Euler's Continued Fraction Formula/Examples

Examples of Euler's Continued Fraction Formula

Example: $\map {\arcsin} x$

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


Example: $\map {\arctan} x$

$\arctan x = \cfrac x {1 + \cfrac {x^2} {3 - x^2 + \cfrac {\paren {3 x}^2} {5 - 3 x^2 + \cfrac {\paren {5 x}^2} {7 - 5 x^2 + \cfrac {\paren {7 x}^2} {\ddots } } } } }$


Example: $e^x$

$e^x = \cfrac 1 {1 - \cfrac x {1 + x - \cfrac x {2 + x - \cfrac {2 x } {3 + x - \cfrac {3 x} {4 + x - \cfrac {\ddots} {\ddots } } } } } }$


Example: $\map {\log} {1 + x}$

$\map \ln {1 + x} = \cfrac x {1 + \cfrac x {2 - x + \cfrac {2^2 x} {3 - 2 x + \cfrac {3^2 x } {4 - 3 x + \cfrac {\ddots} {\ddots } } } } }$
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