The parts of this chapter that deal with Legendre’s integrals
are based in part on
Abramowitz and Stegun (1964, Chapter 17) by L. M. Milne-Thomson. I am greatly
indebted to R. C. Winther for indispensable technical support and to
F. W. J. Olver for long-sustained encouragement of a new approach to
elliptic integrals. I thank
E. Neuman for improvements to §§19.9 and 19.24(i).
The main references used for writing this chapter are
Erdélyi et al. (1953b), Byerly (1888), Cazenave (1969),
and Byrd and Friedman (1971) for Legendre’s integrals, and
Carlson (1977b) for symmetric integrals. For additional bibliographic
reading see Armitage and Eberlein (2006), Cayley (1895), Greenhill (1892),
Legendre (1825–1832), Tricomi (1951), and
Whittaker and Watson (1927).