Arcsine Function in terms of Gaussian Hypergeometric Function
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Theorem
- $\arcsin x = x \map F {\dfrac 1 2, \dfrac 1 2; \dfrac 3 2; x^2}$
where:
- $x$ is a real number with $\size x \le 1$
- $\arcsin$ denotes the arcsine function
- $F$ denotes the Gaussian hypergeometric function.
Proof
\(\ds x \map F {\frac 1 2, \frac 1 2; \frac 3 2; x^2}\) | \(=\) | \(\ds x \sum_{n \mathop = 0}^\infty \frac {\paren {\paren {\frac 1 2}^{\bar n} }^2} {\paren {\frac 3 2}^{\bar n} } \frac {x^{2 n} } {n!}\) | Definition of Gaussian Hypergeometric Function | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{n \mathop = 0}^\infty \frac {\paren {\map \Gamma {\frac 1 2 + n} }^2 \map \Gamma {\frac 3 2} } {\paren {\map \Gamma {\frac 1 2} }^2 \map \Gamma {\frac 3 2 + n} } \frac {x^{2 n + 1} } {n!}\) | Rising Factorial as Quotient of Factorials | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{n \mathop = 0}^\infty \frac {\paren {\map \Gamma {\frac 1 2 + n} }^2 \map \Gamma {\frac 1 2} } {2 \paren {\map \Gamma {\frac 1 2} }^2 \paren {\frac 1 2 + n} \map \Gamma {\frac 1 2 + n} } \frac {x^{2 n + 1} } {n!}\) | Gamma Difference Equation | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{n \mathop = 0}^\infty \frac {\map \Gamma {\frac 1 2 + n} } {\paren {2 n + 1} \sqrt \pi} \frac {x^{2 n + 1} } {n!}\) | Gamma Function of One Half | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{n \mathop = 0}^\infty \paren {\paren {\frac {\paren {2 n}!} {2^{2 n} n!} \sqrt \pi} \times \frac 1 {\paren {2 n + 1} \sqrt \pi} \times \frac 1 {n!} x^{2 n + 1} }\) | Gamma Function of Positive Half-Integer | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{n \mathop = 0}^\infty \frac {\paren {2 n}!} {2^{2 n} \paren {n!}^2} \frac {x^{2 n + 1} } {2 n + 1}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \arcsin x\) | Power Series Expansion for Real Arcsine Function |
$\blacksquare$
Also presented as
This result can also be seen presented in the form:
- $\map F {\dfrac 1 2, \dfrac 1 2; \dfrac 3 2; x^2} = \dfrac {\arcsin x} x$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 31$: Hypergeometric Functions: Special Cases: $31.8$