Following Ramanujan (1913-14), write

(1) |

(2) |

(3) | |||

(4) | |||

(5) | |||

(6) |

and can be derived using the theory of Modular Functions and can always be expressed as roots of algebraic equations when is Rational. For simplicity, Ramanujan tabulated for Even and for Odd. However, (6) allows and to be solved for in terms of and , giving

(7) | |||

(8) |

Using (3) and the above two equations allows to be computed in terms of or

(9) |

In terms of the Parameter and complementary Parameter ,

(10) | |||

(11) |

Here,

(12) |

(13) |

(14) | |||

(15) |

Analytic values for small values of can be found in Ramanujan (1913-1914) and Borwein and Borwein (1987), and have been compiled in Weisstein (1996). Ramanujan (1913-1914) contains a typographical error labeling as .

**References**

Borwein, J. M. and Borwein, P. B. *Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity.*
New York: Wiley, pp. 139 and 298, 1987.

Ramanujan, S. ``Modular Equations and Approximations to .'' *Quart. J. Pure. Appl. Math.* **45**, 350-372, 1913-1914.

Weisstein, E. W. ``Elliptic Singular Values.'' Mathematica notebook EllipticSingular.m.

© 1996-9

1999-05-25