Power Series Expansion for Fresnel Sine Integral Function

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Theorem

$\ds \map {\operatorname S} x = \sqrt {\frac 2 \pi} \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{4 n + 3} } {\paren {4 n + 3} \paren {2 n + 1}!}$

where $\operatorname S$ denotes the Fresnel sine integral function.


Proof

\(\ds \map {\operatorname S} x\) \(=\) \(\ds \sqrt {\frac 2 \pi} \int_0^x \sin u^2 \rd u\) Definition of Fresnel Sine Integral Function
\(\ds \) \(=\) \(\ds \sqrt {\frac 2 \pi} \int_0^x \paren {\sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {\paren {u^2}^{2 n + 1} } {\paren {2 n + 1}!} } \rd u\) Power Series Expansion for Sine Function
\(\ds \) \(=\) \(\ds \sqrt {\frac 2 \pi} \sum_{n \mathop = 0}^\infty \frac {\paren {-1}^n} {\paren {2 n + 1}!} \int_0^x u^{4 n + 2} \rd u\) Power Series is Termwise Integrable within Radius of Convergence
\(\ds \) \(=\) \(\ds \sqrt {\frac 2 \pi} \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{4 n + 3} } {\paren {4 n + 3} \paren {2 n + 1}!}\) Primitive of Power

$\blacksquare$


Sources

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