Laplace Transform of Sine of t over t/Corollary
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Theorem
Let $\sin$ denote the real sine function.
Let $\laptrans f$ denote the Laplace transform of a real function $f$.
Then:
- $\laptrans {\dfrac {\sin a t} t} = \arctan \dfrac a s$
Proof
\(\ds \laptrans {\dfrac {\sin t} t}\) | \(=\) | \(\ds \arctan \dfrac 1 s\) | Laplace Transform of Sine of t over t | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \laptrans {\dfrac {\sin a t} {a t} }\) | \(=\) | \(\ds \dfrac 1 a \arctan \dfrac 1 {s / a}\) | Laplace Transform of Constant Multiple | ||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac 1 a \arctan \dfrac a s\) |
But:
\(\ds \laptrans {\dfrac {\sin a t} {a t} }\) | \(=\) | \(\ds \int_0^\infty e^{-s t} \dfrac {\sin a t} {a t} \rd t\) | Definition of Laplace Transform | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac 1 a \int_0^\infty e^{-s t} \dfrac {\sin a t} t \rd t\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac 1 a \laptrans {\dfrac {\sin a t} t}\) |
The result follows.
$\blacksquare$
Sources
- 1965: Murray R. Spiegel: Theory and Problems of Laplace Transforms ... (previous) ... (next): Chapter $1$: The Laplace Transform: Solved Problems: Translation and Change of Scale Properties: $12$