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symmetric elliptic integrals

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1: 19.16 Definitions
§19.16(i) Symmetric Integrals
19.16.5 R D ( x , y , z ) = R J ( x , y , z , z ) = 3 2 0 d t s ( t ) ( t + z ) ,
which is homogeneous and of degree - a in the z ’s, and unchanged when the same permutation is applied to both sets of subscripts 1 , , n . … …
§19.16(iii) Various Cases of R - a ( b ; z )
2: 19.15 Advantages of Symmetry
§19.15 Advantages of Symmetry
For the many properties of ellipses and triaxial ellipsoids that can be represented by elliptic integrals, any symmetry in the semiaxes remains obvious when symmetric integrals are used (see (19.30.5) and §19.33). …
3: 19.35 Other Applications
4: 19.31 Probability Distributions
§19.31 Probability Distributions
5: 19.21 Connection Formulas
§19.21 Connection Formulas
19.21.3 6 R G ( 0 , y , z ) = y z ( R D ( 0 , y , z ) + R D ( 0 , z , y ) ) = 3 z R F ( 0 , y , z ) + z ( y - z ) R D ( 0 , y , z ) .
19.21.10 2 R G ( x , y , z ) = z R F ( x , y , z ) - 1 3 ( x - z ) ( y - z ) R D ( x , y , z ) + x y / z , z 0 .
§19.21(iii) Change of Parameter of R J
19.21.15 p R J ( 0 , y , z , p ) + q R J ( 0 , y , z , q ) = 3 R F ( 0 , y , z ) , p q = y z .
6: 19.22 Quadratic Transformations
19.22.2 2 R G ( 0 , x 2 , y 2 ) = 4 R G ( 0 , x y , a 2 ) - x y R F ( 0 , x y , a 2 ) ,
Bartky’s Transformation
§19.22(ii) Gauss’s Arithmetic-Geometric Mean (AGM)
7: 19.18 Derivatives and Differential Equations
§19.18(i) Derivatives
19.18.2 d d x R G ( x + a , x + b , x + c ) = 1 2 R F ( x + a , x + b , x + c ) .
§19.18(ii) Differential Equations
and two similar equations obtained by permuting x , y , z in (19.18.10). … The next four differential equations apply to the complete case of R F and R G in the form R - a ( 1 2 , 1 2 ; z 1 , z 2 ) (see (19.16.20) and (19.16.23)). …
8: 19.38 Approximations
9: 19.25 Relations to Other Functions
§19.25(i) Legendre’s Integrals as Symmetric Integrals
§19.25(ii) Bulirsch’s Integrals as Symmetric Integrals
§19.25(v) Jacobian Elliptic Functions
§19.25(vii) Hypergeometric Function
10: 19.24 Inequalities
§19.24(i) Complete Integrals
19.24.7 L ( a , b ) = 8 R G ( 0 , a 2 , b 2 ) .
19.24.9 1 2 g 1 2 R G ( a 0 2 , g 0 2 , 0 ) R F ( a 0 2 , g 0 2 , 0 ) 1 2 a 1 2 ,
§19.24(ii) Incomplete Integrals