# §19.25 Relations to Other Functions

## §19.25(i) Legendre’s Integrals as Symmetric Integrals

Let and . Then

with Cauchy principal value

The transformations in §19.7(ii) result from the symmetry and homogeneity of functions on the right-hand sides of (19.25.5), (19.25.7), and (19.25.14). For example, if we write (19.25.5) as

with

19.25.18

then the five nontrivial permutations of that leave invariant change () into , , , , , and () into , , , , . Thus the five permutations induce five transformations of Legendre’s integrals (and also of the Jacobian elliptic functions).

The three changes of parameter of in §19.7(iii) are unified in (19.21.12) by way of (19.25.14).

## §19.25(iv) Theta Functions

For relations of symmetric integrals to theta functions, see §20.9(i).

## §19.25(v) Jacobian Elliptic Functions

For the notation see §§22.2, 22.15, and 22.16(i).

With and any permutation of the letters , define

which implies

If , then

compare (19.25.35) and (20.9.3).

where we assume if , , or ; if , , or ; real if or ; if ; if ; if ; if .

For the use of -functions with in unifying other properties of Jacobian elliptic functions, see Carlson (2004, 2006a, 2006b, 2008).

Inversions of 12 elliptic integrals of the first kind, producing the 12 Jacobian elliptic functions, are combined and simplified by using the properties of . See (19.29.19), Carlson (2005), and (22.15.11), and compare with Abramowitz and Stegun (1964, Eqs. (17.4.41)–(17.4.52)). For analogous integrals of the second kind, which are not invertible in terms of single-valued functions, see (19.29.20) and (19.29.21) and compare with Gradshteyn and Ryzhik (2000, §3.153,1–10 and §3.156,1–9).

## §19.25(vii) Hypergeometric Function

For these results and extensions to the Appell function 16.13) and Lauricella’s function see Carlson (1963). ( and are equivalent to the -function of 3 and variables, respectively, but lack full symmetry.)