Laplace Transform of Sine Integral Function/Proof 3
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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 $\map f t := \map \Si t = \ds \int_0^t \dfrac {\sin u} u \rd u$.
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 |
$\blacksquare$
Sources
- 1965: Murray R. Spiegel: Theory and Problems of Laplace Transforms ... (previous) ... (next): Chapter $1$: The Laplace Transform: Solved Problems: The Sine, Cosine and Exponential Integrals: $36$: Method $3$