Transforms & Operators — Community Channel — RIDA MATH

Curated prompts

Start here.

Advanced
Where does a Laplace transform live?
For $f(t) = e^{2t}\sin t$ , find $\mathcal{L}\{f\}(s)$ and its region of convergence. Why is the formula meaningful outside that region only through analytic continuation?
A strong contribution: Gives $1/((s-2)^2+1)$ for $\operatorname{Re} s > 2$ and distinguishes the integral from the continued function.
- Lab: Transform Lab
- Read: Why Integral Transforms Can Be Studied Structurally
Advanced
Fourier conventions and where the $2\pi$ goes
Compute the Fourier transform of $e^{-|t|}$ under two conventions: $\int f(t)e^{-i\omega t}\,dt$ and $\int f(t)e^{-2\pi i \xi t}\,dt$ . How do the inversion formulas differ?
A strong contribution: Both transforms computed correctly with the matching inverse formulas stated.
- Lab: Transform Lab
Advanced
Reading $\mathcal{L}\{\sin bx\}$ as geometry
With $r = \sqrt{t^2 + b^2}$ and $\theta = \arctan(b/t)$ , check that $\mathcal{L}\{\sin bx\}(t) = \sin\theta / r$ . What happens to $\theta$ as $t \to 0^+$ , and what does that say about the transform?
A strong contribution: Verifies the identity and interprets $\theta \to \pi/2$ , $r \to b$ in terms of the limiting value $1/b$ .
- Lab: Angular–Hyperbolic Lab
- Read: A Geometric View of Laplace Transform Behavior

Member discussions

Published discussions.

No published discussions yet.

No member discussions have been published in this channel yet.

Contribute

Start a discussion.

Show your attempt, define your notation, and say what kind of answer would help.

Posts are reviewed before publication. Read the guidelines first.

Loading…