19.20 Special Cases19.22 Quadratic Transformations

§19.21 Connection Formulas

Contents

§19.21(i) Complete Integrals

The complete cases of \mathop{R_{F}\/}\nolimits and \mathop{R_{G}\/}\nolimits have connection formulas resulting from those for the Gauss hypergeometric function (Erdélyi et al. (1953a, §2.9)). Upper signs apply if 0<\mathop{\mathrm{ph}\/}\nolimits z<\pi, and lower signs if -\pi<\mathop{\mathrm{ph}\/}\nolimits z<0:

19.21.4 \mathop{R_{F}\/}\nolimits\!\left(0,z-1,z\right)=\mathop{R_{F}\/}\nolimits\!\left(0,1-z,1\right)\mp i\!\mathop{R_{F}\/}\nolimits\!\left(0,z,1\right),
19.21.5 2\!\mathop{R_{G}\/}\nolimits\!\left(0,z-1,z\right)=2\!\mathop{R_{G}\/}\nolimits\!\left(0,1-z,1\right)\pm i2\!\mathop{R_{G}\/}\nolimits\!\left(0,z,1\right)+(z-1)\mathop{R_{F}\/}\nolimits\!\left(0,1-z,1\right)\mp iz\!\mathop{R_{F}\/}\nolimits\!\left(0,z,1\right).

Let y, z, and p be positive and distinct, and permute y and z to ensure that y does not lie between z and p. The complete case of \mathop{R_{J}\/}\nolimits can be expressed in terms of \mathop{R_{F}\/}\nolimits and \mathop{R_{D}\/}\nolimits:

19.21.6 (\sqrt{rp}/z)\mathop{R_{J}\/}\nolimits\!\left(0,y,z,p\right)={(r-1)}\mathop{R_{F}\/}\nolimits\!\left(0,y,z\right)\mathop{R_{D}\/}\nolimits\!\left(p,rz,z\right)+\mathop{R_{D}\/}\nolimits\!\left(0,y,z\right)\mathop{R_{F}\/}\nolimits\!\left(p,rz,z\right), r=(y-p)/(y-z)>0.

If 0<p<z and y=z+1, then as p\to 0 (19.21.6) reduces to Legendre’s relation (19.21.1).

§19.21(ii) Incomplete Integrals

\mathop{R_{D}\/}\nolimits\!\left(x,y,z\right) is symmetric only in x and y, but either (nonzero) x or (nonzero) y can be moved to the third position by using

19.21.7 (x-y)\mathop{R_{D}\/}\nolimits\!\left(y,z,x\right)+(z-y)\mathop{R_{D}\/}\nolimits\!\left(x,y,z\right)=3\!\mathop{R_{F}\/}\nolimits\!\left(x,y,z\right)-3\sqrt{y/(xz)},

or the corresponding equation with x and y interchanged.

19.21.10 2\!\mathop{R_{G}\/}\nolimits\!\left(x,y,z\right)=z\mathop{R_{F}\/}\nolimits\!\left(x,y,z\right)-\tfrac{1}{3}(x-z)(y-z)\mathop{R_{D}\/}\nolimits\!\left(x,y,z\right)+\sqrt{xy/z}, z\neq 0.

Because \mathop{R_{G}\/}\nolimits is completely symmetric, x,y,z can be permuted on the right-hand side of (19.21.10) so that (x-z)(y-z)\leq 0 if the variables are real, thereby avoiding cancellations when \mathop{R_{G}\/}\nolimits is calculated from \mathop{R_{F}\/}\nolimits and \mathop{R_{D}\/}\nolimits (see §19.36(i)).

19.21.11 6\!\mathop{R_{G}\/}\nolimits\!\left(x,y,z\right)=3(x+y+z)\mathop{R_{F}\/}\nolimits\!\left(x,y,z\right)-\sum x^{2}\mathop{R_{D}\/}\nolimits\!\left(y,z,x\right)=\sum x(y+z)\mathop{R_{D}\/}\nolimits\!\left(y,z,x\right),

where both summations extend over the three cyclic permutations of x,y,z.

Connection formulas for \mathop{R_{{-a}}\/}\nolimits\!\left(\mathbf{b};\mathbf{z}\right) are given in Carlson (1977b, pp. 99, 101, and 123–124).

§19.21(iii) Change of Parameter of \mathop{R_{J}\/}\nolimits

Let x,y,z be real and nonnegative, with at most one of them 0. Change-of-parameter relations can be used to shift the parameter p of \mathop{R_{J}\/}\nolimits 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)).

19.21.12 (p-x)\mathop{R_{J}\/}\nolimits\!\left(x,y,z,p\right)+(q-x)\mathop{R_{J}\/}\nolimits\!\left(x,y,z,q\right)=3\!\mathop{R_{F}\/}\nolimits\!\left(x,y,z\right)-3\!\mathop{R_{C}\/}\nolimits\!\left(\xi,\eta\right),

where

19.21.13
(p-x)(q-x)=(y-x)(z-x),
\xi=yz/x,
\eta=pq/x,

and x,y,z may be permuted. Also,

19.21.14 \eta-\xi=p+q-y-z=\frac{(p-y)(p-z)}{p-x}=\frac{(q-y)(q-z)}{q-x}=\frac{(p-y)(q-y)}{x-y}=\frac{(p-z)(q-z)}{x-z}.

For each value of p, permutation of x,y,z produces three values of q, one of which lies in the same region as p and two lie in the other region of the same type. In (19.21.12), if x is the largest (smallest) of x,y, and z, then p and q lie in the same region if it is circular (hyperbolic); otherwise p and q lie in different regions, both circular or both hyperbolic. If x=0, then \xi=\eta=\infty and \mathop{R_{C}\/}\nolimits\!\left(\xi,\eta\right)=0; hence

19.21.15 p\mathop{R_{J}\/}\nolimits\!\left(0,y,z,p\right)+q\mathop{R_{J}\/}\nolimits\!\left(0,y,z,q\right)=3\!\mathop{R_{F}\/}\nolimits\!\left(0,y,z\right), pq=yz.