0
votes
0answers
35 views

Why does s = z+1?

What exactly is Laplace transform? motivated me to ask why unit function is 1/s by Laplace transform and 1/(1-z) by Z-transform? Both seem to be integrals of delta-pulse and secondary integration ...
1
vote
2answers
87 views

IVP with challenging numbers. I've seen it evaluated by non trivial manipulations. Can someone complete it step by step with explanations?

Solve the IVP: $X' = AX+f(t)$ $$\begin{align*} A&= \begin{bmatrix}6/7 & -15/14\\-5/7 & 37/14\end{bmatrix} \\ X(0)&= \begin{bmatrix}4\\-1\end{bmatrix} \\ f(t)&= ...
2
votes
1answer
127 views

Find the probability of certain measurement for a Laplace Operator on a state function

Let $H$ be the operator $ -\frac{d^{2}}{dx^{2}} $ and let its domain be $$\{f\in L^{2}(\mathbb{R},d\lambda)\text{ }:\int_{-\infty}^{\infty}|xF[f(x)]|^{2}dx<\infty\} $$ where $F$ is the Fourier ...
5
votes
2answers
494 views

integral transforms: why do roots in frequency domain correspond to eigenvalues in time domain (and how does it help solve differential equations)?

In Wikipedia you can read about integral transforms, esp. the Laplace transform which maps a differential equation in the time domain into a polynomial equation in the complex frequency domain: ...