Let and . Then
with Cauchy principal value
If , then the Cauchy principal value is
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).
Let . Then
Assume , , and . Let
with . Then
For relations of symmetric integrals to theta functions, see §20.9(i).
With and any permutation of the letters , define
If , then
where we assume if , , or ; if , , or ; real if or ; if ; if ; if ; if .
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).
For the notation see §23.2.
and the left-hand side does not vanish for more than one value of . Also,