Laplace Transform of Sine Integral Function

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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 1

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

Then:

$\map f 0 = 0$

and:

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


By the Initial Value Theorem of Laplace Transform:

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

which leads to:

$c = \dfrac \pi 2$


Thus:

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

$\blacksquare$


Proof 2

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

$\blacksquare$


Proof 3

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

Then:

$\map f 0 = 0$

and:

\(\ds \map \Si t\) \(=\) \(\ds \int_0^t \dfrac {\sin u} u \rd u\) Definition of Sine Integral Function
\(\ds \) \(=\) \(\ds \int_0^t \dfrac 1 u \paren {u - \dfrac {u^3} {3!} + \dfrac {u^5} {5!} - \dfrac {u^7} {7!} + \dotsb} \rd u\) Definition of Real Sine Function
\(\ds \) \(=\) \(\ds t - \dfrac {t^3} {3 \times 3!} + \dfrac {t^5} {5 \times 5!} - \dfrac {t^7} {7 \times 7!} + \dotsb\) Primitive of Power
\(\ds \leadsto \ \ \) \(\ds \laptrans {\map \Si t}\) \(=\) \(\ds \laptrans {t - \dfrac {t^3} {3 \times 3!} + \dfrac {t^5} {5 \times 5!} - \dfrac {t^7} {7 \times 7!} + \dotsb}\)
\(\ds \) \(=\) \(\ds \dfrac 1 {s^2} - \dfrac 1 {3 \times 3!} \dfrac {3!} {s^4} + \dfrac 1 {5 \times 5!} \dfrac {5!} {s^6} - \dfrac 1 {7 \times 7!} \dfrac {7!} {s^8} + \dotsb\) Laplace Transform of Positive Integer Power
\(\ds \) \(=\) \(\ds \dfrac 1 {s^2} - \dfrac 1 {3 s^4} + \dfrac 1 {5 s^6} - \dfrac 1 {7 s^8} + \dotsb\) simplifying
\(\ds \) \(=\) \(\ds \dfrac 1 s \paren {\dfrac {\paren {1 / s} } 1 - \dfrac {\paren {1 / s}^3} 3 + \dfrac {\paren {1 / s}^5} 5 - \dfrac {\paren {1 / s}^7} 7 + \dotsb}\) rearranging
\(\ds \) \(=\) \(\ds \dfrac 1 s \arctan \dfrac 1 s\) Power Series Expansion for Real Arctangent Function

$\blacksquare$


Proof 4

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

Then:

$\map f 0 = 0$

and:

\(\ds \map \Si t\) \(=\) \(\ds \int_0^t \dfrac {\sin u} u \rd u\) Definition of Sine Integral Function
\(\ds \) \(=\) \(\ds \int_0^1 \dfrac {\sin t v} v \rd v\) Integration by Substitution $u = t v$
\(\ds \leadsto \ \ \) \(\ds \laptrans {\map \Si t}\) \(=\) \(\ds \laptrans {\int_0^1 \dfrac {\sin t v} v \rd v}\)
\(\ds \) \(=\) \(\ds \int_0^\infty e^{-s t} \paren {\int_0^1 \dfrac {\sin t v} v \rd v} \rd t\) Definition of Laplace Transform
\(\ds \) \(=\) \(\ds \int_0^1 \dfrac 1 v \paren {\int_0^\infty e^{-s t} \sin t v \rd t} \rd v\) exchanging order of integration
\(\ds \) \(=\) \(\ds \int_0^1 \dfrac {\laptrans {\sin t v} } v \rd v\) Definition of Laplace Transform
\(\ds \) \(=\) \(\ds \int_0^1 \dfrac {\d v} {s^2 + v^2}\) Laplace Transform of Sine
\(\ds \) \(=\) \(\ds \intlimits {\dfrac 1 s \arctan \dfrac v s} 0 1\) Primitive of $\dfrac 1 {x^2 + a^2}$
\(\ds \) \(=\) \(\ds \dfrac 1 s \arctan \dfrac 1 s\)

$\blacksquare$


Sources