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§19.37 Tables

Contents

§19.37(i) Introduction

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

Functions \mathop{K\/}\nolimits\!\left(k\right) and \mathop{E\/}\nolimits\!\left(k\right)

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

Tabulated for k=0(.01)1 to 10D by Fettis and Caslin (1964), and for k=0(.02)1 to 7D by Zhang and Jin (1996, p. 673).

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

Functions \mathop{K\/}\nolimits\!\left(k\right), \mathop{{K^{{\prime}}}\/}\nolimits\!\left(k\right), and i\mathop{{K^{{\prime}}}\/}\nolimits\!\left(k\right)/\mathop{K\/}\nolimits\!% \left(k\right)

Tabulated with k=Re^{{i\theta}} for R=0(.01)1 and \theta=0(1^{\circ})90^{\circ} to 11D by Fettis and Caslin (1969).

Function \mathop{\exp\/}\nolimits\!\left(-\pi\mathop{{K^{{\prime}}}\/}\nolimits\!\left(% k\right)/\mathop{K\/}\nolimits\!\left(k\right)\right)(=q(k))

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 \mathop{\mathrm{arcsin}\/}\nolimits 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

Functions \mathop{F\/}\nolimits\!\left(\phi,k\right) and \mathop{E\/}\nolimits\!\left(\phi,k\right)

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). (\mathop{F\/}\nolimits\!\left(\phi,k\right) is presented as \mathop{\Pi\/}\nolimits\!\left(\phi,0,k\right).)

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

Tabulated for \phi=0(5^{\circ})90^{\circ}, \mathop{\mathrm{arcsin}\/}\nolimits k=0(1^{\circ})90^{\circ} to 6D by Byrd and Friedman (1971), for \phi=0(5^{\circ})90^{\circ}, \mathop{\mathrm{arcsin}\/}\nolimits 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}, \mathop{\mathrm{arcsin}\/}\nolimits k=0(5^{\circ})90^{\circ} to 9D by Zhang and Jin (1996, pp. 674–675).

Function \mathop{\Pi\/}\nolimits\!\left(\phi,\alpha^{2},k\right)

Tabulated (with different notation) for \phi=0(15^{\circ})90^{\circ}, \alpha^{2}=0(.1)1, \mathop{\mathrm{arcsin}\/}\nolimits 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, \mathop{\mathrm{arcsin}\/}\nolimits 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

Functions \mathop{R_{F}\/}\nolimits\!\left(x^{2},1,y^{2}\right) and \mathop{R_{G}\/}\nolimits\!\left(x^{2},1,y^{2}\right)

Tabulated for x=0(.1)1, y=1(.2)6 to 3D by Nellis and Carlson (1966).

Function \mathop{R_{F}\/}\nolimits\!\left(a^{2},b^{2},c^{2}\right) with abc=1

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

Check Values

For check values of symmetric integrals with real or complex variables to 14S see Carlson (1995).

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