Arctangent Function in terms of Gaussian Hypergeometric Function
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Theorem
- $\arctan x = x \map F {\dfrac 1 2, 1; \dfrac 3 2; -x^2}$
where:
- $x$ is a real number with $\size x < 1$
- $\arctan$ denotes the arctangent function
- $F$ denotes the Gaussian hypergeometric function.
Proof
\(\ds x \map F {\frac 1 2, 1; \frac 3 2; -x^2}\) | \(=\) | \(\ds x \sum_{n \mathop = 0}^\infty \frac {\paren {\frac 1 2}^{\bar n} 1^{\bar n} } {\paren {\frac 3 2}^{\bar n} } \frac {\paren {-x^2}^n} {n!}\) | Definition of Gaussian Hypergeometric Function | |||||||||||
\(\ds \) | \(=\) | \(\ds x \sum_{n \mathop = 0}^\infty \frac {\map \Gamma {\frac 1 2 + n} } {\map \Gamma {\frac 1 2} } \times \frac {\map \Gamma {\frac 3 2} } {\map \Gamma {\frac 3 2 + n} } n! \frac {\paren {-x^2}^n} {n!}\) | Rising Factorial as Quotient of Factorials, One to Integer Rising is Integer Factorial | |||||||||||
\(\ds \) | \(=\) | \(\ds x \sum_{n \mathop = 0}^\infty \frac {\map \Gamma {\frac 1 2 + n} } {\map \Gamma {\frac 1 2 + n} } \times \frac {\map \Gamma {\frac 1 2} } {\map \Gamma {\frac 1 2} } \times \frac 1 {2 \paren {n + \frac 1 2} } \paren {-1}^n x^{2 n}\) | Gamma Difference Equation | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n + 1} } {2 n + 1}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \arctan x\) | Power Series Expansion for Real Arctangent Function |
$\blacksquare$
Also presented as
This result can also be seen presented in the form:
- $\map F {\dfrac 1 2, 1; \dfrac 3 2; -x^2} = \dfrac {\arctan x} x$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 31$: Hypergeometric Functions: Special Cases: $31.9$