Partner with ConvertIt.com
 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: 564 HXPERGEOMETRIC FUNCTIONS F. The number c=l-m, m=1, 2, 3, . . . . A fundamental system is 15.5.20 wl(0)=zmF(a+m, b+m; l+m; z) 15.5.21 ~ ~ ( ~ ) = z ~ F ( a + m , b+m; l+m; 2) In z + z r n C 2'' ( a + m ) ~ ( b + m ) m [+(a+m +n)-+(a+ m) ++(b+m +n) n=1 ( 1 + m Inn! 15.6. Riemann's Differential Equation The hypergeometric differential equation 15.5.1 with the (regular) singular points 0, 1, w is a special case of Riemann's differential equation with three (regular) singular points a, b, c 15.6.1 d2w 1-a-a' 1-8-8' =+[ z-a z-b + aa'(a-b)(a-c) & 3 ' ( b - ~ ) ( b - ~ ) +[ 2--a z-b The pairs of the exponents with respect to the singul.ar points a; b; c are a, a'; 8, 8'; y , y' respec- tively subject to the condition 15.6.2 a+a'+B+B'+r+u'=1 The complete set of solutions of 15.6.1 is denoted by the synibol 15.6.3 b c 8' 7' (1z1<1 and a, b#O, -1, -2, . . . -(m-l)) Special Cases of Riemann's P Function (a) The generalized hypergeometric function 15.6.4 (b) The hypergeometric function F(a, b; c; z) 15.6.5 11-c b c-a-b J (c) The Legendre functions P: ( z ) , Q: ( z ) 15.6.6 I w=p{ -Qv 3 p 0 (1-z2)-l 0 W 1 3+3v -ire 3 (d) The confluent hypergeometric function 15.6.7 C J 13-U O k provided lim c--. 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.