Laplace Transform of Exponential

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Theorem

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 $s > 4$.