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symmetric elliptic integrals

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11: 19.34 Mutual Inductance of Coaxial Circles
§19.34 Mutual Inductance of Coaxial Circles
19.34.5 3 c 2 8 π a b M = 3 R F ( 0 , r + 2 , r - 2 ) - 2 r - 2 R D ( 0 , r + 2 , r - 2 ) ,
19.34.6 c 2 2 π M = ( r + 2 + r - 2 ) R F ( 0 , r + 2 , r - 2 ) - 4 R G ( 0 , r + 2 , r - 2 ) .
12: 19.27 Asymptotic Approximations and Expansions
§19.27 Asymptotic Approximations and Expansions
19.27.4 R G ( x , y , z ) = z 2 ( 1 + O ( a z ln z a ) ) , a / z 0 .
19.27.8 R D ( x , y , z ) = 3 x y z - 6 x y R G ( x , y , 0 ) ( 1 + O ( z g ) ) , z / g 0 .
19.27.11 R J ( x , y , z , p ) = 3 p R F ( x , y , z ) - 3 π 2 p 3 / 2 ( 1 + O ( c p ) ) , c / p 0 .
13: 19.1 Special Notation
l , m , n

nonnegative integers.

However, it should be noted that in Chapter 8 of Abramowitz and Stegun (1964) the notation used for elliptic integrals differs from Chapter 17 and is consistent with that used in the present chapter and the rest of the NIST Handbook and DLMF. …
R - a ( b 1 , b 2 , , b n ; z 1 , z 2 , , z n ) .
14: 19.33 Triaxial Ellipsoids
19.33.1 S = 3 V R G ( a - 2 , b - 2 , c - 2 ) ,
§19.33(ii) Potential of a Charged Conducting Ellipsoid
19.33.5 V ( λ ) = Q R F ( a 2 + λ , b 2 + λ , c 2 + λ ) ,
19.33.6 1 / C = R F ( a 2 , b 2 , c 2 ) .
19.33.11 U = 1 2 ( α β γ ) 2 R F ( α 2 , β 2 , γ 2 ) 0 ( g ( r ) ) 2 d r ,
15: 19.28 Integrals of Elliptic Integrals
§19.28 Integrals of Elliptic Integrals
19.28.5 z R D ( x , y , t ) d t = 6 R F ( x , y , z ) ,
19.28.6 0 1 R D ( x , y , v 2 z + ( 1 - v 2 ) p ) d v = R J ( x , y , z , p ) .
19.28.7 0 R J ( x , y , z , r 2 ) d r = 3 2 π R F ( x y , x z , y z ) ,
16: 19.20 Special Cases
§19.20 Special Cases
The general lemniscatic case is … where x , y , z may be permuted. … The general lemniscatic case is …
17: 19.32 Conformal Map onto a Rectangle
§19.32 Conformal Map onto a Rectangle
19.32.1 z ( p ) = R F ( p - x 1 , p - x 2 , p - x 3 ) ,
18: 19.17 Graphics
§19.17 Graphics
See Figures 19.17.119.17.8 for symmetric elliptic integrals with real arguments. …
See accompanying text
Figure 19.17.8: R J ( 0 , y , 1 , p ) , 0 y 1 , - 1 p 2 . … Magnify 3D Help
19: Bille C. Carlson
This symmetry led to the development of symmetric elliptic integrals, which are free from the transformations of modulus and amplitude that complicate the Legendre theory. … This invariance usually replaces sets of twelve equations by sets of three equations and applies also to the relation between the first symmetric elliptic integral and the Jacobian functions. …
20: 19.26 Addition Theorems
§19.26 Addition Theorems
19.26.8 2 R G ( x + λ , y + λ , z + λ ) + 2 R G ( x + μ , y + μ , z + μ ) = 2 R G ( x , y , z ) + λ R F ( x + λ , y + λ , z + λ ) + μ R F ( x + μ , y + μ , z + μ ) + λ + μ + x + y + z .
§19.26(iii) Duplication Formulas
19.26.20 R D ( x , y , z ) = 2 R D ( x + λ , y + λ , z + λ ) + 3 z ( z + λ ) .
19.26.21 2 R G ( x , y , z ) = 4 R G ( x + λ , y + λ , z + λ ) - λ R F ( x , y , z ) - x - y - z .