relation to RC

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1: 19.10 Relations to Other Functions

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§19.10(ii) Elementary Functions

… ►In each case when $y=1$, the quantity multiplying $R_C$ supplies the asymptotic behavior of the left-hand side as the left-hand side tends to 0. …

2: 19.6 Special Cases

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19.6.8 $$F(\varphi ,1)=(\sin \varphi )R_C(1,{\cos }^2\varphi )={\mathrm{gd}}^{-1}\left(\varphi \right).$$

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Symbols:

$R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $\cos z$: cosine function, $F(\varphi ,k)$: Legendre’s incomplete elliptic integral of the first kind, ${\mathrm{gd}}^{-1}\left(x\right)$: inverse Gudermannian function, $\sin z$: sine function and $\varphi$: real or complex argument

Referenced by:

§19.10(ii), §19.10(ii)

Permalink:

http://dlmf.nist.gov/19.6.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §19.6(ii), §19.6 and Ch.19

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3: 19.2 Definitions

… ►Lastly, corresponding to Legendre’s incomplete integral of the third kind we have … ►

§19.2(iv) A Related Function: $R_C(x,y)$

… ►Formulas involving $\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$ that are customarily different for circular cases, ordinary hyperbolic cases, and (hyperbolic) Cauchy principal values, are united in a single formula by using $R_C(x,y)$. … ►When $x$ and $y$ are positive, $R_C(x,y)$ is an inverse circular function if and an inverse hyperbolic function (or logarithm) if $x>y$: …For the special cases of $R_C(x,x)$ and $R_C(0,y)$ see (19.6.15). …

4: 19.36 Methods of Computation

… ►In the Appendix of the last reference it is shown how to compute $R_J$ without computing $R_C$ more than once. … ►Legendre’s integrals can be computed from symmetric integrals by using the relations in §19.25(i). … ►The incomplete integrals $R_F(x,y,z)$ and $R_G(x,y,z)$ can be computed by successive transformations in which two of the three variables converge quadratically to a common value and the integrals reduce to $R_C$, accompanied by two quadratically convergent series in the case of $R_G$; compare Carlson (1965, §§5,6). … ►The step from $n$ to $n+1$ is an ascending Landen transformation if $\theta =1$ (leading ultimately to a hyperbolic case of $R_C$) or a descending Gauss transformation if $\theta =-1$ (leading to a circular case of $R_C$). … ►Here $R_C$ is computed either by the duplication algorithm in Carlson (1995) or via (19.2.19). …

5: 19.21 Connection Formulas

… ►Legendre’s relation (19.7.1) can be written … ►Let $y$, $z$, and $p$ be positive and distinct, and permute $y$ and $z$ to ensure that $y$ does not lie between $z$ and $p$. …If and $y=z+1$, then as $p\to 0$ (19.21.6) reduces to Legendre’s relation (19.21.1). … ►Change-of-parameter relations can be used to shift the parameter $p$ of $R_J$ from either circular region to the other, or from either hyperbolic region to the other (§19.20(iii)). … ►If $x=0$, then $\xi =\eta =\mathrm{\infty }$ and $R_C(\xi ,\eta )=0$; hence …

6: 19.1 Special Notation

… ►In Abramowitz and Stegun (1964, Chapter 17) the functions (19.1.1) and (19.1.2) are denoted, in order, by $K(\alpha )$, $E(\alpha )$, $\mathrm{\Pi }(n\backslash \alpha )$, $F(\varphi \backslash \alpha )$, $E(\varphi \backslash \alpha )$, and $\mathrm{\Pi }(n;\varphi \backslash \alpha )$, where $\alpha =\arcsin k$ and $n$ is the ${\alpha }^2$ (not related to $k$) in (19.1.1) and (19.1.2). … ►

$R_C(x,y)$ ,

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7: Errata

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Section: 15.9(v) Complete Elliptic Integrals. Equations: (11.11.9_5), (11.11.13_5), Intermediate equality in (15.4.27) which relates to $F(a,a;a+1;\frac{1}{2})$, (15.4.34), (19.5.4_1), (19.5.4_2) and (19.5.4_3).

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19.20.11 $$R_J(0,y,z,p)=\frac{3}{2p\sqrt{z}}\ln \left(\frac{16z}{y}\right)-\frac{3}{p}{R}_C(z,p)+O\left(y\ln y\right),$$

as $y\to 0+$, $p$ ($\ne 0$) real, we have added the constant term $\frac{-3}{p}{R}_C(z,p)$ and the order term $O\left(y\ln y\right)$, and hence $\sim$ was replaced by $=$.

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The Kronecker delta symbols have been moved furthest to the right, as is common convention for orthogonality relations.

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A number of additions and changes have been made to the metadata to reflect new and changed references as well as to how some equations have been derived.

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The definition of $R_C(x,y)$ was revised in Notations.

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8: 19.25 Relations to Other Functions

§19.25 Relations to Other Functions

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§19.25(ii) Bulirsch’s Integrals as Symmetric Integrals

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§19.25(vii) Hypergeometric Function

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9: 19.22 Quadratic Transformations

… ►As $n\to \mathrm{\infty }$, $p_n$ and ${\epsilon }_n$ converge quadratically to $M(a_0,g_0)$ and 0, respectively, and $Q_n$ converges to 0 faster than quadratically. … ►Ascent and descent correspond respectively to increase and decrease of $k$ in Legendre’s notation. … ►The transformations inverse to the ones just described are the descending Landen transformations and the ascending Gauss transformations. The equations inverse to (19.22.5) and (19.22.16) are given by …These relations need to be used with caution because $y$ is negative when . …

10: Software Index

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This is software of narrow scope developed as a byproduct of a research project and subsequently made available at no cost to the public. The software is often meant to demonstrate new numerical methods or software engineering strategies which were the subject of a research project. When developed, the software typically contains capabilities unavailable elsewhere. While the software may be quite capable, it is typically not professionally packaged and its use may require some expertise. The software is typically provided as source code or via a web-based service, and no support is provided.

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These are collections of software (e.g. libraries) or interactive systems of a somewhat broad scope. Contents may be adapted from research software or may be contributed by project participants who donate their services to the project. The software is made freely available to the public, typically in source code form. While formal support of the collection may not be provided by its developers, within active projects there is often a core group who donate time to consider bug reports and make updates to the collection.

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An increasing number of published books have included digital media containing software described in the book. Often, the collection of software covers a fairly broad area. Such software is typically developed by the book author. While it is not professionally packaged, it often provides a useful tool for readers to experiment with the concepts discussed in the book. The software itself is typically not formally supported by its authors.

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Such software ranges from a collection of reusable software parts (e.g., a library) to fully functional interactive computing environments with an associated computing language. Such software is usually professionally developed, tested, and maintained to high standards. It is available for purchase, often with accompanying updates and consulting support.

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A cross index of mathematical software in use at NIST.