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Gauss transformation

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1: 19.15 Advantages of Symmetry
Symmetry unifies the Landen transformations of §19.8(ii) with the Gauss transformations of §19.8(iii), as indicated following (19.22.22) and (19.36.9). … …
2: 19.22 Quadratic Transformations
If x , y , z are real and positive, then (19.22.18)–(19.22.21) are ascending Landen transformations when x , y < z (implying a < z - < z + ), and descending Gauss transformations when z < x , y (implying z + < z - < a ). …Descending Gauss transformations include, as special cases, transformations of complete integrals into complete integrals; ascending Landen transformations do not. … The transformations inverse to the ones just described are the descending Landen transformations and the ascending Gauss transformations. …
3: 19.8 Quadratic Transformations
§19.8(iii) Gauss Transformation
We consider only the descending Gauss transformation because its (ascending) inverse moves F ( ϕ , k ) closer to the singularity at k = sin ϕ = 1 . …
4: 35.8 Generalized Hypergeometric Functions of Matrix Argument
Kummer Transformation
Thomae Transformation
5: 19.36 Methods of Computation
The step from n to n + 1 is an ascending Landen transformation if θ = 1 (leading ultimately to a hyperbolic case of R C ) or a descending Gauss transformation if θ = - 1 (leading to a circular case of R C ). … Descending Gauss transformations of Π ( ϕ , α 2 , k ) (see (19.8.20)) are used in Fettis (1965) to compute a large table (see §19.37(iii)). … The function el2 ( x , k c , a , b ) is computed by descending Landen transformations if x is real, or by descending Gauss transformations if x is complex (Bulirsch (1965b)). …
6: 16.6 Transformations of Variable
16.6.2 F 2 3 ( a , 2 b - a - 1 , 2 - 2 b + a b , a - b + 3 2 ; z 4 ) = ( 1 - z ) - a F 2 3 ( 1 3 a , 1 3 a + 1 3 , 1 3 a + 2 3 b , a - b + 3 2 ; - 27 z 4 ( 1 - z ) 3 ) .
For Kummer-type transformations of F 2 2 functions see Miller (2003) and Paris (2005a), and for further transformations see Erdélyi et al. (1953a, §4.5), Miller and Paris (2011), Choi and Rathie (2013) and Wang and Rathie (2013).
7: Bibliography F
  • H. E. Fettis (1965) Calculation of elliptic integrals of the third kind by means of Gausstransformation. Math. Comp. 19 (89), pp. 97–104.
  • 8: Bibliography V
  • R. Vidūnas (2005) Transformations of some Gauss hypergeometric functions. J. Comput. Appl. Math. 178 (1-2), pp. 473–487.
  • 9: 16.4 Argument Unity
    Balanced F 3 4 ( 1 ) series have transformation formulas and three-term relations. … Transformations for both balanced F 3 4 ( 1 ) and very well-poised F 6 7 ( 1 ) are included in Bailey (1964, pp. 56–63). …
    10: 16.5 Integral Representations and Integrals
    16.5.2 F q + 1 p + 1 ( a 0 , , a p b 0 , , b q ; z ) = Γ ( b 0 ) Γ ( a 0 ) Γ ( b 0 - a 0 ) 0 1 t a 0 - 1 ( 1 - t ) b 0 - a 0 - 1 F q p ( a 1 , , a p b 1 , , b q ; z t ) d t , b 0 > a 0 > 0 ,