The Laplace transform is a widely used integral transform, similar to the Fourier transform.
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1answer
23 views
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 ...
2
votes
1answer
177 views
Evaluate the inverse Laplace transform using convolution theorem where the argument is a function of s
We have from convolution theorem:
If $H(s)=F(s)G(s)$ then $$h(t)=L^{-1}\{F(s)G(s)\}=\int_{u=0}^t f(t-u)g(u)du$$
Here, I want to know if $$H(s)=F(P(s))G(P(s))$$ where $P(s)$ is a function of ...
1
vote
1answer
89 views
Laplace Transform Convolution Theorem Applied to Functions without Transforms
My differentials prof taught us the convolution theorem and applied it to a differential equation
$ay'' + by' + cy = e^{t^{2}}$
Then he transformed it and left the transform as $L(e^{t^{2}}) = ...
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1answer
38 views
Laplace inverse of $\frac{e^{-s}(3s^2-s+2)}{(s-1)(s^2+1)}$
I thought maybe you could fist solve $\frac{(3s^2-s+2)}{(s-1)(s^2+1)}$ using partial fractions and later solve the $e^{-s}$ separately as it is a $d(t-1)$ (Dirac delta function).
As you solve the ...
1
vote
1answer
65 views
Lower bounds of laplace transform of characteristic functions
I have the following integral:
\begin{equation}
f(\mu) = \int_0^\infty e^{-\mu t}\varphi_X(t)dt
\end{equation}
where $\varphi_X(t)$ is the characteristic function of some undetermined probability ...
1
vote
1answer
77 views
arguing away - complex analysis
Probably a trivial question but I can't understand how to argue away the value of integrals in complex analysis.
I am trying to find the inverse Laplace transform of $F(s)=\frac{1}{s(s+1)}$. The ...
1
vote
1answer
68 views
Laplace transform of convolution with modified limits
I have an expression such as
$\int_0^{x+l}y(z)g(x-z) dz$
and I want to evaluate its Laplace transform w.r.t $x$ in terms of the Laplace transform of $y(x)$. I know that I can substitute $t=x+l$, and ...
1
vote
1answer
59 views
Step/Impulse function setup confusion
I have a final coming up next week (Tuesday) and a sample question for the test is the following.
I was wondering if I have properly set it up for solving with Laplace as I haven't encountered a ...
4
votes
0answers
165 views
ODE Laplace Transform an impulse bring oscillating system to rest
$2y''+y'+2y=\delta(t-5)$
$y(0)=0, y'(0)=0$
Consider the system given by ODE above in which an oscillation is excited by a unit impulse at $t=5$. Suppose that it is desired to bring the system to ...
3
votes
0answers
125 views
Compare Fourier and Laplace transform
I would like to clarify main difference between Fourier and Laplace transforms and also understand if exponential factor is main difference between this two method. So Fourier transform is ...
3
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0answers
2k views
Relationship Between The Z-Transform And The Laplace Transform
Below I've quoted Wikipedia's entry that relates the Z-Transform to the Laplace Transform. The part I don't understand is $z \ \stackrel{\mathrm{def}}{=}\ e^{s T}$; I thought $z$ was actually an ...
2
votes
0answers
32 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:
$${\left[ {{\mathop{\rm R}\nolimits} {\rm{es}}\frac{{{e^{st}}{{(s - ...
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votes
0answers
80 views
A rational integral with exponential denominator
Prove that:
$$\int_{-\infty }^{+\infty }{\frac{{{x}^{4}}\text{d}x}{\left( \beta +{{\text{e}}^{x}} \right)\left( 1-{{\text{e}}^{-x}} \right)}}=\frac{\left( {{\pi }^{2}}+{{\ln }^{2}}\beta ...
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0answers
52 views
Interpretation of the Laplace transform
Here's my intuitive understanding of the Fourier transform of $f:{\mathbb R}\rightarrow{\mathbb C}$, defined by
$$\mathcal{F}(f)(\omega) = \int_{-\infty}^{\infty}e^{-2 \pi i \, \omega \,x}f(x)dx$$
I ...
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votes
0answers
73 views
How to perform an inverse Laplace transform
I'm trying to work out the inverse Laplace transform
$$f(z)=\mathcal{L}^{-1}\left\{s^2\log\left(1-\frac{z}{s}\right)\right\}.$$
To make sure this was first possible I turned to Mathematica and ...
2
votes
0answers
902 views
Laplace transform of the square root of a generic function
Let $f(t)$ be a function (for example of time $t$).
Is there a general expression of the laplace transform of $\sqrt{f(t)}$ ?
Same question for the inverse Laplace transform :
Let $f(s)$ be the ...
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0answers
21 views
The Square of the Laplace Transform
I have been looking at the Laplace transform $$\mathcal{L}f(s)=\int_0^{\infty}f(t)e^{-st}dt$$ and I'm trying to find
The norm of $\mathcal{L}^2$
The nullspace of The norm of $\mathcal{L}^2$
So ...
1
vote
0answers
36 views
Continuity of the inverse Laplace Transform
If I know $Y(s)$,
can I predict when $\mathscr{L}^{-1}[Y(s)]=y(t)$ will be continuous or continuously differentiable or even stronger conditions?
For example; I'm solving an ODE with the Laplace ...
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0answers
30 views
Find $N$ independent solutions through Laplace Transform
The Laplace Transform method gives us one solution of an Ordinary Differential Equation.
How can we use the same procedure to get $N$ independent solutions, being $N$ the order of the ODE?
Where can ...
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0answers
71 views
Laplace transform of a product of functions
While trying to compute the Laplace transform of a certain product, part of the calculation leaves me with a Bromwich integral which has the form:
...
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0answers
41 views
Inverse Laplace transform is required
I shall be very very thankful if some one can find the inverse Laplace of the function given below. I really need it as early as possible.
$$
\frac{1}{s-a}\exp ...
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vote
0answers
55 views
Laplace inverse transform of a complex function
I have a complex expression and I need to find the laplace inverse transform of this complex term.
The expression is
$$\Large{ e^{\Large{-\frac{x\sqrt{A^2+4Bs}}{2B}}}}$$
In the above expression ...
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vote
0answers
63 views
Identifiability of a state space system
I'm trying to solve assignment 4E.5 from this sheet (ship steering dynamics).
My question are:
Do I need to perform the Laplace Transform in order to check for identifiability?
The state space model ...
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0answers
48 views
Laplace transform curiosity
Experimenting in Mathematica, I see that taking the Laplace transform of certain functions $f(t)\neq 0$ actually gives me a non-zero function $F(s)$. However, for these certain functions, taking the ...
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0answers
127 views
Inverse Laplace Transform by contour integration
In question 1) we get Laplace transform of $$ g(t) = t^a $$ is:
$$
\hat g(t)= {1/s^{a+1}}\int_0^\infty e^{-t}x^a
$$
then I was stuck at question 2) which asks me to evaluate the inverse laplace ...
1
vote
0answers
101 views
How to check solution of Laplace Transform ODE problem
$$\frac{\mathrm dw(t)}{\mathrm dt}+2w(t)=y(t)$$
$$\frac{\mathrm dy(t)}{\mathrm dt}+3y(t)=2w(t)+f(t)$$
The input to the system is $~f(t)$ and the output is $~y(t).$
The initial conditions are ...
1
vote
0answers
140 views
square root of s domain transfer function
I have a question about calculating the square root of the magnitude of a transfer function. When you take the square root, what is happening? My initial idea of a magnitude is a single value, but in ...
1
vote
0answers
276 views
Solving integral equation with Laplace's Transform.
I'm trying to prove the following
$$\int\limits_0^\infty {\frac{{\cos tu}}{{{u^2} + 1}}\log udu} = - \frac{\pi }{2}\int\limits_0^\infty {\frac{{\sin tu}}{{{u^2} + 1}}du} $$
The original problem ...
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31 views
Proof the theorem $\int_0^\infty {{t^n}f(t)dt = {{( - 1)}^{n + 1}}\int_0^\infty {{F^{(n + 1)}}(u)du} } $
Knowing that:
$$ 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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votes
0answers
27 views
How to find out the laplace transform of the product of two functions?
How to find out the laplace transform of $$f(t)\sum_{i=1}^\infty a_i g_i(t)$$ w.r.t the variable t on the domain $[0,\infty)$, where $a_i$'s are constants with value $0\le a_i\le1$.
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votes
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22 views
Triple Laplace transform of a first order partial differential equation
How to find $L_x L_y L_t \{(\frac{\partial}{\partial x}- \frac{\partial}{\partial y}- \frac{\partial}{\partial t}) f(x,y,t)\}$?
I have obtained ...
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votes
0answers
24 views
Theories for Laplace Transform
To my knowledge, Fourier transform is related to some strong and powerful theories. It is a well defined rotation on $\mathcal{L}_2(V, \mathbb{C})$ and can be extended as an isomorphism on ...
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42 views
Laplace transform of a periodic function
Knowing that $$L[f(t)]=\frac1{1-e^{-sp}}\int_0^{p} e^{-st}f(t)dt$$
$p$ indicates the period of the function
If $f$ is a continuous function by segments in $[0,\infty)$ and $F(s)=L[f(t)]$ exists for ...
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0answers
71 views
About evaluating $\mathcal{L}^{-1}_{s\to x}\left\{\dfrac{F(s)}{s}\right\}$ by considering contour integration with different entire functions $F(s)$
Detailedly compare the difficulties of different entire functions $F(s)$ where $F(0)\neq0$ when evaluating $\mathcal{L}^{-1}_{s\to x}\left\{\dfrac{F(s)}{s}\right\}$ by considering contour integration, ...
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0answers
53 views
About the inverse laplace transform of sinc function
How to calculate $\mathcal{L}^{-1}_{s\to x}\{\text{sinc}(s)\}$ ?
Note: $\text{sinc}(s)=\dfrac{\sin s}{s}$ when $s\neq0$ .
Also note that $\lim\limits_{s\to\pm\infty}\dfrac{\sin s}{s}=0$ .
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0answers
48 views
How would I solve this Laplace transform for $Y(s)$ and $y(t)$
How would I solve this Laplace transformation for $Y(s)$ and $y(t)$? I'm lost and I don't know what to do.
Take the Laplace transform of the following initial value problem and solve for
...
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0answers
43 views
Asymptotically expand Laplace transform
Assume $\max_{a\leq t\leq b}{\phi (t)} =\phi (a)$, $\phi '(a)\neq 0,f(a)\neq 0$ and $f(t)$ has a Taylor expansion about $t=a$.
Use integration by parts to show that $I(x)=\int_a^b{f(t)e^{x\phi ...
0
votes
0answers
61 views
About Laplace transform
I dont understand the following working, why the integral becomes double integral?
$$\begin{align}
& \ \ \ \int_{0}^{1}{{{\left( \frac{1}{\ln x}+\frac{1}{1-x} ...
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0answers
72 views
Find the inverse Laplace transformation and limit
The question is related to this post but can be solved independtly.
I am trying to find a general expression in the time domain for the asymptotic behavior when $t \to \infty$ of $\bar{f}(s)$ defined ...
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0answers
105 views
Please help me find this limit and inverse Laplace transform
I need help solving this (I suggest something hereafter but I am not sure if it's ok):
I would like to find an approximate solution of the function $\bar{f}(s)$ defined in the Laplace space. At long ...
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votes
0answers
59 views
Which class of functions can be represented as $F(z)=\int A(t)z^t dt$?
If I have a holomorphic function $f(z)$, then I can write it as $$f(z)=\sum_{n=0}^\infty a_n z^n.$$ So these functions can be viewed as a generating function of the coefficients $\{a_n\}$ which have a ...
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0answers
52 views
Is there a matrix formulation of the Laplace transform?
The matrix formulation of the (discrete) Fourier transform for a signal 5 terms long, can be illustrated as follows:
Signal or time domain
$\left(
\begin{array}{ccccc}
1 & 1 & 1 & 1 ...
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0answers
69 views
Inverse laplace transform - infinite residues
I need to compute the inverse transform of the following, $f(s)=
\dfrac{\sinh(k(l-x))}{\sinh(kl)}\dfrac{\omega}{\omega^2+s^2}$ where $k=\sqrt{\dfrac{s^2}{c^2}+n^2\pi^2},\ 0\leq x\leq l$. I used what ...
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votes
0answers
37 views
Invers laplace transform of resolvant using residues
I got that the inverse laplace of $\frac{1}{(s-λ)}$ is $e^{λt}$. Does this look correct?
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36 views
limit propert of Laplace transform
consider $f\in L^1(R_+)$
and define Laplace transform
$$\mathcal{L}f(z):=\int_0^{\infty} f(s)e^{-zs}\mathbb{d}s. $$
How can I prove $$\lim_{\mathbf{Re}z\rightarrow\infty}\mathcal{L}f(z) = 0?$$
...
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votes
0answers
35 views
Selberg trace and Riemann zeros
Let us suppose we got a modification of the Selberg trace as follows
$$ \sum_{n=0}^{\infty} h(r_n) = \frac{\mu(F)}{4 \pi } \int_{-\infty}^{\infty} r \, h(r) \tanh(\pi r) dr + \sum_{ \{T\} } \frac{ ...
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votes
0answers
116 views
How to find Laplace Stieltjes Transform(LST)
If $F(t)$ is given as
$$F(t)=A(t)+\int_t^\infty e^{-\mu(1-\gamma)(u-t)}a(u) du$$
where $A(t)\{a(t)\}$ is the distribution function {density function} and $\mu, \gamma$ are constants.
Then find the ...
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0answers
262 views
the inverse laplace transform of an exponential function
I was solving differential equations by using laplace transform, but I am stuck here trying to find the laplace inverse below, can anyone help? $$
...
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0answers
81 views
How to solve differential equations of type $x' = x^3 + x^2 + x$ using Laplace Transform?
How do i solve equations like,
$f'(x) = f^3 + f^2 + f$
using laplace transforms?
Any help would be appreciated.....
0
votes
0answers
139 views
convert to system transfer function frequency domain -by Laplace transformation
I need help to convert this differential equation into a transfer function in the frequency domain.
$$Q = H(y-z) + M \frac{dx}{dt} + A\frac{dy}{dt}$$
where
$Q$ = heat input,
$H$ = heat loss ...
