# 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

$\blacksquare$