# §19.9 Inequalities

## §19.9(i) Complete Integrals

The perimeter of an ellipse with semiaxes is given by

Almkvist and Berndt (1988) list thirteen approximations to that have been proposed by various authors. The earliest is due to Kepler and the most accurate to Ramanujan. Ramanujan’s approximation and its leading error term yield the following approximation to :

19.9.10.

Even for the extremely eccentric ellipse with and , this is correct within 0.023%. Barnard et al. (2000) shows that nine of the thirteen approximations, including Ramanujan’s, are from below and four are from above. See also Barnard et al. (2001).

## §19.9(ii) Incomplete Integrals

Throughout this subsection we assume that , , and .

Inequalities for both and involving inverse circular or inverse hyperbolic functions are given in Carlson (1961b, §4). For example,

where

Other inequalities for can be obtained from inequalities for given in Carlson (1966, (2.15)) and Carlson (1970) via (19.25.5).