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1: 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 ,
19.23.3 R D ( 0 , y , z ) = 3 0 π / 2 ( y cos 2 θ + z sin 2 θ ) - 3 / 2 sin 2 θ d θ .
19.23.5 R F ( x , y , z ) = 2 π 0 π / 2 R C ( x , y cos 2 θ + z sin 2 θ ) d θ , y > 0 , z > 0 ,
In (19.23.8) n = 2 , and in (19.23.9) n = 3 . … With l 1 , l 2 , l 3 denoting any permutation of sin θ cos ϕ , sin θ sin ϕ , cos θ , …
2: 19.28 Integrals of Elliptic Integrals
19.28.3 0 1 t σ - 1 ( 1 - t ) R D ( 0 , t , 1 ) d t = 3 4 σ + 2 ( B ( σ , 1 2 ) ) 2 .
19.28.5 z R D ( x , y , t ) d t = 6 R F ( x , y , z ) ,
19.28.7 0 R J ( x , y , z , r 2 ) d r = 3 2 π R F ( x y , x z , y z ) ,
19.28.8 0 R J ( t x , y , z , t p ) d t = 6 p R C ( p , x ) R F ( 0 , y , z ) .
19.28.9 0 π / 2 R F ( sin 2 θ cos 2 ( x + y ) , sin 2 θ cos 2 ( x - y ) , 1 ) d θ = R F ( 0 , cos 2 x , 1 ) R F ( 0 , cos 2 y , 1 ) ,
3: 19.16 Definitions
It should be noted that the integrals (19.16.1)–(19.16.3) have been normalized so that R F ( 1 , 1 , 1 ) = R J ( 1 , 1 , 1 , 1 ) = R G ( 1 , 1 , 1 ) = 1 . …
19.16.5 R D ( x , y , z ) = R J ( x , y , z , z ) = 3 2 0 d t s ( t ) ( t + z ) ,
Just as the elementary function R C ( x , y ) 19.2(iv)) is the degenerate case …and R D is a degenerate case of R J , so is R J a degenerate case of the hyperelliptic integral, … (Note that R C ( x , y ) is not an elliptic integral.) …
4: 19.2 Definitions
19.2.6 D ( ϕ , k ) = 0 ϕ sin 2 θ d θ 1 - k 2 sin 2 θ = 0 sin ϕ t 2 d t 1 - t 2 1 - k 2 t 2 = ( F ( ϕ , k ) - E ( ϕ , k ) ) / k 2 .
The cases with ϕ = π / 2 are the complete integrals: …
§19.2(iv) A Related Function: R C ( x , y )
When x and y are positive, R C ( x , y ) is an inverse circular function if x < y and an inverse hyperbolic function (or logarithm) if x > y : …For the special cases of R C ( x , x ) and R C ( 0 , y ) see (19.6.15). …
5: 19.25 Relations to Other Functions
Equations (19.25.9)–(19.25.11) correspond to three (nonzero) choices for the last variable of R D ; see (19.21.7). … then the five nontrivial permutations of x , y , z that leave R F invariant change k 2 ( = ( z - y ) / ( z - x ) ) into 1 / k 2 , k 2 , 1 / k 2 , - k 2 / k 2 , - k 2 / k 2 , and sin ϕ ( = ( z - x ) / z ) into k sin ϕ , - i tan ϕ , - i k tan ϕ , ( k sin ϕ ) / 1 - k 2 sin 2 ϕ , - i k sin ϕ / 1 - k 2 sin 2 ϕ . … Inversions of 12 elliptic integrals of the first kind, producing the 12 Jacobian elliptic functions, are combined and simplified by using the properties of R F ( x , y , z ) . … For these results and extensions to the Appell function F 1 16.13) and Lauricella’s function F D see Carlson (1963). ( F 1 and F D are equivalent to the R -function of 3 and n variables, respectively, but lack full symmetry.) …
6: 31.2 Differential Equations
F -Homotopic Transformations
Lastly, w ( z ) = ( z - a ) 1 - ϵ w 3 ( z ) satisfies (31.2.1) if w 3 is a solution of (31.2.1) with transformed parameters q 3 = q + γ ( 1 - ϵ ) ; α 3 = α + 1 - ϵ , β 3 = β + 1 - ϵ , ϵ 3 = 2 - ϵ . By composing these three steps, there result 2 3 = 8 possible transformations of the dependent variable (including the identity transformation) that preserve the form of (31.2.1). … There are 4 ! = 24 homographies z ~ ( z ) = ( A z + B ) / ( C z + D ) that take 0 , 1 , a , to some permutation of 0 , 1 , a , , where a may differ from a . … There are 8 24 = 192 automorphisms of equation (31.2.1) by compositions of F -homotopic and homographic transformations. …
7: 19.33 Triaxial Ellipsoids
The surface area of an ellipsoid with semiaxes a , b , c , and volume V = 4 π a b c / 3 is given by …
19.33.2 S 2 π = c 2 + a b sin ϕ ( E ( ϕ , k ) sin 2 ϕ + F ( ϕ , k ) cos 2 ϕ ) , a b c ,
and the electric capacity C = Q / V ( 0 ) is given by
19.33.6 1 / C = R F ( a 2 , b 2 , c 2 ) .
Let a homogeneous magnetic ellipsoid with semiaxes a , b , c , volume V = 4 π a b c / 3 , and susceptibility χ be placed in a previously uniform magnetic field H parallel to the principal axis with semiaxis c . …
8: 1.6 Vectors and Vector-Valued Functions
The divergence of a differentiable vector-valued function F = F 1 i + F 2 j + F 3 k is … The line integral of a vector-valued function F = F 1 i + F 2 j + F 3 k along c is given by … If C is oriented in the positive (anticlockwise) sense, then …Sufficient conditions for this result to hold are that F 1 ( x , y ) and F 2 ( x , y ) are continuously differentiable on S , and C is piecewise differentiable. … For a vector-valued function F , …
9: 31.10 Integral Equations and Representations
for a suitable contour C . …The contour C must be such that … for suitable contours C 1 , C 2 . …where 𝒟 z is given by (31.10.4). The contours C 1 , C 2 must be chosen so that …
10: Bibliography S
  • I. J. Schoenberg (1971) Norm inequalities for a certain class of C  functions. Israel J. Math. 10, pp. 364–372.
  • R. S. Scorer (1950) Numerical evaluation of integrals of the form I = x 1 x 2 f ( x ) e i ϕ ( x ) d x and the tabulation of the function Gi ( z ) = ( 1 / π ) 0 sin ( u z + 1 3 u 3 ) d u . Quart. J. Mech. Appl. Math. 3 (1), pp. 107–112.
  • J. Shapiro (1970) Arbitrary 3 n - j symbols for SU ( 2 ) . Comput. Phys. Comm. 1 (3), pp. 207–215.
  • L. Shen (1998) On an identity of Ramanujan based on the hypergeometric series F 1 2 ( 1 3 , 2 3 ; 1 2 ; x ) . J. Number Theory 69 (2), pp. 125–134.
  • A. Sidi (1997) Computation of infinite integrals involving Bessel functions of arbitrary order by the D ¯ -transformation. J. Comput. Appl. Math. 78 (1), pp. 125–130.