Laplace Transform of Cosine/Proof 3

Theorem

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

Then:

$\laptrans {\cos a t} = \dfrac s {s^2 + a^2}$

where $a \in \R_{>0}$ is constant, and $\map \Re s > a$.

$\ds \laptrans {\cos a t}$

$=$

$\ds \laptrans {\frac {e^{i a t} + e^{-i a t} } 2}$

$\ds $

$=$

$\ds \frac 1 2 \paren {\laptrans {e^{i a t} } + \laptrans {e^{-i a t} } }$

$\ds $

$=$

$\ds \frac 1 2 \paren {\frac 1 {s - i a} + \frac 1 {s + i a} }$

$\ds $

$=$

$\ds \frac 1 2 \paren {\frac {s + i a + s - i a} {s^2 + a^2} }$

simplifying

$\ds $

$=$

$\ds \frac s {s^2 + a^2}$

simplifying

$\blacksquare$