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 New Online Book! Handbook of Mathematical Functions (AMS55) Conversion & Calculation Home >> Reference Information Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables (AMS55) Purchase the electronic edition of this book in Adobe PDF format! FIRST | PREVIOUS | NEXT | LAST | CONTENTS | PAGE | ABOUT | SMALL | MEDIUM | LARGE Below is the OCR-scanned text from this page: HYPERGEOMETRIC FUNCTIONS 565 Transformation Formulas for Riemann's P Function a b c b C 15.6.8 (">"("-">'.{a z-b 2-b 8 Y z}=P { a l k 8-k-1 y+l a' 8' Y' a'+k B'-k-1 y'+l b c bi ci 15.6.9 p {: 8 Y z}=p{: 8 Y a' 8' 7' a' 8' y' where Azl+B Aal+B Abi+B Aci+B 15.6.10 z=- Y a=- Czl+D Cal+Dy b'Cbl+D' ' = W D and A, B, C, D are arbitrary constants such that AD-BC#O. 15.6.11 P { 1, ,". 5, z } = ( E y ( E y P Z-b z-b { 0 Riemann's P function reduced to the hypergeometric function is a b c 0 CD 1 ( ~ - ~ ) ( c - b ) a+8+y a'-Ly a+B'+y 7'-y (z-b)(c-a) The P function on the right hand side is Gauss' hypergeometric function (see 15.6.5). If it is replaced by Kummer's 24 solutions 15.5.3 to 15.5.14 the complete set of 24 solutions for Riemann's differential equation 15.6.1 is obtained. The first of these solutions is for instance by 15.5.3 and 15.6.5 15.7. Asymptotic Expansions The behavior of F(a, b ; c; z) for large 121 is described by the transformation formulas of 15.3. For fixed a, b, z.and large Ic( one has [15.8] 15.7.1 For fixeda, c, z, (c#O, - 1 , -2, . . . , 0<1z1<1) and large Ibl one has [15.2] 15.7.2 F(a, b; C; z>=e-*..[ r ( c ) / r ( c - a ) ] (bz)-"[l +0( I bzJ -91 +tr(c)/r(a>l ebz(bz)o-"[l+0(l bzl-91 (-g< 8% ( b z ) < h ) 15.7.3 F(a, b; c; z ) =e*"[ r (c)/r (c-a)] ( 6 2 ) -"[ 1 + O( 1621 -91 + [r(c)/r(a>lebz( bzY-7" + O( I bz I -7 1 (-!F< arg (bz)<#r) For the case when more than one of the param- eters are large consult I15.21. References 115.11 P. Appell and J. Kamp6 de FCiet, Fonctions hyper- g6om6triques et hyperspheriques (Gauthiers- Vilkrs, Paris, France, 1926). [15.2] A. Erdelyi et al., Higher transcendental functions, vol. 1 (McGraw-Hill Book Co., Inc., New York,' N.Y., 1953). [15.3] E. Goursat, Ann. Sci. fioole Norm. Sup(2)10, 3-142(1881). [15.4] E. Goursat, Propribt6s generales de 1'Bquation d'Euler et de Gauss (Actualit& scientifiques et industriblles 333, Paris, France, 1936). [15.5] J. Kamp6 de Fbriet, La fonction hypergBom6trique (Gauthiers-Villars, Paris, France, 1937). [15.6] E. Klein, Vorlesungen uber die hypergeometrische Funktion (B. G. Teubner, Berlin, Germany, 1933). The page scan image above, and the text in the text box above, are contributions of the National Institute of Standards and Technology that are not subject to copyright in the United States.