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

learn more… | top users | synonyms

3
votes
1answer
14 views

Moment generating function and convergent random variables

denote by $X$ and $X_n$, $n\in \mathbb{N}$, random variables and $r\in\mathbb{R}_+$ with $E=\mathbb{E}\left[ e^{rX} \right] < \infty$ and $E_n=\mathbb{E}\left[ e^{rX_n} \right] < \infty$ for all ...
0
votes
1answer
15 views

Laplace transform and value in x(0)

Somebody told me that if i have something like this: $x''(t) + x'(t) = -2x(t) + u$ $x(0) = 7$ and use laplace transform on it i will get $s^2X(s) + sX(s) = -2X(s) + U(s)$ next i'm getting ...
19
votes
3answers
5k 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 ...
0
votes
3answers
59 views

Evaluation of $\int_{0}^{\infty}t^3e^{-3t}dt$

I have to evaluate the integral $\int_{0}^{\infty}t^3e^{-3t}dt$ using complex analysis techniques (the laplace transform). Can you check my steps, please? $$\int_{0}^{\infty}t^3e^{-3t}dt =\Rightarrow ...
0
votes
0answers
23 views

Laplace transform of a definite integral

I'm having some troubles with what follows. I am interested in finding the Laplace transform w.r.t. $x$ of some real-valued, positive, continuous (in general well-behaved) function $f(x,t),x,t>0$. ...
4
votes
2answers
58 views

Show that $\int\limits_0^\infty\frac{1}{t}(\cos(at)-\cos(bt))dt=\ln(b/a),\,a,b>0$.

Show that $$\int\limits_0^\infty\frac{1}{t}(\cos(at)-\cos(bt))dt=\ln(b/a),\,a,b>0.$$ Thanks to wikipedia I know that $$\int\limits_0^\infty\frac{1}{t}(\cos(at)-\cos(bt))\,dt ...
0
votes
0answers
13 views

On Laplace transforms - Applications in Probability Theory

I'm trying to find good bibliography on Laplace transforms for Applications in Probability Theory. I can't understand deeply the importance of this tool; nor I was taught very much on the subject. ...
-2
votes
0answers
16 views

A name of abook with exercises of laplace methode [closed]

I really nead a name of abook which contains exercises concern Laplace methods
2
votes
1answer
26 views

Restrictions on repeated use of initial conditions in ODE

It seems to be common practice when solving ODE's to keep a count of what conditions you have used. I was under the impression that once a condition has been used it cannot be used again. However, I ...
0
votes
0answers
21 views

Existence of solutions in time and Laplace domains

I have not made use of Laplace transforms for many years since my education and I am a bit rusty on applying the various theorems associated with the transform. I have an equation $f(t)=0$ and I am ...
0
votes
0answers
7 views

Interpretation of diagonal detail in Haar Wavelet Transforms

I am a statistics grad student, and I have just begun exploring the topic of wavelet regression (specifically, Haar wavelets for discrete functions). I understand the generalization from a one ...
0
votes
1answer
43 views

fourier transform of $f(x) = x^2+\frac{1}{1+2x^4}$

I really have no thought on this. I can't seem to use residue thm., nor could I find a inverse transform for it. by some Fourier Calculator I know it's solvable but how?
2
votes
0answers
17 views

Differential Equation by Laplace Transform [closed]

I was solving normal IVP problmes but I have no idea as how to solve this problem with $u(t)$ present in the question. Please help with this one.
4
votes
2answers
75 views

Can Laplace solve every lineair differential equation?

I'm learning about laplace tranform method to solve lineair differential equations but i'm wondering if laplace transformations can be used to solve every linear differential equations there is. Or ...
2
votes
1answer
35 views

express as contour integral $ f(x) = \int_0^\infty dt \; e^{-t/g} \; \frac{1}{\sqrt{1 - 2 t x}} $

Let $0 < x < 1$, I have to compute this Laplace transform: $$ f(x) = \int_0^\infty dt \; e^{-t/g} \; \frac{1}{\sqrt{1 - 2 t x}} $$ I am not 100% this interal is defined. If $t > ...
1
vote
0answers
39 views

What are disadvantages/limitations of Laplace?

I was curious about what limitations the famous Laplace theorem for solving ODE had and what drawbacks it may have. PS: I am NOT familiar with Fourier
2
votes
0answers
800 views

inverse Laplace transform of $e^\sqrt{as}$

I am trying to find the inverse Laplace transform of $e^\sqrt{as}$ for $a>0$. So we need to solve $\oint_B dz \: e^\sqrt{az} e^{z t}$ (Bromwich contour), but not sure how to start. How do we even ...
1
vote
0answers
27 views
0
votes
3answers
24 views

Verify second order Cauchy Riemann equations

How do I differentiate the equations in 12? I understand the hint, but I'm not sure how to act on it.
0
votes
1answer
19 views

Where are the particular and homogeneous solution of the ODE when using Laplace?

When solving an ODE with Laplace, it seems as if there is no distinction between the homogeneous and particular solution. As if you calculated both at once. Is this correct? How does it come? Where ...
4
votes
2answers
164 views

Book on applied mathematics

My Applied Mathematics course covers these subjects: -Calculus of Variations -Laplace Transform -Fourier Analysis -Special Functions -Integral Equations And as an introduction to the subject it has ...
0
votes
0answers
31 views

Comparison between Laplace, operator calculus and system of first order ODE

I am trying to understand those three methods to solve differential equations. I would like to know what actually are the differences between the three: Laplace calculus operator conversion to a ...
1
vote
1answer
33 views

Given integral equation, find $y(1)$

Let $y(t)$ be a continuous function on $[0,\infty)$ whose Laplace transforms exists. If $y(t)$ satisfies $$\int\limits_0^t(1-\cos(t-\tau))y(\tau)d\tau=t^4\to(1)$$ then $y(1)=$ I was able to find ...
0
votes
2answers
26 views

To solve given differential equation using laplace transform

I am solving following diff eqn using laplace transform: \begin{eqnarray} y''+y= \begin{cases} 0, & \text{if 0<t<2 $\pi$}\\ \sin t, & \text{t>$2\pi$} ...
0
votes
1answer
62 views

Laplace transform of a signal over t

I tried to identify a formula that is appropriate for computing the Laplace transform of $$f(t)=\displaystyle\frac{\cos 2t-\cos 3t}{t}$$ but I couldn't find one. Give me a suggestion please. Thanks, ...
1
vote
2answers
50 views

What is the difference between an impulse response and a transferfunction?

An imupulse response, is the output you get when you apply an impulse, like a delta dirac function, to your system (only for LTI?). By knowing the impulse response you know the system. The ...
2
votes
1answer
100 views

A very simple question: what spaces of function does the Laplace transform map from and into?

Given a function $f$, we can write $f\colon\mathbb{R} \to \mathbb{R}$ to denote that $f$ takes a number from $\mathbb{R}$ into $\mathbb{R}$. Easy enough. Given the Laplace transform operator ...
1
vote
1answer
80 views

Why does the Laplace transform of $t^2 \exp(at)$ exist?

My book states a theorem : "Let $f(t)$ be a function piecewise continuous on $[0, A]$ for $ A > 0$ and have an exponential order at infinity with $|f(t)| \leq M \exp(at)$. Then the Laplace ...
1
vote
0answers
31 views

To find the value of a constant when the Laplace transform of a function is given

This question is regarding my previous post Find the value of a constant when the Laplace transform of a function is given where the hint was given by Moo to find the laplace transform of ...
0
votes
0answers
23 views

Laplace transform of $g(x)=\begin{cases}0.5\sin{4t}&\text{if }t<4\\0&\text{if }t\geq 4\end{cases}$ using second shift theorem

Using the Second Shifting Theorem to compute the Laplace transform of $$g(t)= \begin{cases} 0.5\sin{4t}&\text{if }t<4\\ 0&\text{if }t\geq 4 \end{cases}$$. I try to write ...
0
votes
0answers
32 views

Find the value of a constant when the Laplace transform of a function is given

I am given that $F(s) = \tan^{-1}{s} + k$ is the laplace transform of some function $f(t)$ $t\geq 0$ . I have to find the value of $k$. What I get is: $F(s) = L(f(t))$ $\Rightarrow L(f(t)) = ...
1
vote
0answers
27 views

Convolution of complex functions (Laplace Domain)

Convolution of functions in the time domain is equivalent to multiplication in the frequency domain. However, I am interested in multiplication of functions in the time domain, which is convolution in ...
1
vote
1answer
25 views

Laplace Transform of Shifted Function

Why do we need to multiply the shifted function $f(t - a)$ by the shifted step function $u(t - a)$ to obtain the Laplace transform? $$ \mathcal{L\{f(t - a)\}} = \int_0^\infty u(t - a)f(t - ...
1
vote
1answer
23 views

Using Laplace Transform to solve this ODE

How to solve this ODE, with Laplace Transform: $$ \begin{cases} 20y'(x)+y(x)+4y''(x)=20\\ y(0)=10\\ 4y'(0)=-2 \end{cases} $$ Thanks in advance. My work: ...
2
votes
2answers
56 views

To find Inverse Laplace of $\,F(s)=\log\dfrac{s+1}{(s+2)(s+3)}$

To find Inverse Laplace of $$F(s)= \log\frac{s+1}{(s+2)(s+3)}.$$ I have tried to use shifting theorems, but of no use. Should I apply series for log and take inverse laplace of individual terms, if ...
0
votes
1answer
62 views

Solve the IVP $xy'' + y' + 4xy = 0, y(0) = 3, y'(0) = 0$

It has to be solved with Laplace transform and then converted to Bessel equation. $L(xy'') = -\frac{dL(y'')}{ds}$ $L(4xy) = -\frac{4dL(y)}{ds}$ $L(y'') = s²L(y) - sy(0) - y'(0) = s²L(y) -3s$ ...
0
votes
0answers
12 views

The need for two laplace transforms

So I have recently come across Laplace transforms, but I have seen one sided and two sided laplace transforms, my question is why do we need two kinds of transforms, when do we use which transform?
0
votes
1answer
52 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 ...
0
votes
0answers
47 views

Laplace trasform

i am trying to do this exercise but i do not get it. The laplace trasform is: \begin{equation} T(f)(s)= \int_{0}^{\infty} f(t)e^{-st} dt \end{equation} The exercise is: a) If $f$ is the ...
0
votes
1answer
34 views

2-sided Laplace transform of $\exp(-(t + e^{-t}))$

I'm having trouble finding an analytic solution to the 2-sided Laplace transform of; $$f(t) = \exp(-(t + e^{-t}))$$ Integration by parts doesn't seem to help. Any pointers appreciated. It seems like ...
1
vote
1answer
46 views

Solve $y''-xy'+y = 1 , y(0)=1, y'(0) = 2 $ with Laplace transform

What's making me get stuck is the Laplace transform of $xy'$. I'm aware of different methods of solving this, but it's asking specifically for Laplace transform.
2
votes
1answer
19 views

Closed-loop transfer function in the time domain

In a simple linear system with feedback (figure 1), the closed-loop transfer function $H(s)$ can be written as $$ H(s)=\frac{X_o(s)}{X_i(s)} = \frac{G(s)}{1+G(s)F(s)} $$ by solving the equations $$ ...
6
votes
1answer
74 views

Inverse Laplace transform of $1/\sqrt{s^2-a^2}$ using complex integration

I want to find the inverse Laplace transform of $$F(s) = \frac{1}{\sqrt{s^2-a^2}}$$ preferably using the Bromwich integral: $$f(t) = \frac{1}{2\pi i}\int_{\beta -I \infty}^{\beta +i ...
6
votes
1answer
2k 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 ...
4
votes
3answers
228 views

Finding the inverse Laplace transform of $ \ln \! \left( 1 + \frac{1}{s^{2}} \right) $.

Can someone help me find the inverse Laplace transform of $ \ln \! \left( 1 + \dfrac{1}{s^{2}} \right) $? I have no idea where to start.
2
votes
2answers
48 views

Inverse Laplace Transform of $\ln[\frac{s^2+a^2}{s^2+b^2}]$

How does one find $\mathcal{L}^{-1}\{\ln[\frac{s^2+a^2}{s^2+b^2}]\}$? I've tried splitting it up into $\mathcal{L}^{-1}\{\ln(s^2+a^2)\}-\mathcal{L}^{-1}\{\ln(s^2+b^2)\}$. However, I can't think of ...
0
votes
1answer
24 views

Calculating Laplace inverse

I'm having difficulties calculating a simple Laplace inverse : $$ \frac{S-4}{S^2-2S-11} $$ I'm new at this and couldn't find good examples for this case. could you please guide me ?
0
votes
2answers
18 views

How to get the Laplace transform of $t \cdot f(t) \cdot e^t$

Is there a formula to get the Laplace transform of $t \cdot f(t) \cdot e^t$ ? I tried integration, but that got me nowhere, because I'm probably missing something. Any ideas?
0
votes
0answers
28 views

The inverse Laplace transform of $\Gamma\left(\zeta\right) \, W_{\zeta,\mu}(z) $

Someone has a reference that addresses an integral of the followns type $$I = \frac{1}{2i\pi} \int_{\sigma-i\infty}^{\sigma+i\infty} e^{t\zeta} \, \Gamma\left(\zeta\right) \, W_{\zeta,\mu}(z) \, ...
0
votes
1answer
51 views

How can I find the Fourier transform of constant value like $1$.

The textbook told me that $\mathbb F[1] = \delta(f)$ and $\mathbb F[\delta(t)]=1$. It is easy to prove that $\mathbb F[\delta(t)] = 1$. $$ \mathbb F[\delta(t)] = \int_{-\infty}^\infty ...