Laplace Transform of Cosine/Proof 1

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Theorem

Let $\cos$ be the real cosine function.

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 $\map \Re s > a$.


Proof

\(\displaystyle \map {\laptrans {\cos {a t} } } s\) \(=\) \(\displaystyle \int_0^{\to +\infty} e^{-s t} \cos {a t} \rd t\) $\quad$ Definition of Laplace Transform $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \intlimits {\frac {e^{-s t} \paren {-s \cos a t + a \sin a t} } {\paren {-s}^2 + a^2} } {t \mathop = 0} {t \mathop \to +\infty}\) $\quad$ Primitive of $e^{a x} \cos b x$ $\quad$
\(\displaystyle \) \(=\) \(\displaystyle 0 - \frac {-s \, \map \cos {0 \times a} + a \, \map \sin {0 \times a} } {s^2 + a^2}\) $\quad$ Exponential Tends to Zero, Exponential of Zero $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \frac {s \cos 0 - a \sin 0} {s^2 + a^2}\) $\quad$ simplifying $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \frac s {s^2 + a^2}\) $\quad$ Sine of Zero is Zero, Cosine of Zero is One $\quad$

$\blacksquare$