Laplace Transform of Sine Integral Function/Proof 2

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Theorem

$\laptrans {\map \Si t} = \dfrac 1 s \, \arctan \dfrac 1 s$

where:

$\laptrans f$ denotes the Laplace transform of the function $f$
$\Si$ denotes the sine integral function


Proof

\(\displaystyle \laptrans {\dfrac {\sin t} t}\) \(=\) \(\displaystyle \arctan \dfrac 1 s\) Laplace Transform of $\dfrac {\sin t} t$
\(\displaystyle \leadsto \ \ \) \(\displaystyle \laptrans {\int_{u \mathop \to 0}^{u \mathop = t} \frac {\sin u} u \rd u}\) \(=\) \(\displaystyle \dfrac 1 s \arctan \dfrac 1 s\) Laplace Transform of Integral
\(\displaystyle \leadsto \ \ \) \(\displaystyle \laptrans {\map \Si t}\) \(=\) \(\displaystyle \dfrac 1 s \arctan \dfrac 1 s\) Definition of Sine Integral

$\blacksquare$


Sources