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1: 19.1 Special Notation
(For other notation see Notation for the Special Functions.) … The first set of main functions treated in this chapter are Legendre’s complete integrals … The second set of main functions treated in this chapter is … R - a ( b 1 , b 2 , , b n ; z 1 , z 2 , , z n ) is a multivariate hypergeometric function that includes all the functions in (19.1.3). A third set of functions, introduced by Bulirsch (1965b, a, 1969a), is …
2: 19.2 Definitions
Assume 1 - sin 2 ϕ ( - , 0 ] and 1 - k 2 sin 2 ϕ ( - , 0 ] , except that one of them may be 0, and 1 - α 2 sin 2 ϕ { 0 } . …
§19.2(iv) A Related Function: R C ( x , y )
Formulas involving Π ( ϕ , α 2 , k ) that are customarily different for circular cases, ordinary hyperbolic cases, and (hyperbolic) Cauchy principal values, are united in a single formula by using R C ( x , y ) . … When x and y are positive, R C ( x , y ) is an inverse circular function if x < y and an inverse hyperbolic function (or logarithm) if x > y : …For the special cases of R C ( x , x ) and R C ( 0 , y ) see (19.6.15). …
3: 19.12 Asymptotic Approximations
With ψ ( x ) denoting the digamma function5.2(i)) in this subsection, the asymptotic behavior of K ( k ) and E ( k ) near the singularity at k = 1 is given by the following convergent series: …where … For the asymptotic behavior of F ( ϕ , k ) and E ( ϕ , k ) as ϕ 1 2 π - and k 1 - see Kaplan (1948, §2), Van de Vel (1969), and Karp and Sitnik (2007). As k 2 1 - They are useful primarily when ( 1 - k ) / ( 1 - sin ϕ ) is either small or large compared with 1. …
4: 19.16 Definitions