The Laplace transform is a widely used integral transform, similar to the Fourier transform.
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votes
3answers
574 views
What exactly is Laplace transform?
I've been working on Laplace transform for a while. I can carry it out on calculation and it's amazingly helpful. But I don't understand what exactly is it and how it works. I google and found out ...
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votes
1answer
202 views
Compute the inverse Laplace transform of $e^{-\sqrt{z}}$
I want to compute the inverse Laplace transform of a function
$$
F(z) = e^{-\sqrt{z}}.
$$
This problem seems very nontrivial to me. Here one can find the answer: the inverse Laplace transform of ...
9
votes
1answer
138 views
Laplace transform identity
Is there a function equal to its Laplace transform?
I mean
$$ \int_{0}^{\infty}dt\exp(-st)f(t)= f(s).$$
Of course I know $f(t)=0 $ satisfy the equation.
For the case of the Fourier transform, I ...
7
votes
4answers
253 views
How to find the Laplace transform of $\frac{1-\cos(t)}{t^2}$?
$$ f(t)=\frac{1-\cos(t)}{t^2} $$ $$ F(S)= ? $$
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votes
2answers
204 views
Inverse Laplace Transform help
Is the information below correct?
Find the inverse Laplace transform of $$ F(s) = \frac{s}{s^2 + 4s + 13}$$
Soln:
a) Complete the squares to simplify our denominator
$$ s^2 + 4s + 13 = (s+2)^2 + 9 ...
5
votes
5answers
569 views
How can I evaluate $\int_{-\infty}^{\infty}\frac{e^{-x^2}(2x^2-1)}{1+x^2}dx$?
How can I solve this integral: $$\int_{-\infty}^{\infty}\frac{e^{-x^2}(2x^2-1)}{1+x^2}dx.$$ Can I solve this problem using the Laplace transform? How can I do this?
5
votes
3answers
2k views
Differential equations and Fourier and Laplace transforms
Why do both the Fourier transform and the Laplace transform appear in the study of differential equations? I've never understood why there are some situations where the Fourier transform is used and ...
5
votes
2answers
137 views
What kind of book would show where the inspiration for the Laplace transform came from?
I'm trying to find out where to learn about integral transforms and inversions like the Laplace transform and the Bromwich integral.
I'm looking for a book that describes how you can find (derive) ...
5
votes
2answers
112 views
Usage of inverse Laplace transform
At my current study level in college, use of inverse Laplace transform is not mentioned well - textbooks say "use tables." So, can anyone show me how to use inverse Lapalce transform? And also proof?
...
5
votes
2answers
45 views
Laplace Transform
How can one show that 1/$e^s$ is not the laplace transform of any function?
Note that function here does not include distributions like dirac delta function.
5
votes
1answer
431 views
inverse Laplace Transform: $ L^{-1} \{\log \frac{s^2 - a^2}{s^2} \}$.
I am styding Laplace transforms and for some reason I have stuck in the followning exercise.
Find the inverse Laplace Transform $ L^{-1} \{\log \frac{s^2 - a^2}{s^2} \}$.
Any help?
Thank's in ...
5
votes
1answer
83 views
Find the Laplace transform of $f(t) = \begin{cases} 0, & \text{if $t<5$} \\ t^2−10t+31, & \text{if $t\ge 5$} \\ \end{cases} $
Find the Laplace transform of
$$f(t) =
\begin{cases}
0, & \text{if $t<5$} \\
t^2−10t+31, & \text{if $t\ge 5$} \\
\end{cases}
$$
$F(s)=$ __________?
Here is my work. I went wrong ...
5
votes
1answer
79 views
A Laplace transform question
Suppose I have a positive integrable random variable $X$ s.t. $$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}$$
and $U=\ln Z$ ...
5
votes
2answers
236 views
Laplace transform of integrated geometric Brownian motion
Is there any closed form of the Laplace transform of an integrated geometric Brownian motion ?
A geometric Brownian motion $X=(X_t)_{t \geq 0}$ satisifies $dX_t = \sigma X_t \, dW_t$ where ...
5
votes
1answer
241 views
$\mathcal{B}^{-1}_{s\to x}\{e^{as^2+bs}\}$ and $\mathcal{L}^{-1}_{s\to x}\{e^{as^2+bs}\}$ , where $a\neq0$
http://en.wikipedia.org/wiki/Integral_transform#Table_of_transforms claims than the integral form of inverse bilateral Laplace transform and inverse Laplace transform are both the same. But are they ...
4
votes
2answers
621 views
Why the Fourier and Laplace transforms of the Heaviside (unit) step function do not match?
The Fourier transform of the Heaviside step function $u(t)$ is $\dfrac{1}{iω} + π δ(ω)$.
The Laplace transform of the same function is $\dfrac{1}{s}$.
I remember the proof came from derivatives and ...
4
votes
2answers
97 views
Laplace transform:$\int_0^\infty \frac{\sin^4 x}{x^3} \, dx $
I have a trouble with a integral:
Using this Laplace trasform equation:
$$\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]
...
4
votes
2answers
433 views
integral transforms: why do roots in frequency domain correspond to eigenvalues in time domain (and how does it help solve differential equations)?
In Wikipedia you can read about integral transforms, esp. the Laplace transform which maps a differential equation in the time domain into a polynomial equation in the complex frequency domain:
...
4
votes
3answers
238 views
Physical meaning behind Frequency domain?
I understand its usage and why is it important because It transforms differential equations to algebraic ones..
But I can't get the physical meaning of the new form of the equation and the meaning of ...
4
votes
2answers
297 views
Laplace transform. How to derive
I have this integral related to a Laplace transform and I was wondering if anyone knows of a clever way to derive it. I know we usually look these up in a table, but this form is not in a table I ...
4
votes
1answer
204 views
Finding the Laplace transform of $f(x)=|\cos(x)|$
I have function $f(x)=|\cos(x)|, x≥0$ and like to derive its Laplace transform. I am told that $f(x+\pi)=f(x)$. Help me please.
4
votes
1answer
233 views
How to solve $t-2f(t) = \int_0^t(e^\tau- e^{-\tau})f(t-\tau)d\tau$
I want to solve this equation. It reminds me something about Laplace transform.
I am sure that I must use it order to solve it.
$$t-2f(t) = \int_0^t(e^\tau- e^{-\tau})f(t-\tau)d\tau$$
How to do it?
...
4
votes
2answers
292 views
Laplace Transformations
can someone kindly help me with these few questions? :)
Find $L{e^tf(t)}$ in terms of f*(s) and state a range of s which this is defined.
I couldn't figure this out. You use the definition but i ...
4
votes
1answer
43 views
Uniqueness of the Laplace Transformation
Today; when I was doing some Inverse Laplace transformation in the class, I encountered the following problem cited in Zill's book:
The inverse Laplace transformation may be not unique. In ...
4
votes
1answer
66 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.
4
votes
2answers
501 views
Evaluating $\int_0^{\infty}\!\sin(ax)\,dx$
I've been told by somebody (who is more advanced in math than me) that
$$
\int_0^{\infty}\!\sin(ax)\,dx = \frac{1}{a}
$$
But my basic intuition that the integral is the area under the curve $\sin(ax)$ ...
4
votes
2answers
137 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.
4
votes
1answer
52 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?
4
votes
1answer
381 views
Solving inverse Laplace Transform with convolution theroem.
Ok so I have recently found a transform that produced.
$$x\left( s \right) = \frac{\pi }{2}\frac{{\log s}}{{{s^2} - 1}}$$
However the function was given in an integral parametric form so to call it, ...
4
votes
1answer
243 views
Laplace transform and differentiation
Let $F(s)$ be the Laplace transform of $f(t)$:
$$F\left(s\right)=\int_{0}^{\infty}e^{-st}f\left(t\right)dt$$
It then follows that $f(t)$ can be recovered from $F(s)$ by the inverse Laplace ...
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
3answers
138 views
Having problem in Laplace
I have a question:
Find the Laplace transform of
$$ \frac{1}{s^2(s^2+4)}. $$
Now I also do have its solution. I wanted to know why do we Laplace inverse integral method (Please correct me if the name ...
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
98 views
Finding the Laplace Transform of sin(t)/t
I'm in a Differential Equations class, and I'm having trouble solving a Laplace Transformation problem.
This is the problem:
Consider the function
$$f(t) = \{\begin{align}&\frac{\sin(t)}{t} ...
3
votes
2answers
128 views
Find the inverse Laplace transform of $F(s)=\frac{e^{−6s}}{s^2+0s−16}$
Find the inverse Laplace transform of
$$F(s)=\frac{e^{−6s}}{s^2+0s−16}$$
Here is my work:
$$F(s)=\frac{e^{−6s}}{s^2+0s−16}$$
$$s^2+0s−16 = (s+4)(s-4)$$
$$\frac{1}{(s+4)(s-4)} = \frac{A}{s+4} + ...
3
votes
1answer
88 views
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 ...
3
votes
2answers
146 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 ...
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 ...
3
votes
1answer
151 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 ...
3
votes
2answers
319 views
Laplace transform of a Gaussian function with complex variable
The bilateral Laplace transform of a Gaussian function could be established as:
$$e^{x^2/2}=\frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{\infty}e^{-xy}e^{-y^2/2} dy$$
Then what should be a similar relation ...
3
votes
1answer
93 views
Laplace Transform $s$ sufficiently large [duplicate]
Possible Duplicate:
How does partial fraction decomposition avoid division by zero?
I've got a question about the domain restriction of $s$ when taking the Laplace Transform. I think my ...
3
votes
1answer
64 views
How to solve partial integro-differential equation?
Suppose the following partial integro-differential equation for a function $u(x,t)$ with $t\geq0$, $x \in [0,L]$:
$\partial_t u = \partial_{xx} u + f(u,\lambda)$
$\lambda = B\left(u_0 - \int_{x=0}^L ...
3
votes
3answers
120 views
Integrating using Laplace Transforms
$$\int_{0}^\infty {\cos(xt)\over 1+t^2}dt $$
I'm supposed to solve this using Laplace Transformations.
I've been trying this since this morning but I haven't figured it out. Any pointers to push me ...
3
votes
1answer
89 views
Unclear step on proof of Laplace transform of a derivative
Reading the article on the Laplace Transform in Wolfram MathWorld, I found the proof that $\mathcal{L}[f'(t)] = sF(s) - f(0)$.
I understand the first and second steps, but I don't understand the ...
3
votes
1answer
124 views
Inverse Laplace transform computation
Calculate the inverse Laplace transform
$$ \displaystyle{ \mathcal{L^{-1}} \{ s\log \frac{s^2 + a^2}{s^2 - a^2} \} }$$
I know that is boring but I would really appreciate some help.
Thank's in ...
3
votes
1answer
69 views
Approximating the logarithm of a Laplace transform
Suppose $X$ is a random variable on $\mathbb R_+$ with finite mean, i.e. $\mathbb E X <+\infty$.
Let $F_X(t)$ be its c.d.f. and $\mathcal{L}_X(\cdot)$ its Laplace transform, i.e.
...
3
votes
1answer
1k views
Solving a differential equation with the Dirac-Delta function without Laplace transformations
So I'm trying to solve the following differential equation:
$y''+3y'+2y=\delta(t-1)$, $y(0)=0$, $y'(0)=0$. (where $\delta$ is the Dirac's delta function)
Everything I've read in my ...
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
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 ...
3
votes
1answer
92 views
Help with a partial fraction decomposition
One of my homework problems last week was to find the inverse Laplace transform of the following:
$$F(s)=\frac{2s+1}{s^2-2s+2}.$$
The answer is $f(t)= 2e^t \cos t + 3e^t \sin t$.
Obviously once ...
