Power Series Expansion for Sine Integral Function
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Theorem
- $\ds \map \Si x = \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n + 1} } {\paren {2 n + 1} \times \paren {2 n + 1}!}$
where:
- $\Si$ denotes the sine integral function
- $x$ is a real number.
Proof
\(\ds \map \Si x\) | \(=\) | \(\ds \int_0^x \frac {\sin u} u \rd u\) | Definition of Sine Integral Function | |||||||||||
\(\ds \) | \(=\) | \(\ds \int_0^x \frac {\paren {-1}^n} u \paren {\sum_{n \mathop = 0}^\infty \frac {u^{2 n + 1} } {\paren {2 n + 1}!} } \rd u\) | Power Series Expansion for Sine Function | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{n \mathop = 0}^\infty \paren {-1}^n \paren {\int_0^x \frac {u^{2 n} } {\paren {2 n + 1}!} \rd u}\) | Power Series is Termwise Integrable within Radius of Convergence | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n + 1} } {\paren {2 n + 1} \times \paren {2 n + 1}!}\) | Primitive of Power |
$\blacksquare$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 35$: Sine Integral $\ds \map \Si x = \int_0^x \frac {\sin u} u \rd u$: $35.11$