Laplace transform [Jaeyoung Wiki]

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====== Laplace transform ======

Laplace transforms turn time-domain functions, where $t$ is the variable (time), into frequency-domain functions, where $s$ is the variable (complex frequency).

===== Formal definition =====

$$F(s) = \int_{0}^{\infty} f(t)e^{-st} dt$$

===== Inverse Laplace transform =====

Similar to the [[kb:z-transform|Z-transform]], we usually calculate the inverse Laplace transform by reorganizing the Laplace representation into a form we recognize with partial fractions and then pattern matching. Again, the time-domain representation depends on the desired [[kb:region_of_convergence|region of convergence]] - the same Laplace domain representation can result in different time-domain representations, depending on the RoC.


^ Laplace domain representation ^ Region of convergence ^ Time-domain representation ^
| $H(s)=\frac{1}{s-p}$ | $s > p$ | $h(t)=e^{pt}u(t)$ |
| $H(s)=\frac{1}{s-p}$ | $s < p$ | $h(t)=-e^{pt}u(-t)$|
| $H(s) = 1 $ | All | $ h(t) = \delta(t) $ | 
===== Why Laplace transforms are cool =====

Integration in the time domain becomes division by $s$ in the Laplace domain, and differentiation in the time domain becomes multiplication by $s$ in the Laplace domain. This is useful for block diagrams.

=== References ===

  * [[https://mathvault.ca/laplace-transform/]]