Articles of laplace transform

What is the laplace transform and how is it performed? (detailed explanation)

I am a high school student and I became interested after someone mentioned it. Although I am not quite at the level where I am taught this it just captured my attention. Could someone give me an explanation of what the laplace transform is, and perhaps even a relatively basic example?

Laplace transform of $f(t^2)$

Suppose we know the Laplace transform of a function $f(t)$: $$F(s) = \int_0^\infty f(t) \mathrm e^{-st} \mathrm d t$$ Can we connect this to the Laplace transform of $h(t) = f(t^2)$? Is there a useful relation here? Generalization: This is related to, and, which asks a more general question about Fourier transforms. I […]

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 $$f(t) = \{\begin{align}&\frac{\sin(t)}{t} \;\;\;\;\;\;\;\; t \neq 0\\& 1\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; t = 0\end{align}$$ a) Using the power series (Maclaurin) for $\sin(t)$ – Find the power series representation for $f(t)$ for $t > 0.$ b) Because […]

Finding Inverse Laplace Transform

Can somebody help me to find the inverse Laplace transform of $$F(s)\exp\left(-\sqrt{\frac{s}{a}}\right)$$ or at least $$\exp\left(-\sqrt{\frac{s}{a}}\right)?$$ I’ll be so grateful.

Laplace transform of $\cos(at)$

I need to find the Laplace transform of $\cos(at)$ I know that $L\{\cos(at)\}= \int_{0}^{\infty} e^{-st} \cos (at) dt$ but I am having trouble finding the integral Thank you

Functional form of a series of a product of Bessels

This question arises from my answer to an inverse Laplace transform question. The result I got took the form $$ f(t)= e^{-r_0 t/2} H(t-a) \left [ J_0\left(\frac{1}{2} a r_0\right) I_0\left(\frac{1}{2} r_0 t\right) \\+ 2 \sum_{k=1}^{\infty} J_{2 k}\left(\frac{1}{2} a r_0\right) I_{2 k}\left(\frac{1}{2} r_0 t\right)\right ] $$ where $H$ is the Heaviside step function: $$H(x) = \begin{cases} […]

$\int_0^\infty x e^{-\mathrm i x\cos(\varphi)}\mathrm dx=-\frac{1}{\cos (\varphi )^2}$ is that correct?

Good day. This integral looks very simple, yet I don’t know how to start. $$\int_0^\infty x e^{-\mathrm i x\cos(\varphi)}\mathrm dx$$ I know that if the lower integration limit was $-\infty$ it would be a derivative of a Dirac delta function $2\pi \mathrm i\;\delta'(\cos(\varphi))$. But it isn’t. So I guess it should somehow relate to Dirac […]

Inversion of Laplace transform $F(s)=\log(\frac{s+1}{s})$ (Bromwich integral)

I am looking for the inversion of Laplace transform $F(s)=\log(\frac{s+1}{s})$. I started by using the general formula of the Bromwich integral: $\displaystyle \lim_{R\to\infty} \int_{a-iR}^{a+iR} \frac{1}{2\pi i}\log\left(\frac{s+1}{s}\right) e^{st}ds $ Then, I used that: $\displaystyle \log\left(\frac{s+1}{s}\right)=\sum_{n=1}^{\infty} \frac{ (-1)^{n+1} }{n} (1/s)^n $ for $|s|>1$. Since $|s|>1$ the Bromwich line should be to the right of $1$. So: $\displaystyle […]

Laplace transform of $\sin(\sqrt{t})$

I’m tired and completely blank on how to find a solution of Laplace transformation of $\sin(\sqrt{t})$.Please specify the method used and if possible something other than using series method. thank you

Evaluating improper integrals using laplace transform

I want to calculate the following improper integral using Laplace and transforms (and laplace transforms only). $$\int_0^{\infty} x e^{-3x} \sin{x}\, dx$$ I propose the following method. I plan to use $\mathcal{L}(\int_0^x f(t) dt)=\frac{\mathcal{L}(f)(s)}{s}$. So, first I need to find out the laplace transform of the integrand, which can be done using $\mathcal{L}({xf})=-\frac{dL(f)}{ds}=-\frac{d}{ds}(e^{-3x} \sin{x})=-\frac{d}{ds}(\frac{1}{(s+3)^2+1})=\frac{2(s+3)}{((s+3)^2)+1)^2}$ So, my […]