1
vote
0answers
59 views

Transformed Laplace “solution space”

From my own knowledge I can tell that when we take the Laplace transformation of a function we are in essence transforming our f(t) into a F(s). I've looked at several Q/A here asking for the ...
2
votes
1answer
273 views

Understanding Laplace Transforms

The Laplace transform of a function $f(t)$ is a function that maps $\mathbb{C} \mapsto \mathbb{C}$. $$f(s) = \int_0^\infty f(t)e^{-st}dt, \text{ with } s=x + iy$$ Since $s = x + iy$ is complex, ...
14
votes
3answers
4k views

Differential equations and Fourier and Laplace transforms

Why do both the Fourier transform and the Laplace transform appear in the study of differential equations? I've never understood why there are some situations where the Fourier transform is used and ...
2
votes
4answers
756 views

Why is the derivative multiplication by frequency in Laplace transform?

Why is the time-domain derivative equivalent to multiplication by frequency ($s$) in the Laplace transform? Why is the time-domain integral equivalent to division by frequency ($\frac{1}{s}$) in the ...