ is an approximate relation between frequency ...
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When does the Laplace Transform converge?

Consider the following function in the $s$-domain: $$F(s) = \frac{1}{(s^2 + 1)(2s-1)}$$ My book concludes that the ROC (Region of convergence) must be $\Re(s) > 1/2$ because a) The ROC can't ...
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39 views

Prove the laplace transform of $\sinh(at)$?

My problem today is as above. Here is what I have done: Use integral definition of laplace transform to get $$\int_0^\infty \sinh(at)\exp(-st)dt$$ $$= \lim_{b \to \infty}\int_0^b \sinh(at)\exp(-st)...
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Find Laplace inverse

Let $${{{{x^{\ast}(s) = \left( \frac{1}{(s+\mu_1 + \mu_2) (s + \hat{\lambda}_2) (s + \lambda_1 +\lambda_2 )}\right)}}}}$$ be the laplace transform in question, where $\mu_1,\mu_2, \lambda_1,\lambda_2, ...
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80 views

Laplace Transform

Suppose that $F(s)=L[f(t)]$ and $G(s)=L[g(t)]$, where $L$ is the Laplace transformation $$F(s)=L[f(t)]=\int_0^{+\infty}e^{-st}f(t)dt.$$ I'm trying to prove that: $$\textrm{If}\ \ \lim_{t\to 0^+} \frac{...
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134 views

Solving forced undamped vibration using Laplace transforms

I'm heaving trouble solving the following undamped forced vibration problem using Laplace transforms: $$\ddot{q}(t) + \omega_n^2 q(t) = \cos(\omega t).$$ I will show what I have done so far, and I'd ...
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3answers
89 views

Generalized Fresnel integral $\int_0^\infty \sin x^p \, {\rm d}x$

I am stuck at this question. Find a closed form (that may actually contain the Gamma function) of the integral $$\int_0^\infty \sin (x^p)\, {\rm d}x$$ I am interested in a Laplace approach, double ...
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49 views

Find the inverse laplace transform (step-function)

Find the inverse laplace transform for $$F(s) = \frac{e^{-2s}}{s^2+s-2}$$ I have narrowed it down to that it has something to do with step-functions but I can't see it, can I do something with the ...
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1answer
93 views

How do we solve the laplace transform of the Heat Kernel?

I am interested in the value of $$\int_0^\infty e^{-\alpha t}\frac{e^{-\frac{|x-y|^2}{2t}}}{\sqrt{2\pi t}}\, dt $$ this is the laplace transform of the Heat kernel (changing the time variable) This ...
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56 views

How to solve an integral with a fractional order.

How should I find a value of these integrals: $$ A:=\int_{0}^{\infty}\frac{\sin(x)}{(x+1)^{2-\nu}x^{\nu}}dx \quad\text{and}\quad B:=\int_{0}^{\infty}\frac{\sin(x)}{(x+1)^{1-\nu}x^{\nu}}dx, $$ where $\...