Continued Fraction for Real Arcsine Function

Theorem

Let $-1 \le x \le 1$

Then:

$\arcsin x = \cfrac x {1 - \cfrac {x^2} {2 \times 3 + x^2 - \cfrac {2 \times 3 \times \paren {3 x}^2} {4 \times 5 + \paren {3 x}^2 - \cfrac {4 \times 5 \times \paren {5 x}^2} {6 \times 7 + \paren {5 x}^2 - \cfrac {6 \times 7 \times \paren {7 x}^2} {\ddots } } } } }$

Proof

$\ds \arcsin x$

$=$

$\ds \sum_{n \mathop = 0}^\infty \frac {\paren {2 n}!} {2^{2 n} \paren {n!}^2} \frac {x^{2 n + 1} } {2 n + 1}$

Power Series Expansion for Real Arcsine Function for $-1 \le x \le 1$

$\ds $

$=$

$\ds x + \frac {x^3} {2 \times 3} + \frac {\paren {1 \times 3} x^5} {2 \times 4 \times 5} + \frac {\paren {1 \times 3 \times 5} x^7} {2 \times 4 \times 6 \times 7} + \cdots$

$\ds $

$=$

$\ds x + x \paren {\dfrac {x^2} {2 \times 3} } + x \paren {\dfrac {x^2} {2 \times 3} } \paren {\dfrac {\paren {3 x}^2} {4 \times 5} } + x \paren {\dfrac {x^2} {2 \times 3} } \paren {\dfrac {\paren {3 x}^2} {4 \times 5} } \paren {\dfrac {\paren {5 x}^2} {6 \times 7} } + \cdots$

$\ds $

$=$

$\ds a_0 + a_0 a_1 + a_0 a_1 a_2 + a_0 a_1 a_2 a_3 + \cdots$

From Euler's Continued Fraction Formula, we have:

$a_0 + a_0 a_1 + a_0 a_1 a_2 + a_0 a_1 a_2 a_3 + \cdots = \cfrac {a_0} {1 - \cfrac {a_1} {1 + a_1 - \cfrac {a_2} {1 + a_2 - \cfrac {a_3} {1 + a_3 - \cfrac {\ddots} {\ddots } } } } }$

Therefore:

$\ds x + x \paren {\dfrac {x^2} {2 \times 3} } + x \paren {\dfrac {x^2} {2 \times 3} } \paren {\dfrac {\paren {3 x}^2} {4 \times 5} } + x \paren {\dfrac {x^2} {2 \times 3} } \paren {\dfrac {\paren {3 x}^2} {4 \times 5} } \paren {\dfrac {\paren {5 x}^2} {6 \times 7} } + \cdots$

$=$

$\ds \cfrac x {1 - \cfrac {\paren {\dfrac {x^2} {2 \times 3} } } {1 + \paren {\dfrac {x^2} {2 \times 3} } - \cfrac {\paren {\dfrac {\paren {3 x}^2} {4 \times 5} } } {1 + \paren {\dfrac {\paren {3 x}^2} {4 \times 5} } - \cfrac {\paren {\dfrac {\paren {5 x}^2} {6 \times 7} } } {1 + \paren {\dfrac {\paren {5 x}^2} {6 \times 7} } - \cfrac {\ddots} {\ddots } } } } }$

$\ds $

$=$

$\ds \cfrac x {1 - \cfrac {\paren {\dfrac {x^2} {2 \times 3} } \times 2 \times 3 } {2 \times 3 \times \paren {1 + \paren {\dfrac {x^2} {2 \times 3} } } - \cfrac {\paren {\dfrac {\paren {3 x}^2} {4 \times 5} } \times 2 \times 3 \times 4 \times 5 } {4 \times 5 \times \paren {1 + \paren {\dfrac {\paren {3 x}^2} {4 \times 5} } } - \cfrac {\paren {\dfrac {\paren {5 x}^2} {6 \times 7} } \times 4 \times 5 \times 6 \times 7 } {6 \times 7 \times \paren {1 + \paren {\dfrac {\paren {5 x}^2} {6 \times 7} } } - \cfrac {\ddots} {\ddots } } } } }$

multiplying by $1$

$\ds $

$=$

$\ds \cfrac x {1 - \cfrac {x^2} {2 \times 3 + x^2 - \cfrac {2 \times 3 \times \paren {3 x}^2} {4 \times 5 + \paren {3 x}^2 - \cfrac {4 \times 5 \times \paren {5 x}^2} {6 \times 7 + \paren {5 x}^2 - \cfrac {6 \times 7 \times \paren {7 x}^2} {\ddots } } } } }$

$\blacksquare$

Also see

Sources

1750: Leonhard Paul Euler: De fractionibus continuis Observationes: Pages $\text {32 - 81}$