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1: 5.18 q -Gamma and q -Beta Functions
For q -integrals see §17.2(v).
2: 19.33 Triaxial Ellipsoids
19.33.5 V ( λ ) = Q R F ( a 2 + λ , b 2 + λ , c 2 + λ ) ,
3: 20.9 Relations to Other Functions
§20.9(i) Elliptic Integrals
and the notation of §19.2(ii), the complete Legendre integrals of the first kind may be expressed as theta functions: … In the case of the symmetric integrals, with the notation of §19.16(i) we have
20.9.3 R F ( θ 2 2 ( z , q ) θ 2 2 ( 0 , q ) , θ 3 2 ( z , q ) θ 3 2 ( 0 , q ) , θ 4 2 ( z , q ) θ 4 2 ( 0 , q ) ) = θ 1 ( 0 , q ) θ 1 ( z , q ) z ,
The relations (20.9.1) and (20.9.2) between k and τ (or q ) are solutions of Jacobi’s inversion problem; see Baker (1995) and Whittaker and Watson (1927, pp. 480–485). …
4: 17.2 Calculus
§17.2(v) Integrals
If f ( x ) is continuous at x = 0 , then …
5: Bibliography H
  • L. Habsieger (1988) Une q -intégrale de Selberg et Askey. SIAM J. Math. Anal. 19 (6), pp. 1475–1489.
  • 6: 22.20 Methods of Computation
    If either τ or q = e i π τ is given, then we use k = θ 2 2 ( 0 , q ) / θ 3 2 ( 0 , q ) , k = θ 4 2 ( 0 , q ) / θ 3 2 ( 0 , q ) , K = 1 2 π θ 3 2 ( 0 , q ) , and K = - i τ K , obtaining the values of the theta functions as in §20.14. …
    7: 19.21 Connection Formulas
    19.21.15 p R J ( 0 , y , z , p ) + q R J ( 0 , y , z , q ) = 3 R F ( 0 , y , z ) , p q = y z .