The tag has no wiki summary.

learn more… | top users | synonyms

0
votes
1answer
21 views

Find the Laplace transform of integral(from 0 to x) sin(2t) dt

Find the Laplace transform of $\int_0^x\,\sin\,(2t)\,dt$ So basically, $$\int_0^x\,\sin\,(2t)\,dt = -\frac{1}{2}(\cos\,(2x) - 1)$$ So $$\mathcal{L}\{\cos\,(2x)\} = \dfrac{s}{s^2 + 4}$$ ...
2
votes
1answer
145 views

Proof of Laplace expansion using minors

I've come across with the following proof of the Laplace expansion: Let $\Delta=\sum_{j=1}^n (-1)^{1+j} a_{1j}\bar M_j^1$ and $\tilde{\Delta}= \sum_{j=1}^n (-1)^{i+j} a_{ij}\bar ...
0
votes
1answer
31 views

Laplacian transform of division by square root of t?

In this formula: $$f(t)=e^{-3t}t^{\frac{-1}2}$$ I saw examples on $t^n$ where $n>0$. But in above example $n<0$. I don't know how to deal with the $t^{\frac{-1}2}$. I know that ...
0
votes
0answers
34 views

Amount of sub-matrices created by Laplace expansion

I have created a program that solves a matrices determinant using the Laplace expansion method, and I was wondering if there is a equation which provides how many sub-matrices are created and used in ...
0
votes
0answers
54 views

Lagrange interpolation polynomial and Vandermonde matrix?

I am trying to solve a problem about Lagrange interpolation polynomial, but i have the feeling i am stuck in something very trivial...So, may i ask you for a little help? Given is the polynomial ...
0
votes
1answer
57 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
3answers
79 views

Evaluate determinant of an $n \times n$-Matrix

I have the following task: Let $K$ be a field, $n \in \mathbb{N}$ and $a,b \in K^n$. Evaluate the determinant of the following matrix: $$\begin{pmatrix} a_1+b_1 & b_2 & b_3 & \dots ...
1
vote
1answer
47 views

How does Laplace expansion work?

\begin{bmatrix} 1 & 2 & 0 & 0 & a\\ 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 1 & 1 & 0\\ 0 & 0 & 0 & 1 & 0\\ 1 & 0 & 1 & 1 & 1 ...
1
vote
0answers
59 views

How to find Laplace approximation for following integral?

Let's have integral $$ I(x) = \frac{1}{2\pi} \int \limits_{-\pi}^{\pi}e^{xcos(\theta )}d \theta, \quad x \to +\infty . $$ How to use Laplace approximation for this integral and find first two ...
0
votes
0answers
19 views

Laplace s Domain Simplification involving shifting

This is probably a straight-forward question (forgive me - it has been a while) - I would like to solve the following equation for $V_c(s)$: $$ {V_{DC} \over s} + {\omega V_{AC}\over {s^2 + \omega^2} ...
0
votes
2answers
1k views

Determinant of 4x4 Matrix by Expansion Method

Find det(B) = \begin{bmatrix} 2 & 5 & -3 & -2 \\ -2 & -3 & 2 & -5 \\ 1 & 3 & -2 & 0 \\ -1 & -6 & 4 & 0 \\ \end{bmatrix} I chose the 4th column because ...
0
votes
1answer
24 views

solve laplace equation by fourier tranform

If $$ ∇^2 u=0$$ ,for $$ x≥0$$ and if $$u=f(y)$$on $$x=0$$ show that $$u(x,y)=x/π ∫_-∞^∞]〖f(ξ)/(x^2+(y-ξ)^2 ) dξ〗$$ solve by fourier tranform
1
vote
0answers
118 views

Real approximation to the maximum using Laplace's method integral

The Laplace's Method states that under some conditions, it holds that: $ \sqrt{\frac{2\pi}{M(-g''(x_0))}} h(x_0) e^{M g(x_0)} \approx \int_a^b\! h(x) e^{M g(x)}\, dx \text { as } M\to\infty$ Where ...
5
votes
1answer
66 views

Is this matrix positive-semidefinite in general?

for the matrix written below I was wondering if one can show that it is positive-semidefinite for $n>3$ and $0< \alpha<1$. (Or not. For $n=2, 3$ it works by showing that all principal minors ...
0
votes
2answers
49 views

More Laplace! - help needed

Here is the exam question that I am practicing: I have completed the first two parts to this question (thankfully to stackexchange) Laplace question - help needed Laplace question continued ...
1
vote
2answers
67 views

Laplace question - help needed

I am currently studying the Laplace transformation and came across this question: I have no idea of how to start this and am completely lost. If anyone could help I would be really grateful. ...
0
votes
0answers
52 views

laplace transform and O notation

I am studying about asymptotic expansions of integrals and I am confused with the following. Assume we have a function $f$ smooth enough with $\displaystyle{ \lim_{t \to \infty} f(t) < \infty }$ ...
-2
votes
1answer
96 views

Prove that L[f' ' ](s)$ = $sL[f](s)

Can anyone prove this question ? Let $f$:$\mathbb{R}$$→$$\mathbb{C}$ be continuous function such that $f$$(0)$ $=$ $0$ and that $f'$ be a piecewise continuous function and absolutely integrable on ...
0
votes
1answer
59 views

Finding the determinant of a $3 \times 3$ matrix via Laplace Expansion

I have a matrix here where I need to calculate the determinant using Laplace expansion. $$ \begin{pmatrix} 4 & 0 & 1\\19 & 1 & -3\\7 & 1 & 0 \end{pmatrix} $$ So I did the ...
3
votes
1answer
131 views

Laplace method help

$$\int_{0}^{\infty} \frac{e^{-x \cosh t}}{\sqrt{(\sinh t)}}dt$$ I'm trying to use Laplace's method to find the leading asymptotic behavior as $x$ goes to positive infinity, but I'm having some ...
4
votes
1answer
270 views

Laplace integral and leading order behavior

Consider the integral: $$ \int_0^{\pi/2}\sqrt{\sin t}e^{-x\sin^4 t} \, dt $$ I'm trying to use Laplace's method to find its leading asymptotic behavior as $x\rightarrow\infty$, but I'm running into ...
0
votes
1answer
145 views

How do I understand the proof of Laplace's Theorem in wikipedia?

See http://en.wikipedia.org/wiki/Laplace_expansion What does $\tau\,=(n,n-1,\ldots,i)\sigma'(j,j+1,\ldots,n)$ stand for as well as the statements follow? "Since the two cycles can be written ...
0
votes
1answer
220 views

Help With Difficult Proof

Suppose we have the following equation 1: $$\tag{1} A_G(x,y,z) = \frac{A_1}{q(z)} e^{-ik \frac{x^2 + y^2}{2q(z)}} $$ where $$ q(z) = z+iz_0 $$ and $i$ is equal to $\sqrt{-1}$. Suppose we have ...
1
vote
1answer
80 views

Complete expansion of Laplace integral

Let $\varphi \in C^\infty (\mathbb R^n ;\mathbb R)$ such that 1) $\varphi(0)=0$ 2) $\varphi(x)>0$ on $\mathbb R^n\setminus 0$ 3) $\text{Hess}_{\varphi}(0)>0 $ and let $B_1(0)$ be the ...
3
votes
1answer
355 views

Laplace expansion

This statement is from the book of Winitzki Linear Algebra via Exterior Products. (Section 3.4, page 123) Let $V$ be finite dimensional vector space, $\dim(V)=N$. The determinant of the matrix ...