Laplace Transform
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- tags
- Math, DiffEQ
- source
- https://en.wikipedia.org/wiki/Laplace_transform, Elementary Differential Equations and Boundary Value Problems
The Laplace Transform is a particular type of Integral Transform.
Let \(f(t)\) be given for \(t\ge0\)
\[ \mathcal{L}\{ f(t) \} = F(s) = \int_{0}^{\infty} e^{-st} f(t) dt \]
The Laplace Transform $F(s) for \(s>\alpha\) exists if
- \(f\) is a piecewise continuous function on the interval \(0\leq t \leq A \in \reals^{+}\)
- \(|f(t)| \leq k e^{\alpha t}\) when \(t \geq \mu\). \(k, \mu \in \reals^{+}\), and \(\alpha \in \reals\).