# What is Laplace transform: Definition and 776 Discussions

In mathematics, the Laplace transform, named after its inventor Pierre-Simon Laplace (), is an integral transform that converts a function of a real variable

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{\displaystyle t}
(often time) to a function of a complex variable

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{\displaystyle s}
(complex frequency). The transform has many applications in science and engineering because it is a tool for solving differential equations. In particular, it transforms linear differential equations into algebraic equations and convolution into multiplication.For suitable functions f, the Laplace transform is the integral

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{\displaystyle {\mathcal {L}}\{f\}(s)=\int _{0}^{\infty }f(t)e^{-st}\,dt.}

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1. ### Laplace transform of ##f(t)=(u(t)-u(t-2\pi))\sin{t}##

I tried to solve this as follows $$f(t)=(u(t)-u(t-2\pi))\sin{t}$$ $$=u(t)\sin{t}-u(t-2\pi)\sin{t}$$ $$\mathcal{L}(f(t))=e^{0\cdot s}\mathcal{sin{t}}-e^{-2pi s}\mathcal{L}(\sin{(t+2\pi)})$$ $$=\frac{1-e^{-2\pi s}}{s^2+1}$$ where I used the fact that ##\sin{(t+2\pi)}=\sin{t}##. Then I looked...
2. ### Partial fractions with complex linear terms

I am interested specifically in solving this problem by factoring the quadratic term into complex linear factors. $$s^2+4=0$$ $$\implies s=\pm 2i$$ $$\frac{5s+6}{(s-2i)(s+2i)(s-2)}=\frac{A}{s-2i}+\frac{B}{s+2i}+\frac{C}{s-2}$$ We can solve for ##C## using the cover-up method with ##s=2## to...
3. ### I Getting zeros and poles for Laplace transform

I'm following the intuition behind getting the zero's and poles of a damped cosine function with this video At around 11:50, he shows some graphics pertaining to multiplying the probing function with the impulse response, but the graphics don't seem correct. For example, in the B+B' graphic...
4. ### I On convolution theorem of Laplace transform: Schiff

Here follows the theorem and proof: Questions: 1. I do not understand the following part "...and hence, in view of the preceding calculation, ##\int_0^\infty \int_0^\infty |e^{-st}f(\tau)g(t-\tau)|dtd\tau## converges". We know that ##\mathcal{L}\big(f(t)\big)## and...
5. ### I On Laplace transform of derivative

The following three results are used in the proof of the theorem I have a question about. Now follows the theorem and its proof I have a question about. I do not understand why ##e^{-st}f(t)\rvert_0^\infty=-f(0)##. By Lemma 2, this is only possible if ##f## is of exponential type of order...
6. ### Laplace transform vs phasor analysis in circuit analysis

I recently acquainted myself with Laplace transform, and it appears that it has some relations with phasor analysis. This observation stems from the fact that while in Laplace transform, we have ##s = \sigma + j \omega## as the variable, in phasor analysis, we just use ##j\omega,## apparently...
7. ### I Laplace Transform of Sign() or sgn() functions

Trying to model friction of a linear motor in the process of creating a state space model of my system. I've found it easy to model friction solely as viscous friction in the form b * x_dot, where b is the coefficient of viscous friction (N/m/s) and x_dot represents the motor linear velocity...
8. ### Engineering How to implement a transfer function in Simulink with variable coefficients?

The implementations for the two filters in simulink are as follow: For the first filter: For the second one: The obtained results have values of 10^-12, while the expected results should be between 10^-3 - 10. Since it's the first time when I try t implement a tf with variable coefficients I...

12. ### Mellin transform of Dirac delta function ##\delta(t-a)##

Hi, I found Laplace transform of this Dirac delta function which is ##F(s) = e^{-st}## since ##\int_{\infty}^{-\infty} f(t) \delta (t-a) dt = f(a)## and that ##\delta(x) = 0## if ##x \neq 0## Then the Mellin transform ##f(t) = \frac{1}{2 \pi i} \int_{\gamma - i \omega}^{\gamma +i \omega}...
13. ### I Laplace transform of a simple equation (Simple question)

Lets consider very simple equation ##x''(t)=0## for ##x(0)=0##, ##x'(0)=0##. By employing Laplace transform I will get s^2X(s)=0 where ##X(s)## is Laplace transform of ##x(t)##. Why then this is equivalent to X(s)=0 why we do not consider ##s=0##?
14. ### Inverse Laplace transform

\mathcal{L}^{-1}[\frac{e^{-5s}}{s^2-4}]=Res[e^{-5s}\frac{1}{s^2-4}e^{st},s=2]+Res[e^{-5s}\frac{1}{s^2-4}e^{st},s=-2] From that I am getting f(t)=\frac{1}{4}e^{2(t-5)}-\frac{1}{4}e^{-2(t-5)}. And this is not correct. Result should be f(t)=\theta(t-5)(\frac{1}{4}e^{2(t-5)}-\frac{1}{4}e^{-2(t-5)})...
15. ### A Laplace transform of derivatives

I have a question regarding Laplace transforms of derivatives \mathcal{L}[f'(t)]=p\mathcal{L}[f(t)]−f(0^−) Can anyone explain me why ##0^-##?
16. ### A Why is the MGF the Laplace transform?

The Laplace transform gives information about the exponential components in a function, as well as oscillatory components. To do so there is a need for the complex plane (complex exponentials). I get why the MGF of a distribution is very useful (moment extraction and classification of the...
17. ### I Understanding the Laplace Transform of cos(t)/t

So, I know the direct definition of the Laplace Transform: $$\mathcal{L}\{f(t) \} = \int_0^\infty e^{-st}f(t)dt$$ So when I plug in: $$\frac{\cos(t)}{t}$$ I get a divergent integral. however:https://www.wolframalpha.com/input/?i=+Laplace+transform+cos%28t%29%2F%28t%29 is supposed to be the...
18. ### What is the Inverse Laplace Transform of e^(-sx^2/2)?

My attempt at finding this was via convolution theorem, where we take F(s) = 1/s^2 and G(s) = e^(-sx^2/2). Then to use convolution we need to find the inverses of those transforms. From a table of Laplace transforms we know that f(t) = t. But I am sort of struggling with e^(-sx^2/2). My 'guess'...
19. ### MHB Inverse laplace transform pf infinite product

I have to do inverse laplace transform of infinite product that is shown below. Can somebody help me with that?
20. ### Laplace Transform Finding Open-Circuit Voltage

I am interested in modeling a battery charging/discharging. I am starting off with a simple model using a voltage source in series with a parallel RC branch which is in series with a resistor. I will be measuring the open circuit voltage between the last series resistor and the bottom of the...

41. ### I Solving a differential equation using Laplace transform

Hi, I was trying to see if the following differential equation could be solved using Laplace transform; its solution is y=x^4/16. You can see below that I'm not able to proceed because I don't know the Laplace pair of xy^(1/2). Is it possible to solve the above equation using Laplace...
42. ### Can someone double check my solution to this Laplace Transform problem?

My solution is in the file shown here
43. ### Help Proving a Complex Laplace Transform

So I could just try using the definition by taking the limit as T goes to infinity of ∫ from 0 to T of that entire function but that would be a mess. I tried breaking it down into separate pieces and seeing if I could use anything from the table but I honestly have no clue I'm really stuck. I'd...
44. ### I Why is the Laplace transform unchanged when t is replaced with -t?

In Mathematical Methods in the Physical Sciences by Mary Boas, the author defines the Laplace transform as... $${L(f)=}\int_0^\infty{f(t)}e^{-pt}{dt=F(p)}$$ The author then states that "...since we integrate from 0 to ##\infty##, ##{L(f)}## is the same no matter how ##{f(t)}## is defined for...
45. ### Physical Significance of the Laplace Transform

I have used Laplace transform during my EE studies to solve differential equations and in control system analysis, but we were taught that as a tool kit to make the math easier. The physical meaning was never explained. I know basic time and frequency domain concepts (thanks to Fourier series)...
46. ### Engineering Advanced Circuits, Laplace Transform, Find Initial Conditions

Vo(S) = [ N(s)Vi(s) + (- s2 + s - 2) ] / s3 + s2 + 1 ; can ignore (-s^2 + s - 2). From relevant equations: Vo(S) = [N(s)*Vi(s)]/(s^3 + s^2 + 1); -> (d3Vo(t)/dt3) + (d2Vo(t)/dt2) + Vo(t) = N(t)(dvi)/dt L[vi(t)] = t to s domain: [s3Vo(s) - s2Vo(0-) - SV'o(0-) - Vo''(0-)]Vo(s) + s2 - SVo -...
47. ### Laplace transform of sin(ωt)/[1+cos^2(ωt)]

Homework Statement L{sin(ωt)/[1+cos^2(ωt)]} = Homework Equations d {arctan[cos(ωt)]} /dt = - ω•sin(ωt)/[1+cos^2(ωt)] The Attempt at a Solution ∫e^(-st)•[sin(ωt)/(1+cos²(ωt)] dt = -(1/ω)•∫e^(-st)•{arctan[cos(ωt)]}' dt = = (integrating by parts and taking Re(s) > 0) = = π/(4ω) -(s/ω)•∫...
48. ### Simplifying Laplace Transform of Cosine with Angular Frequency and Phase Shift

Homework Statement I have to find the L-transform of ##f(x) = cos(\omega t + \phi)## Homework Equations . The Attempt at a Solution The straightforward approach is to write ##cos(\omega t + \phi)## as ##cos(\omega t)cos(\phi) - sin(\omega t)sin(\phi)## and it becomes: Lf(s) = \frac {s...
49. ### I Transfer Function relating momentum and force

Hey all, I hope this is the correct forum section to post this in. I heard about this problem from a youtube video but I've not been able to simulate it because the video was meant only for an introduction into PID control. Here's the problem: A remote control helicopter is hovering just...
50. ### Solving a 2D PDE using the Fourier Transform

Homework Statement Solve the following partial differential equation , using Fourier Transform: Given the following: And a initial condition: Homework EquationsThe Attempt at a Solution First , i associate spectral variables to the x and t variables: ## k ## is the spectral variable...