19 Elliptic Integrals Computation 19.36 Methods of Computation 19.38 Approximations

§19.37 Tables

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Keywords:: Legendre’s elliptic integrals, tables
Permalink:: http://dlmf.nist.gov/19.37
See also:: Annotations for Ch.19

§19.37(i) Introduction
§19.37(ii) Legendre’s Complete Integrals
§19.37(iii) Legendre’s Incomplete Integrals
§19.37(iv) Symmetric Integrals

§19.37(i) Introduction

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Permalink:: http://dlmf.nist.gov/19.37.i
See also:: Annotations for §19.37 and Ch.19

Only tables published since 1960 are included. For earlier tables see Fletcher (1948), Lebedev and Fedorova (1960), and Fletcher et al. (1962).

§19.37(ii) Legendre’s Complete Integrals

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Permalink:: http://dlmf.nist.gov/19.37.ii
See also:: Annotations for §19.37 and Ch.19

Functions $K\left(k\right)$ and $E\left(k\right)$

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See also:: Annotations for §19.37(ii), §19.37 and Ch.19

Tabulated for $k^{2}=0(.01)1$ to 6D by Byrd and Friedman (1971), to 15D for $K\left(k\right)$ and 9D for $E\left(k\right)$ by Abramowitz and Stegun (1964, Chapter 17), and to 10D by Fettis and Caslin (1964).

Tabulated for to 10D by Fettis and Caslin (1964), and for to 7D by Zhang and Jin (1996, p. 673).

Tabulated for $\operatorname{arcsin}k=0(1^{\circ})90^{\circ}$ to 6D by Byrd and Friedman (1971) and to 15D by Abramowitz and Stegun (1964, Chapter 17).

Functions $K\left(k\right)$ , ${K^{\prime}}\left(k\right)$ , and $i{K^{\prime}}\left(k\right)/K\left(k\right)$

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See also:: Annotations for §19.37(ii), §19.37 and Ch.19

Tabulated with $k=R{\mathrm{e}}^{i\theta}$ for and $\theta=0(1^{\circ})90^{\circ}$ to 11D by Fettis and Caslin (1969).

Function $\exp\left(-\pi{K^{\prime}}\left(k\right)/K\left(k\right)\right)(=q(k))$

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See also:: Annotations for §19.37(ii), §19.37 and Ch.19

Tabulated for $k^{2}=0(.01)1$ to 6D by Byrd and Friedman (1971) and to 15D by Abramowitz and Stegun (1964, Chapter 17).

Tabulated for $\operatorname{arcsin}k=0(1^{\circ})90^{\circ}$ to 6D by Byrd and Friedman (1971) and to 15D by Abramowitz and Stegun (1964, Chapter 17).

Tabulated for $k^{2}=0(.001)1$ to 8D by Beli͡akov et al. (1962).

§19.37(iii) Legendre’s Incomplete Integrals

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Referenced by:: §19.36(ii)
Permalink:: http://dlmf.nist.gov/19.37.iii
See also:: Annotations for §19.37 and Ch.19

Functions $F\left(\phi,k\right)$ and $E\left(\phi,k\right)$

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See also:: Annotations for §19.37(iii), §19.37 and Ch.19

Tabulated for $\phi=0(5^{\circ})90^{\circ}$ , $k^{2}=0(.01)1$ to 10D by Fettis and Caslin (1964).

Tabulated for $\phi=0(1^{\circ})90^{\circ}$ , $k^{2}=0(.01)1$ to 7S by Beli͡akov et al. (1962). ( $F\left(\phi,k\right)$ is presented as $\Pi\left(\phi,0,k\right)$ .)

Tabulated for $\phi=0(5^{\circ})90^{\circ}$ , to 10D by Fettis and Caslin (1964).

Tabulated for $\phi=0(5^{\circ})90^{\circ}$ , $\operatorname{arcsin}k=0(1^{\circ})90^{\circ}$ to 6D by Byrd and Friedman (1971), for $\phi=0(5^{\circ})90^{\circ}$ , $\operatorname{arcsin}k=0(2^{\circ})90^{\circ}$ and $5^{\circ}(10^{\circ})85^{\circ}$ to 8D by Abramowitz and Stegun (1964, Chapter 17), and for $\phi=0(10^{\circ})90^{\circ}$ , $\operatorname{arcsin}k=0(5^{\circ})90^{\circ}$ to 9D by Zhang and Jin (1996, pp. 674–675).

Function $\Pi\left(\phi,\alpha^{2},k\right)$

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Keywords:: Legendre’s elliptic integrals, tables
See also:: Annotations for §19.37(iii), §19.37 and Ch.19

Tabulated (with different notation) for $\phi=0(15^{\circ})90^{\circ}$ , $\alpha^{2}=0(.1)1$ , $\operatorname{arcsin}k=0(15^{\circ})90^{\circ}$ to 5D by Abramowitz and Stegun (1964, Chapter 17), and for $\phi=0(15^{\circ})90^{\circ}$ , $\alpha^{2}=0(.1)1$ , $\operatorname{arcsin}k=0(15^{\circ})90^{\circ}$ to 7D by Zhang and Jin (1996, pp. 676–677).

Tabulated for $\phi=5^{\circ}(5^{\circ})80^{\circ}(2.5^{\circ})90^{\circ}$ , $\alpha^{2}=-1(.1)-0.1,0.1(.1)1$ , $k^{2}=0(.05)0.9(.02)1$ to 10D by Fettis and Caslin (1964) (and warns of inaccuracies in Selfridge and Maxfield (1958) and Paxton and Rollin (1959)).

Tabulated for $\phi=0(1^{\circ})90^{\circ}$ , $\alpha^{2}=0(.05)0.85,0.88(.02)0.94(.01)0.98(.005)1$ , $k^{2}=0(.01)1$ to 7S by Beli͡akov et al. (1962).

§19.37(iv) Symmetric Integrals

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Keywords:: symmetric elliptic integrals, tables
Permalink:: http://dlmf.nist.gov/19.37.iv
See also:: Annotations for §19.37 and Ch.19

Functions $R_{F}\left(x^{2},1,y^{2}\right)$ and $R_{G}\left(x^{2},1,y^{2}\right)$

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See also:: Annotations for §19.37(iv), §19.37 and Ch.19

Tabulated for , to 3D by Nellis and Carlson (1966).

Function $R_{F}\left(a^{2},b^{2},c^{2}\right)$ with

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See also:: Annotations for §19.37(iv), §19.37 and Ch.19

Tabulated for $\sigma=0(.05)0.5(.1)1(.2)2(.5)5$ , $\cos\left(3\gamma\right)=-1(.2)1$ to 5D by Carlson (1961a). Here $\sigma^{2}=\tfrac{2}{3}((\ln a)^{2}+(\ln b)^{2}+(\ln c)^{2})$ , $\cos\left(3\gamma\right)=(4/\sigma^{3})(\ln a)(\ln b)(\ln c)$ , and are semiaxes of an ellipsoid with the same volume as the unit sphere.