2
votes
0answers
56 views

Contour integral: different answers with different contours

Good day to everyone. I have a following contour integral problem. I have to find a solution for the integral $$\underset{\gamma_r }{\oint }\frac{e^{\lambda s} }{(1-s) s^{a-b} \left(s-\theta ...
2
votes
1answer
79 views

Laplace transform of : $t^{\gamma-1} F(\alpha,\beta,\delta,\frac{t}{d})$, where $F$ is the Gauss' hypergeometric function

What is the Laplace transform of : $t^{\gamma-1} F(\alpha,\beta,\delta,\frac{t}{d})$, where $\gamma >0 $ and $F$ is the Gauss' hypergeometric function. Note that I have the Laplace transform of : ...
0
votes
1answer
39 views

Solve Laplace Integral (3 factors) [closed]

Please provide steps to solve this integral: $$3\int^t_0{\sin{u}(t-u)e^{-(t-u)}du}.$$
2
votes
2answers
96 views

Inverse Laplace with $\ln$

How can I compute the inverse Laplace of 1) $\ln\left(\dfrac{s+1}{s-1}\right)$ 2) $\ln\left(\dfrac{s-1}{s}\right)$. Can someone please hep me to do these two problems
2
votes
0answers
55 views

inverse laplace transform of $$p^{-3/2}e^{-\sqrt{pa}}(\cos(\sqrt{ap})+\sin(\sqrt{ap}))$$

I used the Residue theorem to solve this problem. But, I could not obtain the solution given by $$\mathscr{L}^{-1}\left( { p^{-3/2}e^{-\sqrt{pa}}\over{2\sqrt{2}}} [\cos(\sqrt{ap})+\sin(\sqrt{ap})] ...
2
votes
2answers
188 views

Prove that the Laplace trasform is a Linear trasformation

Could you help me prove that the Laplace Trasform is a Linear trasformation? Thank you.
1
vote
0answers
43 views

Laplace transform $y''''+37y''+36y=g(t)$

Hey this problem is making me insane so have at it and let me know what I keep screwing up. Express the solution of the initial value problem in terms of a convolution integral: ...
1
vote
1answer
83 views

Evaluate $\int_{0}^{\infty}\sqrt{\frac{\sqrt{(a^2-y^2)^2+4y^2}+a^2-y^2}{(a^2-y^2)^2+4y^2}}dy=\sqrt{2}\pi$

Prove or disprove that$$\int_{0}^{\infty}\sqrt{\frac{\sqrt{(a^2-y^2)^2+4y^2}+a^2-y^2}{(a^2-y^2)^2+4y^2}}dy=\sqrt{2}\pi$$ for any $a>1$. I came across with this integral evaluating inverse ...
1
vote
1answer
109 views

inverse Laplace transfor by using maple or matlab

I want to use inverse Laplace transform to F function by using maple or matlab. However I cannot get any answer. I know the answer from table but I want to use one of softwares. from table: ...
2
votes
2answers
59 views

Solving an integral using Laplace transform and inverse Laplace transform

I want to solve this integral equation using Laplace: $$ Y(t) + 3{\int\limits_0^t Y(t)}\operatorname d\!t = 2cos(2t)$$ if $$ \mathcal{L}\{Y(t)\} = f(s)$$ then, $$ f(s) + 3 \frac{f(s)}{s} = ...
0
votes
2answers
103 views

Demonstration with Laplace

How can I demonstrate this? If $F(t)$ is a periodic function with a period of $T>0$, then $$ \mathcal{L}\{F(t)\} = \frac{\int\limits_0^T e^{-st} F(t)\operatorname d\!t}{1-e^{-sT}}\operatorname ...
1
vote
2answers
76 views

Integral equation solve using Laplace transform

How can I solve this integral equation using Laplace transform? $${\int\limits_0^{\infty}\ }\frac{e^{-t}(1-\cos t)}{t}\operatorname d\!t$$ Knowing that $$ \mathcal{L}\{\cos t\} = \frac{s}{s^2+1} $$ ...
1
vote
2answers
38 views

integro-differential equation with application in quantum mechanics

I am trying to solve for the time dynamics for a simple quantum system (two-site system with sinusoidal coupling and a decay parameter on one site) and the math is looking not so simple. Here is the ...
2
votes
2answers
68 views

Laplace transform with initial value problem $y''+4y=12\sin(2t)$.

Using Laplace transforms solve the initial value problem. $$y''+4y = 12\text{sin}(2t); \qquad\qquad y(\pi)=-3, \quad y'(\pi)=-3$$ I have begun with writing: $\mathcal{L} (y'') = s^2y(s) -s y(\pi) ...
0
votes
1answer
51 views

Laplace's Method Integration

Consider the integral \begin{equation} I_n(x)=\int^2_1 (\log_{e}t) e^{-x(t-1)^{n}} \, dt \end{equation} Use Laplace's Method to show that \begin{equation} I_n(x) \sim \frac{1}{nx^\frac{2}{n}} ...
0
votes
1answer
56 views

Laplace's Method (Integration)

Consider the integral \begin{equation} I(x)=\int^{2}_{0} (1+t) \exp\left(x\cos\left(\frac{\pi(t-1)}{2}\right)\right) dt \end{equation} Use Laplace's Method to show that \begin{equation} I(x) \sim ...
2
votes
2answers
33 views

How can we take the LaPlace transform of a piecewise function?

How can we take the LaPlace transform of a function, given piece-wise function notation? For example, $f(t)=\begin{cases} 0 &\mbox{for } 0<t<2\\ t&\mbox{ for } 2<t \end{cases}$ ...
2
votes
1answer
82 views

Evaluating an integral with Laplace

We need to evaluate the following integral: $$\int_{0}^{\infty}\frac{\cos(tx)}{x^2+a^2}dx$$ There is the following note: "You may interchange taking the Laplace transform and integrating." I have ...
1
vote
1answer
67 views

Bromwich integral of $1/s^k$ with k real (non integer) and $1<k$

Is there a simple way to compute the inverse laplace transform of $1/s^k$ with k non integer using Bromwich integral (basically without using the known laplace transform of $t^n$)?
1
vote
1answer
33 views

Find the Laplace transform of $(t-\pi/2)\sin(t-\pi/2)$ using the time shift

What is the Laplace transform of $(t-\pi/2)\sin(t-\pi/2)$? I used the relationship $\mathcal{L}((t-a)f(t-a))=e^{-as}F(s)$ Hence I get $\dfrac{2e^{-(\pi/2)s}}{s^2+1}$. Would this be correct?
1
vote
1answer
78 views

Laplace transform of $g_n(t)=\begin{cases}\frac{(1-e^{-t})^n}{t^n}&:t>0,\\0&:t\le0.\end{cases}$

Find Laplace transform for this function "$g$" $$g_n(t)=\begin{cases}\frac{(1-e^{-t})^n}{t^n}&:t>0,\\0&:t\le0.\end{cases}$$ Then Take advantage of it to calculate the following ...
0
votes
0answers
76 views

Contour integral (inverse Laplace transform) with arctan

I have what I think is a relatively simple contour integral involving arctan, but it is giving me difficulty. I would really appreciate any help. The integral itself is, with τ, λ, and k all real and ...
0
votes
1answer
321 views

Laplace Transform $f(t)=2\cos(3t)$

Determine the laplace transform of the function $f(t)=2\cos(3t)$, without using the table of Laplace transforms. I use by part integration to solve it, with $u=e^{-st},\, du/dt=-se^{-st}$ and ...
0
votes
3answers
77 views

Decompose integral of derivative and $e^{st}$ (laplace transform)

I'm reading on Laplace transform and stumbled upon the transform of a derived function. Could someone explain me this? $$ \begin{equation} \int_{0^{-}}^\infty \frac{d}{dt}f(t)e^{-st} dt = ...
2
votes
3answers
81 views

Laplace transform of the following function

find the laplace transform of the function : $$f(t) =\begin{cases} t^2, & 0<t<1 \\ 2\cos t+2, & t>1 \\ \end{cases}$$ My attempt: $$L\{f(t)\}=\int_{0}^{1}e^{-st} \ t^2 \ ...
2
votes
1answer
138 views

Integral Equation without solution?

working on a physical problem I arrived at the following equation $$ y(x) + A \int_{0}^{x} e^{\lambda (t-x)} y(t) \mathrm{d}t = 0$$ and after some struggling (not that easy to apply the basic Laplace ...
1
vote
1answer
302 views

Laplace transformation of a polynomial function involving square root (or approximation of the integral)

Could somebody suggest how to solve this Laplace transform: $$ \int_0^\infty{e^{-at}\over\sqrt{A+Bt+Ct^2}}{\rm d\,}t $$ ? The real coefficients $A,B,C$ are chosen such that the roots of $A+Bt+Ct^2$ ...
2
votes
1answer
73 views

How to approach/solve this integral?

Could somebody suggest how to approach or solve this integral: $$ \int_{0}^\infty e^{-a t}{2+t-2\sqrt{1+t}\over t^2}{\rm d\,}t, $$ where $a>0$ ? It is not a homework. I tried to use residuum ...
0
votes
1answer
28 views

On Laplace transforming an integrand

Suppose I have the following function: $$ M(s) = \int_{0}^{\infty}\frac{f(E)}{s+2\pi i /T(E)}\text{d}E \tag{1} $$ Where $f(E)$ is some potentially complex function, $T(E)$ is some real function and ...
2
votes
0answers
306 views

How Heaviside step function changes limits of integration

This question involves the Laplace transform of the convolution of two functions. The derivation in my textbook has a step that really confuses me. First I'll lay out their argument. $$ f(t) = f_1(t) ...
3
votes
1answer
124 views

$\int_0^\infty x e^{-\mathrm i x\cos(\varphi)}\mathrm dx=-\frac{1}{\cos (\varphi )^2}$ 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 ...
6
votes
1answer
349 views

Inverse Laplace Transform of $\bar p_D = \frac{K_0(\sqrt[]s r_D)}{sK_0(\sqrt[]s)}$

I solved the following partial differential equation using Laplace Transform: $\LARGE \frac{1}{r_D}\frac{\partial}{\partial r_D}(r_D\frac{\partial p_D}{\partial r_D})=\frac{\partial p_D}{\partial ...
3
votes
2answers
57 views

Evaluating Laplace Transform

I have a Laplace transform function of the following form and I'm trying to evaluate it. From my research I think I need to take the Inverse Laplace Transform and then integrate, but I'm having ...
1
vote
2answers
112 views

Laplace transform and integration together

The question, given in the textbook, is somewhat different. However, I am rephrasing it as follows: $$ \frac{2}{\pi}\int_{0}^{\frac{\pi}{2}} \left[ \mathcal L \lbrace \cos(t\cos\theta) \rbrace ...
1
vote
2answers
58 views

Integration by Parts confusion

I am using this video to learn Laplace Transform. The example used is a fairly basic one: $$ \int_{0}^{\infty}t.e^{-st}dt $$ Simple enough, you need to integrate by ...
3
votes
2answers
400 views

Continuity of integral function

How to show that the following function is right continuous at $0$ (that is, when $a\to0+$): $I(a) = \int_0^{\infty}\frac{\sin x}{x}e^{-ax}dx$? I know that Lebesgue integral $I(0) = \frac{\pi}{2}$. ...
1
vote
3answers
164 views

Find the inverse Laplace transform of $f(t) = \int_t^\infty \frac{e^{ - u}}{u}du$

Find the inverse Laplace transform of the integral:$$f(t) = \int_t^\infty {\frac{{{e^{ - u}}}}{u}du} $$ If the integral: $$f(t) = \int_0^\infty {\frac{{{e^{ - u}}}}{u}du} $$ I had done. However the ...
1
vote
2answers
75 views

Laplace transforms please help

I really need help to find to Laplace transforms of $f(x)=x+e^{-x}$, and $g(x)=xe^x$. I'm having big troubles on the calculations. Thanks.
0
votes
2answers
267 views

Laplace Transform of $tf(t)$

Q. prove that $\mathfrak{L}\{tf(x)\}=-\frac{d\mathfrak{L}\{f(x)\}}{ds}$ where the notation used is standard one. Attempt I tried what would seem obvious way to start: ...
1
vote
2answers
151 views

Laplace of two functions multiplied together?

I'm trying to figure out how to do the Laplace transform of $\delta(t-\pi) \sin{t}$. I know the Laplace transform of each of these. But how would I find it when they are multiplied together?
2
votes
0answers
116 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 ...
0
votes
1answer
52 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,\ ...
0
votes
0answers
76 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} ...
1
vote
1answer
52 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 ...
1
vote
1answer
82 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 ...
1
vote
1answer
839 views

Find the Laplace transform of $g(t) = 1+cos^2(2t)$ by direct integration

I'm having trouble finding out how to directly integrate the function $f(t)$ because of the $\cos^2(2t)$ term. I understand that $\cos^2(2t) = \frac{1}{2} + \frac{1}{2}\cos(4t)$ but I don't ...
5
votes
1answer
992 views

How to figure of the Laplace transform for $\log x$?

I was looking at a table of common Laplace transforms of functions when I came across the transform for $\log x$. Apparently, the transform is as follows: $$\mathcal{L} \left\{ \log ...
0
votes
1answer
320 views

Numerically calculating inverse Laplace via the inverse Laplace transformation formula

I'm trying to simulate a control system whose transfer function is $H(s)$. I'm comparing different numerical methods for this. I have already used these two methods: - Converting the transfer function ...
7
votes
2answers
802 views

Bringing a limit inside of an integral

Under what conditions does $$ \lim_{a \to 0^{+}} \int_{0}^{\infty} f(x) e^{-ax} \ dx = \int_{0}^{\infty} f(x) \ dx \ ?$$ For example, for $a>0$, $$ \int_{0}^{\infty} J_{0}(x) e^{-ax} \ dx = ...
3
votes
1answer
321 views

Solving integral equation with Laplace's Transform.

I'm trying to prove the following $$\int\limits_0^\infty {\frac{{\cos tu}}{{{u^2} + 1}}\log udu} = - \frac{\pi }{2}\int\limits_0^\infty {\frac{{\sin tu}}{{{u^2} + 1}}du} $$ The original problem ...