Arctangent Function in terms of Gaussian Hypergeometric Function

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Theorem

Let $\map F {a, b; c; x}$ denote the Gaussian hypergeometric function of $x$:

$\ds \sum_{k \mathop = 0}^\infty \dfrac {a^{\overline k} b^{\overline k} } {c^{\overline k} } \dfrac {x^k} {k!}$


Then:

$\arctan x = x \map F {\dfrac 1 2, 1; \dfrac 3 2; -x^2}$

where:

$x$ is a real number such that $\size x < 1$
$\arctan$ denotes the arctangent 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

Arctangent Function in terms of Gaussian Hypergeometric Function can also be seen presented in the form:

$\map F {\dfrac 1 2, 1; \dfrac 3 2; -x^2} = \dfrac {\arctan x} x$

or:

$\map F {\dfrac 1 2, 1; \dfrac 3 2; -x^2} = \dfrac {\inv \tan x} x$


Sources

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