1
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0answers
29 views

Laplace transform, when $s \rightarrow \infty$

I'm reviewing lecture notes on Laplace Transform and there's one step that I don't understand: Find the solution to: $$x y'' + y' + xy = 0, y(0) = 1, y'(0) \mbox{ finite}$$ Taking the Laplace ...
1
vote
1answer
57 views

Inverse Laplace transform of $s^{k}$

How can I find the inverse Laplace transform of $s^{k}$ where $k$ is non-integer and negative? I know that $$\mathcal{L}^{-1}[s^k] = \frac{1}{2\pi i}\int e^{st} s^k ds$$ and since we have ...
0
votes
0answers
28 views

Laplace transform of a majorated function

I have the following problem. I have an analytic function and I want to show that it is majorated by a convenient function. To do that, it is very helpful to solve the transformed equation. I have a ...
1
vote
1answer
51 views

Deducing Laplace Formulas

I have to compute the followings integrals $\forall\; b\in \mathbb{C},\; \text{Re} \;b \gt0,p\gt 0$ $$ \int_{-\infty}^\infty \frac{e^{ipx}}{x-ib}$$ $$ \int_{-\infty}^\infty \frac{e^{ipx}}{x+ib}$$ ...
1
vote
1answer
52 views

Laplace Transform Damp Harmonic Motion

http://gyazo.com/19d18f085731c6dbc304fefdaece4f3c.png I'm currently on (a) where so far I have gotten; $ y'' + 2y' + 5y = f(t) $ Using Laplace transforms, I get; $ Y(s)$ = $ F(s) + s+2\over(s^2 ...
2
votes
0answers
45 views

Laplace transform to describe a bounded function

It is easy to show that if a real function $f:\mathbb{R}\rightarrow\mathbb{R}$ is contained in a strip $[a,b]$, that is if $\forall_{x}\, a\le f(x)\le b$, then its Laplace transform is bouned by ...
4
votes
1answer
173 views

Laplace transform of and impulse sampled function using “frequency” convolution

This is a long question, but assume we have this: The book uses the frequency convolution theorem to solve this problem. To solve the integral, it uses a contour + residue theorem to solve it. The ...
1
vote
1answer
66 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
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
68 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 ...
2
votes
2answers
140 views

Inverse Laplace transform of $\frac{s}{\sqrt{(s+a)^3}}$

Trying to find the inverse Laplace transform of $\frac{s}{\sqrt{(s+a)^3}}$. So solving $\oint_B dz \: \frac{z}{\sqrt{(z+a)^3}} e^{z t}$ (Bromwich contour). I tried doing a u-substitution with $u=z+a$ ...
0
votes
1answer
324 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 ...
4
votes
1answer
198 views

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} ...
0
votes
1answer
54 views

Showing an inequality

I wish to show $$|{(Re^{i \theta})^{-\frac{1}{2}}}\exp(\frac{-1}{Re^{i \theta}})| < \frac{M}{R^k}$$ for some M, k > 0 I've managed to reduce it to $$|R^{-\frac{1}{2}}| |\exp(\frac{-1}{Re^{i ...
0
votes
1answer
40 views

inverse transform of $Z(\omega) =\frac{a}{\alpha-i\omega}$

I am stuck at calculating the inverse transorm of $Z(\omega) =\frac{a}{\alpha-i\omega}$. Can someone help me please? thanks
0
votes
0answers
46 views

laplace transform and infinitely differentiation

This fact appears in my statistics textbook (Pg 543, statistical decision theory and bayesian analysis). it says : for normal distribution the generalized bayes estimator becomes \begin{align*} ...
2
votes
0answers
545 views

Inverse Laplace Transform as Bromwich Integral

I am seeking a references that provide a rigorous treatment of the inverse Laplace transform (Bromwich integrals), and how to compute them (beyond using tabled solutions - they don't cover my needs, ...
1
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1answer
55 views

Domain of the Laplace transform.

The unilateral Laplace transform of an $f:[0,\infty]\rightarrow \mathbb{C}$ is defined as $$F(s)=\int_{0}^{\infty}e^{-st}f(t)dt$$ My lecturer didn't go into detail on the domain of the transform, ...
3
votes
1answer
387 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 ...
1
vote
1answer
575 views

Condition for the inverse laplace transform of a function to exist and bromwich integral

Given any function, is there any way of determining from the nature of the function, if it is the laplace transform of a piecewise continuous function of exponential order? For e.g. say the function ...
1
vote
1answer
100 views

Inverse Laplace transformation of a complex function

Consider the complex function $\displaystyle f(s)=\frac{1}{\frac lc\sqrt{(s(s+r_0)}}$ where $r_0, l, c$ are positive real number and s is a complex variable. How I can obtain the inverse Laplace ...
1
vote
1answer
220 views

Calculation of the Inverse Laplace Transform of $\frac{1}{p}$ by contour integration.

I am always told in my lessons of control engineering that the inverse Laplace Transform of $\frac{1}{p}$ is the Heaviside step function $\theta(t)$. But I have a problem when I calculate the inverse ...
3
votes
1answer
704 views

Inverse Laplace transform using contour integration

I want to show by contour integration that $\displaystyle\mathcal{L}^{-1} \{\text{arccot}(s) \}(t)= \frac{\sin t\ }{t}$. In other words, I want to evaluate $\displaystyle \frac{1}{2 \pi i} \int_{a - ...
4
votes
0answers
115 views

Interpretation of the Laplace transform

Here's my intuitive understanding of the Fourier transform of $f:{\mathbb R}\rightarrow{\mathbb C}$, defined by $$\mathcal{F}(f)(\omega) = \int_{-\infty}^{\infty}e^{-2 \pi i \, \omega \,x}f(x)dx$$ I ...
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votes
0answers
61 views

Which class of functions can be represented as $F(z)=\int A(t)z^t dt$?

If I have a holomorphic function $f(z)$, then I can write it as $$f(z)=\sum_{n=0}^\infty a_n z^n.$$ So these functions can be viewed as a generating function of the coefficients $\{a_n\}$ which have a ...
2
votes
1answer
255 views

Inverse Laplace transform and Jordan's Lemma

I'm trying find the inverse Laplace transform $f(t)$ where I have $F(p)=\dfrac{9}{p(p+3)^2}$. I know $f(t)$ already to be $1-3te^{-3t}-e^{-3t}$. I have the integral $$f(t)=\dfrac{9}{2 \pi ...
1
vote
1answer
90 views

arguing away - complex analysis

Probably a trivial question but I can't understand how to argue away the value of integrals in complex analysis. I am trying to find the inverse Laplace transform of $F(s)=\frac{1}{s(s+1)}$. The ...
2
votes
2answers
296 views

Inverse Laplace transform of $\frac{\log(s)}{1 + s}$

Is it possible to find the inverse laplace transform $$\mathcal{L}^{-1}\frac{\log(s)}{1 + s}$$ using the Bromwich integral formula $$\mathcal{L}^{-1} \{F(s)\}(t) = f(t) = \frac{1}{2\pi ...
2
votes
2answers
163 views

Laplace transform exercise

I found this on Priestley's Complex Analysis in the Laplace transforms bit. Suppose $f$ satisfies $f'(t)=f(kt)$ for $t>0$, where $0<k<1$ and $f(0)=1$. Prove that ...
1
vote
1answer
137 views

Laplace inverse of the sine function

I was wondering if there is a closed-form Laplace inverse of the sine function. I have tried the following: $$ \sin(as)=\sum_{n=0}^{\infty}\frac{(-1)^{n}(as)^{2n+1}}{(2n+1)!} $$ an $n$-th power of ...