Laplace Transform of Sine of Root/Proof 1

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Theorem

$\laptrans {\sin \sqrt t} = \dfrac {\sqrt \pi} {2 s^{3/2} } \map \exp {-\dfrac 1 {4 s} }$

where $\laptrans f$ denotes the Laplace transform of the function $f$.


Proof

\(\ds \sin \sqrt t\) \(=\) \(\ds \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {\paren {\sqrt t}^{2 n + 1} } {\paren {2 n + 1}!}\) Definition of Real Sine Function
\(\ds \) \(=\) \(\ds \sum_{n \mathop = 0}^\infty \frac {\paren {-1}^n} {\paren {2 n + 1}!} t^{n + \frac 1 2}\)
\(\ds \leadsto \ \ \) \(\ds \laptrans {\sin \sqrt t}\) \(=\) \(\ds \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {\map \Gamma {n + \frac 3 2} } {\paren {2 n + 1}! s^{n + \frac 3 2} }\) Laplace Transform of Power, Linear Combination of Laplace Transforms
\(\ds \) \(=\) \(\ds \frac 1 {s^{3/2} } \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {\paren {n + \frac 1 2} \map \Gamma {n + \frac 1 2} } {\paren {2 n + 1}! s^n}\) Gamma Difference Equation
\(\ds \) \(=\) \(\ds \frac {\sqrt \pi} {2 s^{3/2} } \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {\paren {2 n + 1} \paren {2 n}!} {2^{2 n} n! \paren {2 n + 1}! s^n}\) Gamma Function of Positive Half-Integer
\(\ds \) \(=\) \(\ds \frac {\sqrt \pi} {2 s^{3/2} } \sum_{n \mathop = 0}^\infty \frac {\paren {-1}^n} {2^{2 n} n! s^n}\)
\(\ds \) \(=\) \(\ds \frac {\sqrt \pi} {2 s^{3/2} } \sum_{n \mathop = 0}^\infty \frac 1 {n!} \paren {-\frac 1 {2^2 s} }^n\)
\(\ds \) \(=\) \(\ds \dfrac {\sqrt \pi} {2 s^{3/2} } e^{-1/\paren {2^2 s} }\) Definition of Exponential Function
\(\ds \) \(=\) \(\ds \dfrac {\sqrt \pi} {2 s^{3/2} } \map \exp {-\dfrac 1 {4 s} }\) simplifying

$\blacksquare$


Sources

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