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symmetric elliptic integrals

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21: 19.23 Integral Representations
§19.23 Integral Representations
19.23.1 R F ( 0 , y , z ) = 0 π / 2 ( y cos 2 θ + z sin 2 θ ) - 1 / 2 d θ ,
19.23.2 R G ( 0 , y , z ) = 1 2 0 π / 2 ( y cos 2 θ + z sin 2 θ ) 1 / 2 d θ ,
19.23.3 R D ( 0 , y , z ) = 3 0 π / 2 ( y cos 2 θ + z sin 2 θ ) - 3 / 2 sin 2 θ d θ .
19.23.4 R F ( 0 , y , z ) = 2 π 0 π / 2 R C ( y , z cos 2 θ ) d θ = 2 π 0 R C ( y cosh 2 t , z ) d t .
22: 19.29 Reduction of General Elliptic Integrals
§19.29 Reduction of General Elliptic Integrals
§19.29(i) Reduction Theorems
19.29.4 y x d t s ( t ) = 2 R F ( U 12 2 , U 13 2 , U 23 2 ) ,
19.29.32 y x d t t 4 + a 4 = 2 R F ( U , U + 2 a 2 , U - 2 a 2 ) ,
19.29.33 ( x - y ) 2 U = ( x 4 + a 4 + y 4 + a 4 ) 2 - ( x 2 - y 2 ) 2 .
23: 19.37 Tables
§19.37(iv) Symmetric Integrals
24: 19.19 Taylor and Related Series
§19.19 Taylor and Related Series
19.19.6 R J ( x , y , z , p ) = R - 3 2 ( 1 2 , 1 2 , 1 2 , 1 2 , 1 2 ; x , y , z , p , p )
25: 19.30 Lengths of Plane Curves
19.30.5 L ( a , b ) = 4 a E ( k ) = 8 a R G ( 0 , b 2 / a 2 , 1 ) = 8 R G ( 0 , a 2 , b 2 ) = 8 a b R G ( 0 , a - 2 , b - 2 ) ,
19.30.9 s = 1 2 I ( e 1 ) = - 1 3 a 2 b 2 R D ( r , r + b 2 + a 2 , r + b 2 ) + y r + b 2 + a 2 r + b 2 , r = b 4 / y 2 .
19.30.11 s = 2 a 2 0 r d t 4 a 4 - t 4 = 2 a 2 R F ( q - 1 , q , q + 1 ) , q = 2 a 2 / r 2 = sec ( 2 θ ) ,
19.30.13 P = 4 2 a 2 R F ( 0 , 1 , 2 ) = 2 a 2 × 5.24411 51 = 4 a K ( 1 / 2 ) = a × 7.41629 87 .
26: 20.9 Relations to Other Functions
20.9.3 R F ( θ 2 2 ( z , q ) θ 2 2 ( 0 , q ) , θ 3 2 ( z , q ) θ 3 2 ( 0 , q ) , θ 4 2 ( z , q ) θ 4 2 ( 0 , q ) ) = θ 1 ( 0 , q ) θ 1 ( z , q ) z ,
20.9.4 R F ( 0 , θ 3 4 ( 0 , q ) , θ 4 4 ( 0 , q ) ) = 1 2 π ,
20.9.5 exp ( - π R F ( 0 , k 2 , 1 ) R F ( 0 , k 2 , 1 ) ) = q .
27: 19.36 Methods of Computation
§19.36 Methods of Computation
19.36.3 R F ( 1 , 2 , 4 ) = R F ( z 1 , z 2 , z 3 ) ,
19.36.5 R F ( 1 , 2 , 4 ) = 0.68508 58166 .
19.36.8 R F ( t n 2 , t n 2 + θ c n 2 , t n 2 + θ a n 2 )
28: 22.15 Inverse Functions
For representations of the inverse functions as symmetric elliptic integrals see §19.25(v). …
29: Bibliography Z
  • D. G. Zill and B. C. Carlson (1970) Symmetric elliptic integrals of the third kind. Math. Comp. 24 (109), pp. 199–214.
  • 30: Bibliography N
  • E. Neuman (2003) Bounds for symmetric elliptic integrals. J. Approx. Theory 122 (2), pp. 249–259.