Laplace Transform of Sine Integral Function/Proof 1

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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

Let $\map f t := \map \Si t = \displaystyle \int_0^t \dfrac {\sin u} u \rd u$.

Then:

$\map f 0 = 0$

and:

\(\displaystyle \map {f'} t\) \(=\) \(\displaystyle \dfrac {\sin t} t\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle t \map {f'} t\) \(=\) \(\displaystyle \sin t\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle \laptrans {t \map {f'} t}\) \(=\) \(\displaystyle \laptrans {\sin t}\)
\(\displaystyle \) \(=\) \(\displaystyle \dfrac 1 {s^2 + 1}\) Laplace Transform of Sine
\(\displaystyle \leadsto \ \ \) \(\displaystyle -\dfrac \d {\d s} \laptrans {\map {f'} t}\) \(=\) \(\displaystyle \dfrac 1 {s^2 + 1}\) Derivative of Laplace Transform
\(\displaystyle \leadsto \ \ \) \(\displaystyle \map {\dfrac \d {\d s} } {s \laptrans {\map f t} - \map f 0}\) \(=\) \(\displaystyle -\dfrac 1 {s^2 + 1}\) Laplace Transform of Derivative
\(\displaystyle \leadsto \ \ \) \(\displaystyle s \laptrans {\map f t}\) \(=\) \(\displaystyle -\int \dfrac 1 {s^2 + 1} \rd s\) $\map f 0 = 0$, and integrating both sides with respect to $s$
\(\displaystyle \leadsto \ \ \) \(\displaystyle s \laptrans {\map f t}\) \(=\) \(\displaystyle -\arctan s + C\) Primitive of $\dfrac 1 {x^2 + a^2}$


By the Initial Value Theorem of Laplace Transform:

$\displaystyle \lim_{s \mathop \to \infty} s \laptrans {\map f t} = \lim_{t \mathop \to 0} \map f t = \map f 0 = 0$

which leads to:

$c = \dfrac \pi 2$


Thus:

\(\displaystyle s \laptrans {\map f t}\) \(=\) \(\displaystyle \dfrac \pi 2 - \arctan s\)
\(\displaystyle \) \(=\) \(\displaystyle \arccot s\) Sum of Arctangent and Arccotangent
\(\displaystyle \) \(=\) \(\displaystyle \arctan \dfrac 1 s\) Arctangent of Reciprocal equals Arccotangent
\(\displaystyle \leadsto \ \ \) \(\displaystyle \laptrans {\map f t}\) \(=\) \(\displaystyle \dfrac 1 s \arctan \dfrac 1 s\)

$\blacksquare$


Sources