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
- 1965: Murray R. Spiegel: Theory and Problems of Laplace Transforms ... (previous) ... (next): Chapter $1$: The Laplace Transform: Laplace Transforms of Special Functions: $8$