# §19.39(i) Introduction

In this section we provide links to the research literature describing the implementation of algorithms in software for the evaluation of functions described in this chapter. Citations in bulleted lists refer to papers for which research software has been made available and can be downloaded via the Web. References to research software that is available in other ways is listed separately.

A more complete list of available software for computing these functions is found in the Software Index. For another listing of Web-accessible software for the functions in this chapter, see GAMS Class C14.

# §19.39(ii) Legendre’s and Bulirsch’s Complete Integrals

Unless otherwise stated, the functions are $\mathop{K\/}\nolimits\!\left(k\right)$ and $\mathop{E\/}\nolimits\!\left(k\right)$, with $0\leq k^{2}(=m)\leq 1$.

For research software see Bulirsch (1969b, function $\mathop{\mathrm{cel}\/}\nolimits$), Herndon (1961a, b), Merner (1962), Morita (1978, complex modulus $k$), and Thacher Jr. (1963). For other software, sometimes with $\mathop{\Pi\/}\nolimits\!\left(\alpha^{2},k\right)$ and complex variables, see the Software Index.

# §19.39(iii) Legendre’s and Bulirsch’s Incomplete Integrals

Unless otherwise stated, the variables are real, and the functions are $\mathop{F\/}\nolimits\!\left(\phi,k\right)$ and $\mathop{E\/}\nolimits\!\left(\phi,k\right)$.

For research software see Bulirsch (1965b, function $\mathop{\mathrm{el2}\/}\nolimits$), Bulirsch (1969b, function $\mathop{\mathrm{el3}\/}\nolimits$), Jefferson (1961), and Neuman (1969a, functions $\mathop{E\/}\nolimits\!\left(\phi,k\right)$ and $\mathop{\Pi\/}\nolimits\!\left(\phi,k^{2},k\right)$). For other software, sometimes with $\mathop{\Pi\/}\nolimits\!\left(\phi,\alpha^{2},k\right)$ and complex variables, see the Software Index.

# §19.39(iv) Symmetric Integrals

The variables are real and the functions are $R$-functions.

• Carlson and Notis (1981). Fortran.

• Press and Teukolsky (1990).