# §19.37 Tables

## §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).