# Bulirsch elliptic integrals

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##### 1: 19.2 Definitions
###### §19.2(iii) Bulirsch’s Integrals
19.2.11_5 $\operatorname{el1}\left(x,k_{c}\right)=\int_{0}^{\operatorname{arctan}x}\frac{% 1}{\sqrt{{\cos}^{2}\theta+k_{c}^{2}{\sin}^{2}\theta}}\,\mathrm{d}\theta,$
$K\left(k\right)=\operatorname{cel}\left(k_{c},1,1,1\right),$
$E\left(k\right)=\operatorname{cel}\left(k_{c},1,1,k_{c}^{2}\right),$
$D\left(k\right)=\operatorname{cel}\left(k_{c},1,0,1\right),$
##### 2: 19.1 Special Notation
 $l,m,n$ nonnegative integers. …
$R_{-a}\left(b_{1},b_{2},\dots,b_{n};z_{1},z_{2},\dots,z_{n}\right)$ is a multivariate hypergeometric function that includes all the functions in (19.1.3). …
##### 3: 19.25 Relations to Other Functions
###### §19.25(ii) Bulirsch’s Integrals as Symmetric Integrals
19.25.19 $\operatorname{cel}\left(k_{c},p,a,b\right)=aR_{F}\left(0,k_{c}^{2},1\right)+% \tfrac{1}{3}{(b-pa)}R_{J}\left(0,k_{c}^{2},1,p\right),$
19.25.20 $\operatorname{el1}\left(x,k_{c}\right)=R_{F}\left(r,r+k_{c}^{2},r+1\right),$
19.25.21 $\operatorname{el2}\left(x,k_{c},a,b\right)=a\operatorname{el1}\left(x,k_{c}% \right)+\tfrac{1}{3}{(b-a)}R_{D}\left(r,r+k_{c}^{2},r+1\right),$
19.25.22 $\operatorname{el3}\left(x,k_{c},p\right)=\operatorname{el1}\left(x,k_{c}\right% )+\tfrac{1}{3}{(1-p)}R_{J}\left(r,r+k_{c}^{2},r+1,r+p\right).$
##### 4: 19.39 Software
For research software see Bulirsch (1965b, function $\operatorname{el2}$), Bulirsch (1969b, function $\operatorname{el3}$), Jefferson (1961), and Neuman (1969a, functions $E\left(\phi,k\right)$ and $\Pi\left(\phi,k^{2},k\right)$). …
##### 5: 19.36 Methods of Computation
Computation of Legendre’s integrals of all three kinds by quadratic transformation is described by Cazenave (1969, pp. 128–159, 208–230). …
##### 6: Bibliography B
• R. Bulirsch (1969a) An extension of the Bartky-transformation to incomplete elliptic integrals of the third kind. Numer. Math. 13 (3), pp. 266–284.
• R. Bulirsch (1969b) Numerical calculation of elliptic integrals and elliptic functions. III. Numer. Math. 13 (4), pp. 305–315.
• R. Bulirsch (1965a) Numerical calculation of elliptic integrals and elliptic functions. II. Numer. Math. 7 (4), pp. 353–354.
• R. Bulirsch (1965b) Numerical calculation of elliptic integrals and elliptic functions. Numer. Math. 7 (1), pp. 78–90.
• ##### 7: Bibliography R
• K. Reinsch and W. Raab (2000) Elliptic Integrals of the First and Second Kind – Comparison of Bulirsch’s and Carlson’s Algorithms for Numerical Calculation. In Special Functions (Hong Kong, 1999), C. Dunkl, M. Ismail, and R. Wong (Eds.), pp. 293–308.
• ##### 8: Bibliography M
• P. Midy (1975) An improved calculation of the general elliptic integral of the second kind in the neighbourhood of $x=0$ . Numer. Math. 25 (1), pp. 99–101.