Laplace Transform of Exponential

Theorem

This article is complete as far as it goes, but it could do with expansion. In particular: three theorems: $\laptrans {e^{a t} }$ for $a \in \R$; $\laptrans {e^{i b t} }$ for $b \in \R$; $\laptrans {e^{\psi t} }$ for $\psi \in \C$ In progress: this page has been renamed, and a master page created. The pages for the second two instances need to be written. In due course, unless someone gets there before me. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding this information. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Expand}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

Real Argument

Let $\laptrans f$ denote the Laplace transform of a function $f$.

Let $e^x$ be the real exponential.

Then:

$\map {\laptrans {e^{a t} } } s = \dfrac 1 {s - a}$

where $a \in \R$ is constant, and $\map \Re s > \map \Re a$.

Imaginary Argument

Laplace Transform of Exponential/Imaginary Argument

Complex Argument

Laplace Transform of Exponential/Complex Argument

Examples

Laplace Transform of $2 e^{4 t}$

$\laptrans {2 e^{4 t} } = \dfrac 2 {s - 4}$

for $s > 4$.