Throughout this subsection , except in (19.9.4).
for . The lower bound in (19.9.4) is sharper than when .
Other inequalities are:
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 :
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).
Throughout this subsection we assume that , , and .
Sharper inequalities for are:
Inequalities for both and involving inverse circular or inverse hyperbolic functions are given in Carlson (1961b, §4). For example,