# RC-function

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##### 1: 19.1 Special Notation
(For other notation see Notation for the Special Functions.) … The first set of main functions treated in this chapter are Legendre’s complete integrals … The second set of main functions treated in this chapter is … $R_{-a}\left(b_{1},b_{2},\dots,b_{n};z_{1},z_{2},\dots,z_{n}\right)$ is a multivariate hypergeometric function that includes all the functions in (19.1.3). A third set of functions, introduced by Bulirsch (1965b, a, 1969a), is …
##### 2: 19.2 Definitions
Assume $1-{\sin}^{2}\phi\in\mathbb{C}\setminus(-\infty,0]$ and $1-k^{2}{\sin}^{2}\phi\in\mathbb{C}\setminus(-\infty,0]$, except that one of them may be 0, and $1-\alpha^{2}{\sin}^{2}\phi\in\mathbb{C}\setminus\{0\}$. …
###### §19.2(iv) A Related Function: $R_{C}\left(x,y\right)$
Formulas involving $\Pi\left(\phi,\alpha^{2},k\right)$ that are customarily different for circular cases, ordinary hyperbolic cases, and (hyperbolic) Cauchy principal values, are united in a single formula by using $R_{C}\left(x,y\right)$. … When $x$ and $y$ are positive, $R_{C}\left(x,y\right)$ is an inverse circular function if $x and an inverse hyperbolic function (or logarithm) if $x>y$: …For the special cases of $R_{C}\left(x,x\right)$ and $R_{C}\left(0,y\right)$ see (19.6.15). …
##### 3: 19.12 Asymptotic Approximations
With $\psi\left(x\right)$ denoting the digamma function5.2(i)) in this subsection, the asymptotic behavior of $K\left(k\right)$ and $E\left(k\right)$ near the singularity at $k=1$ is given by the following convergent series: …where … For the asymptotic behavior of $F\left(\phi,k\right)$ and $E\left(\phi,k\right)$ as $\phi\to\tfrac{1}{2}\pi-$ and $k\to 1-$ see Kaplan (1948, §2), Van de Vel (1969), and Karp and Sitnik (2007). As $k^{2}\to 1-$They are useful primarily when $\ifrac{(1-k)}{(1-\sin\phi)}$ is either small or large compared with 1. …
##### 4: 19.16 Definitions
19.16.18 $R_{C}\left(x,y\right)=R_{-\frac{1}{2}}\left(\tfrac{1}{2},1;x,y\right).$