# §19.21 Connection Formulas

## §19.21(i) Complete Integrals

The complete cases of and have connection formulas resulting from those for the Gauss hypergeometric function (Erdélyi et al. (1953a, §2.9)). Upper signs apply if , and lower signs if :

Let , , and be positive and distinct, and permute and to ensure that does not lie between and . The complete case of can be expressed in terms of and :

If and , then as (19.21.6) reduces to Legendre’s relation (19.21.1).

## §19.21(ii) Incomplete Integrals

is symmetric only in and , but either (nonzero) or (nonzero) can be moved to the third position by using

or the corresponding equation with and interchanged.

Because is completely symmetric, can be permuted on the right-hand side of (19.21.10) so that if the variables are real, thereby avoiding cancellations when is calculated from and (see §19.36(i)).

where both summations extend over the three cyclic permutations of .

Connection formulas for are given in Carlson (1977b, pp. 99, 101, and 123–124).

## §19.21(iii) Change of Parameter of

Let be real and nonnegative, with at most one of them 0. Change-of-parameter relations can be used to shift the parameter of from either circular region to the other, or from either hyperbolic region to the other (§19.20(iii)). The latter case allows evaluation of Cauchy principal values (see (19.20.14)).

where

19.21.13

and may be permuted. Also,

19.21.14

For each value of , permutation of produces three values of , one of which lies in the same region as and two lie in the other region of the same type. In (19.21.12), if is the largest (smallest) of , and , then and lie in the same region if it is circular (hyperbolic); otherwise and lie in different regions, both circular or both hyperbolic. If , then and ; hence