Laplace Transform of Sine of Root

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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 1

\(\ds \sin \sqrt t\) \(=\) \(\ds \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {\paren {\sqrt t}^{2 n + 1} } {\paren {2 n + 1}!}\) 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

$\blacksquare$


Proof 2

Let $\map y t := \sin \sqrt t$.

Differentiating twice with respect to $t$, we get:

$(1): \quad 4 t y'' + 2 y'' + y = 0$

Let $\map Y s = \laptrans {\map t y}$ be the Laplace transform of $y$.


Then taking the Laplace transform of $(1)$:

\(\ds -4 \map {\dfrac \d {\d s} } {\laptrans {\map {y''} t} } + 2 \laptrans {\map {y'} t} + \laptrans {\map y t}\) \(=\) \(\ds 0\) Derivative of Laplace Transform
\(\ds \leadsto \ \ \) \(\ds -4 \map {\dfrac \d {\d s} } {s^2 \, \map Y s - s \, \map y 0 - \map {y'} 0} + 2 \paren {s \, \map Y s - \map y 0} + \map Y s\) \(=\) \(\ds 0\) Laplace Transform of Derivative, Laplace Transform of Second Derivative
\(\ds \leadsto \ \ \) \(\ds 4 s^2 \, \map {Y'} s - \paren {6 s - 1} \map Y s\) \(=\) \(\ds 0\) simplifying
\(\ds \leadsto \ \ \) \(\ds \map Y s\) \(=\) \(\ds \dfrac c {s^{3/2} } \, \map \exp {\dfrac 1 {4 s} }\) solving the differential equation

For small $t$, we have:

$\sin \sqrt t \sim \sqrt t$

and:

$\laptrans {\sqrt t} = \dfrac {\sqrt \pi} {2 s^{3/2} }$

For large $t$, we have:

$Y \sim \dfrac c {2 s^{3/2} }$

Hence by comparison:

$c = \dfrac {\sqrt \pi} 2$

Hence:

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

$\blacksquare$


Sources