The Laplace transform is a widely used integral transform, similar to the Fourier transform.
4
votes
1answer
65 views
Inverse Laplace Transform of $(s+1)/z^s$
I'm trying to compute this ILT
$$\mathcal{L}^{-1}\left\{\frac{s+1}{z^s}\right\},$$
where $|z|>1$. However, I'm not sure this is possile? Any help would be appreciated.
1
vote
1answer
86 views
Solving Differential equation with laplace transformation
$$y''-4y'+9y=9\quad,\quad y(0)=0 \quad ,\quad y'(0)=-8
$$
solve the differential equation with the laplace transformation.
i will solve this question to this state, but i cannot continue.
s^2 ...
0
votes
0answers
42 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 ...
2
votes
1answer
114 views
Special Laplace Inversion
Use complex analysis to show $$\frac1{2\pi i}\int_{a- i\infty}^{a+i\infty} e^{st}/s^{1/2} ds = \frac1{\sqrt{\pi t}}\ ,\quad a >0, t> 0 .$$ This is a special case of Bromwich's integral for the ...
0
votes
3answers
114 views
Functions without Laplace transform?
We have just started working with Laplace transformations at our university course. One of the I came across as following:
Provide three examples of functions for which the Laplace transform does ...
2
votes
1answer
174 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
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
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 ...
2
votes
0answers
80 views
A rational integral with exponential denominator
Prove that:
$$\int_{-\infty }^{+\infty }{\frac{{{x}^{4}}\text{d}x}{\left( \beta +{{\text{e}}^{x}} \right)\left( 1-{{\text{e}}^{-x}} \right)}}=\frac{\left( {{\pi }^{2}}+{{\ln }^{2}}\beta ...
2
votes
2answers
156 views
improper integral involving $e^x$
Show that :
$$\int_{-\infty }^{+\infty }{\frac{{{x}^{2}}\, \text{d}x}{\left( \beta +{{\text{e}}^{x}} \right)\left( 1-{{\text{e}}^{-x}} \right)}}=\frac{\left( {{\pi }^{2}}+{{\ln }^{2}}\beta \right)\ln ...
0
votes
1answer
44 views
A improper integral on expontential
Evaluate:
$$\int_{0}^{\infty }{\frac{\left( 1-{{\text{e}}^{-px}} \right)\left( 1-{{\text{e}}^{-qx}} \right)\left( 1-{{\text{e}}^{-rx}} \right)}{{{\text{e}}^{x}}}}\text{d}x,\ \ \ p>0,\ q>0,\ ...
2
votes
0answers
52 views
Interpretation of the Laplace transform
Here's my intuitive understanding of the Fourier transform of $f:{\mathbb R}\rightarrow{\mathbb C}$, defined by
$$\mathcal{F}(f)(\omega) = \int_{-\infty}^{\infty}e^{-2 \pi i \, \omega \,x}f(x)dx$$
I ...
1
vote
1answer
34 views
Show that Laplace transform of measure belongs to $C^{\infty}(0,\mathbb{R}^n_{+})$
Let $\mu$ be an exponentially decreasing Borel measure on $\mathbb{R}^n_{+}$, i.e. there exists $r>0$ such that
$$
\int\limits_{\mathbb{R}^n_{+}} e^{r|x|} \, \mu(dx) < \infty.
$$
I want to ...
1
vote
0answers
48 views
Laplace transform curiosity
Experimenting in Mathematica, I see that taking the Laplace transform of certain functions $f(t)\neq 0$ actually gives me a non-zero function $F(s)$. However, for these certain functions, taking the ...
0
votes
0answers
61 views
About Laplace transform
I dont understand the following working, why the integral becomes double integral?
$$\begin{align}
& \ \ \ \int_{0}^{1}{{{\left( \frac{1}{\ln x}+\frac{1}{1-x} ...
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votes
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 ...
0
votes
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 ...
2
votes
1answer
116 views
Understanding Laplace Transforms
The Laplace transform of a function $f(t)$ is a function that maps $\mathbb{C} \mapsto \mathbb{C}$.
$$f(s) = \int_0^\infty f(t)e^{-st}dt, \text{ with } s=x + iy$$
Since $s = x + iy$ is complex, ...
1
vote
1answer
38 views
Convergence integral causal function
I have an exercise where there is the following given:
$f$ is a causal function.
$f$ is Laplace transformable:$\int_{0}^{\infty} f(t)e^{-zt}
\, dt = L(z) $ with $Real(z)> -1$
I have to ...
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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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votes
1answer
73 views
Can you do this Laplace Transform
Can you do this? This is part of my final year EE work. I need to solve this in order to figure out how my sensor is behaving. Please help, and stop down voting. If it is too difficult for you, just ...
0
votes
1answer
55 views
Laplace Transform of this function
Find $L\{F(t)\}$ if
$$F(t) = \begin{cases} \sin t & \text{between }0 < t < \pi \\
0 & \text{between } \pi < t < 2\pi
\end{cases}$$
Really stumped by this one. Please can you ...
2
votes
1answer
82 views
Inverse Laplace transform and Jordan's Lemma
I'm trying find the inverse Laplace transform $f(t)$ where I have $F(p)=\dfrac{9}{p(p+3)^2}$. I know $f(t)$ already to be $1-3te^{-3t}-e^{-3t}$. I have the integral $$f(t)=\dfrac{9}{2 \pi ...
2
votes
0answers
73 views
How to perform an inverse Laplace transform
I'm trying to work out the inverse Laplace transform
$$f(z)=\mathcal{L}^{-1}\left\{s^2\log\left(1-\frac{z}{s}\right)\right\}.$$
To make sure this was first possible I turned to Mathematica and ...
2
votes
2answers
156 views
Find the inverse Laplace transform of…
I'm trying to find a limit in a function and need to calculate the following inverse Laplace transform:
$$
\mathcal{L}^{-1}\left\{\cfrac{1}{\sqrt{s}+s}\right\}
$$
2
votes
1answer
124 views
Laplace Transform of $\cosh^2(3t)$
Could someone help me on laplace transfrom ?
Using Laplace transform of derivative of $f(t)$,
Find the Laplace transform for
A) $\cosh^2(3t)$
How to derive it using Laplace transform of ...
5
votes
1answer
79 views
A Laplace transform question
Suppose I have a positive integrable random variable $X$ s.t. $$E[e^X]=+\infty$$
Now let's take a series with general term $p_n$, summing to one, and define $$Z=\sum_{n>0}p_ne^{X_n}$$
and $U=\ln Z$ ...
3
votes
1answer
47 views
Example of a function
I am looking for an example of a function $f$ such that $\lim_{t\to x_n}f(t)=\infty$ for infinitely many points $x_n$ and for which the Laplace transform $\mathscr{L}(f)$ exists. I am sure it must be ...
3
votes
1answer
148 views
How can I efficiently sketch a Nyquist diagram?
I have the following transfer function:
$$P(s) = \frac{3}{(s-1)(s+2)(s+3)}, s= j\omega$$
I got the starting and endpoints:
$$\omega_0 = -\frac{1}{2}, \omega_\infty = 0$$
When I split the ...
4
votes
2answers
134 views
Laplace transform of $y''' - 3y'' + 3y' - y = (t^2)e^t$ where $y(0)=1$, $y'(0)=0$, $y'' = -2$
Any ideas?
I got:-
$$s^3 - 2s^2 + 3s - 4/(s(s^2 + 3) + 1))$$
but I got it wrong, obviously, because it does not simplify into any inverse laplaces.
2
votes
2answers
106 views
Inverse Laplace of $\dfrac{(s+1)e^{-\pi s}}{s^2 + s + 1}$
Does anyone know how to calculate the Inverse Laplace transform of $\;\;\dfrac{(s+1)e^{-\pi s}}{s^2 + s + 1}\;\,$ ?
I've tried it and got (u is the unit step function):
...
0
votes
1answer
62 views
Laplace Transform of multiplied term like $u(t)u(4-t)$
First of all,
If this is a two-terms function I'd be simple. It will produce $$ \mathcal{L}[u(t)] = \frac{1}{s} $$
Except, I'm not sure what to do with $u(4-t)$. If it was $u(t-4)$, it would be ...
1
vote
1answer
77 views
arguing away - complex analysis
Probably a trivial question but I can't understand how to argue away the value of integrals in complex analysis.
I am trying to find the inverse Laplace transform of $F(s)=\frac{1}{s(s+1)}$. The ...
3
votes
1answer
72 views
Show that if $L\{F(t)\} = f(s)$ then $L\{F(at)\} = \frac{1}{a} f(\frac{s}{a})$
I'm trying to answer this question and I just don't know how to finish it.
I've tried integrating the $te^{-st}$ by parts and then multiplying it by $\frac{1}{a}$ but it doesn't show the answer I ...
1
vote
2answers
151 views
How to compute Inverse Laplace transform using Convolution
How do you evaluate the inverse transform below using convolution ?
$$
\mathcal{ L^{-1} } \left[ {\frac{s}{(s^2 + a^2)^2}} \right]
$$
I tried
$$\begin{align} \mathcal{ L^{-1} } \left[ ...
3
votes
2answers
81 views
Laplace transform of $x^2$
I can't seem to be able to understand why $$\mathcal{L}(x^n)(s) = \frac{n}{s} \mathcal{L}(x^{n-1})(s)$$ This one line has got me stuck! I know that $$\mathcal{L}(f(x)g(x)) \neq F(s)G(s)$$ so how could ...
2
votes
2answers
121 views
Inverse Laplace transform of $\frac{\log(s)}{1 + s}$
Is it possible to find the inverse laplace transform $$\mathcal{L}^{-1}\frac{\log(s)}{1 + s}$$ using the Bromwich integral formula $$\mathcal{L}^{-1} \{F(s)\}(t) = f(t) = \frac{1}{2\pi ...
1
vote
1answer
150 views
Inverse Laplace Transform involving $\cosh$.
While doing an assignment on solving a PDE I stumbled into the following inverse Laplace transform question (involving $\cosh$? I can't believe it). Mathematica gives no solution and I have no idea ...
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votes
2answers
164 views
Inverse Laplace Transform : $ F(s)=\frac{2 \omega^3}{(s^2+\omega^2)^2} $
Please help to find inverse laplace transform :
$$ F(s)=\frac{2 \omega^3}{(s^2+\omega^2)^2} $$
4
votes
1answer
51 views
What do you do if you need the Laplace transform of a diverging function?
How would I manage $\scr L \{e^{t^2}\}$? Does it even make sense to ask? Is it just a given that there are diverging Laplace functions that can't be handled?
1
vote
3answers
100 views
How to Find Inverse Laplace Transform of $ F(s)=\frac{1}{\pi} \cot^{-1}(\frac{10s}{\pi}) $
$$ F(s)=\frac{\cot^{-1}(\frac{10s}{\pi})}{\pi} $$ $$ f(t) = ?$$
0
votes
0answers
52 views
Is there a matrix formulation of the Laplace transform?
The matrix formulation of the (discrete) Fourier transform for a signal 5 terms long, can be illustrated as follows:
Signal or time domain
$\left(
\begin{array}{ccccc}
1 & 1 & 1 & 1 ...
1
vote
2answers
77 views
Question about the Laplace of a step function.
I'm just now learning how to take the Laplace of a simple step function, but I have a question about the terms. I'll show my work so far and hopefully someone can step in and answer the question I ...
1
vote
0answers
122 views
Inverse Laplace Transform by contour integration
In question 1) we get Laplace transform of $$ g(t) = t^a $$ is:
$$
\hat g(t)= {1/s^{a+1}}\int_0^\infty e^{-t}x^a
$$
then I was stuck at question 2) which asks me to evaluate the inverse laplace ...
1
vote
1answer
80 views
How do I evaluate $\lim_{h \to \infty} e^{h(1-s)}$?
I'm messing around with Laplace, and was trying to find the transform of $e^{t}$ and I have to evaluate $$\lim_{h \to \infty} e^{h(1-s)}$$ I figure if $s=1$, the limit is $1$. If $0≤s<1$, the ...
3
votes
4answers
102 views
Laplace transform of $x^a$
How to prove that the Laplace transform of $x^a$ is:
$$\mathcal{L}\{x^a\}(s)=\frac{\Gamma(a+1)}{ s^{a+1}}$$
Also how to prove that the inverse Laplace transform of $\frac{\Gamma(a+1)}{ s^{a+1}}$ is ...
3
votes
2answers
140 views
Laplace transform of $t \cos(t)$ by definition
I want to find the Laplace transform of $t \cos(t)$ by the definition $$\int e^{-st} t \cos(2t)dt$$
The solution manual just say try the $$u = t, dv = e^{-st} \cos(2t)$$
I use the integration by ...
7
votes
4answers
252 views
How to find the Laplace transform of $\frac{1-\cos(t)}{t^2}$?
$$ f(t)=\frac{1-\cos(t)}{t^2} $$ $$ F(S)= ? $$
0
votes
0answers
68 views
Inverse laplace transform - infinite residues
I need to compute the inverse transform of the following, $f(s)=
\dfrac{\sinh(k(l-x))}{\sinh(kl)}\dfrac{\omega}{\omega^2+s^2}$ where $k=\sqrt{\dfrac{s^2}{c^2}+n^2\pi^2},\ 0\leq x\leq l$. I used what ...
1
vote
1answer
140 views
Finding the inverse Laplace of $e^{-3s}\frac{1}{(s-1)^2}$
I know I can use the following:
$$\mathcal{L}^{-1}\{e^{-as}F(s)\} = u(t-a)f(t-a)$$
$$\mathcal{L}^{-1}\{\frac{n!}{s^{n+1}}\} = t^n$$
$$\mathcal{L}^{-1}\{F(s-a)\} = e^{at}f(t)$$
but I'm confused as how ...
