Laplace Transform of Cosine/Proof 2
Jump to navigation
Jump to search
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
\(\ds \laptrans {e^{i a t} }\) | \(=\) | \(\ds \frac 1 {s - i a}\) | Laplace Transform of Exponential | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {s + i a} {s^2 + a^2}\) | multiply top and bottom by $s + i a$ |
Also:
\(\ds \laptrans {e^{i a t} }\) | \(=\) | \(\ds \laptrans {\cos a t + i \sin a t}\) | Euler's Formula | |||||||||||
\(\ds \) | \(=\) | \(\ds \laptrans {\cos a t} + i \laptrans {\sin a t}\) | Linear Combination of Laplace Transforms |
So:
\(\ds \laptrans {\cos a t}\) | \(=\) | \(\ds \map \Re {\laptrans {e^{i a t} } }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \map \Re {\frac {s + i a} {s^2 + a^2} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac s {s^2 + a^2}\) |
$\blacksquare$
Sources
- 1965: Murray R. Spiegel: Theory and Problems of Laplace Transforms ... (previous) ... (next): Chapter $1$: The Laplace Transform: Solved Problems: Laplace Transforms of some Elementary Functions: $2$