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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 563 has three (regular) singular points z=O, 1, m . The pairs of exponents at these points are respectively. The general theory of differential equations of the Fuchsian type distinguishes between the following cases. A. None of the numbers c, c-a-b; a-b is equal to an integer. Then two linearly independent solutions of 15.5.1 in the neighborhood of the singular points 0, 1, m are respectively 15.5.3 15.5.4 15.5.5 15.5.6 15.5.7 15.5.8 wi(o)=F(a, b; C; z)=(l-z)"-"-'F(c-~, c-b; C; Z ) w ~ C O ) = z'-'F(u-c+ 1 , b -C + 1 ; 2-C; Z ) = z'-'( 1 - Z ) '-'-'F( 1 -a, 1 - b ; 2 -c; Z ) W1(1)=F(a, b ; a+b+l-C; 1-Z)=z1-'F(1+b-c, l+a-c; a+b+I-c; 1-2) wZ(1) = ( 1 - Z ) '-"-'F(c - b, c-a ; c -a- b + 1 ; 1 - Z ) = z'-'( 1 - Z ) '-"-'F( 1 -a, 1 - b ; c -a- b + 1 ; 1 - Z) ~ q , (m) = z-"F(u, a-c+ 1 ; a-b+ 1 ; Z-') = z'-'(z- 1) '--"-'F(1 - b, C- b ; a- b+ 1 ; Z-') ~ 2 ( ~ ) =z-*F(b, b-c+ 1; b-a+ 1; z-') =Z.-'(Z-~) '--"-'F(l -a, C-U; b-a+ 1 ; Z-') The second set of the above expressions is obtained by applying 15.3.3 to the first set. Another set of representations is obtained by applying 15.3.4 to \$5.5.3 through 15.5.8. This gives 15.5.9-15.5.14. =(I-z)-'F b, C-U; C; - ( 2-1 "> ( 2-1 " > 15.5.9 w~(O)=(~-Z)-'F a, c-b; C; - 15.5.10 w~(O)=Z'-'(~-Z)'-"-'F a-c+I, 1-b; 2-C; - =Z~-'(~-Z)'-'-~F b-c+l, 1-U; 2-c;- ( 2-1 " > ( 2-1 " > 15.5.11 wi(1) = z-"F(a, a-c+ 1 ; a+ b -c+ 1 ; 1 - Z-') = z-'F(b, b -c+ 1 ; a+ b -c+ 1 ; 1 - Z-') 15.5.12 wZ(1) = 9- '( 1 - Z ) "-"-'F(c -a, 1 -u; c -a - b + 1 ; 1 - Z- ') = 2- '( 1 - Z ) c-a-'F(~ - b, 1 - b ; c -G - b + 1 ; 1 - Z- ') ( 1-2 ' > 1 15.5.13 W ~ ( ~ ) = ( Z - ~ ) - ~ F a, c-b; a-b+l; =)=(z-l)-'F b, c-a; b-u+l; - 15.5.14 ( (z-l)'-'-'F b-c+l, 1-u; b-a+l; - - z l - c a-c+I, 1-b; a-b+l; - - ( 1-2 '> ( 1-2 '> w ~ ~ ~ ) = z ' - ' ( z - ~ ) ~ - " - ' F 15.5.3 to 15.5.14 constitute Kummer's 24 solutions of the hypergeometric equation. The analytic con- tinuation of w1 ,2(0)(z) can then be obtained by means of 15.3.3 to 15.3.9. Then one of the hypergeometric series for instance w1 B. One of the numbers a, b, c-a, c-b is an integer. 15.5.3, 15.5.4 terminates and the corresponding solution is of the form 15.5.15 w=Z"(l- z)\$.(z) where p,(z) is a polynomial in z of degree n. This case is referred to as the degenerate case of the hypergeometric differential equation and its solutions are listed and discussed in great detail in D5.21. C. The number c-a-b is an integer, c nonintegral. Then 15.3.10 to 15.3.12 give the analytic continu- ation of w1 ,2(o) into the neighborhood of z=1. Similarly 15.3.13 and 15.3.14 give the analytic continu- ation of ~ 1 . 2 ( ~ ) into the neighborhood of z= o~ in case a-b is an integer but not c, subject of course to the further restrictions c-u=O, f l , f 2 . . . (For a detailed discussion of all poasible cases, see [15.2]). D. The number c= 1. Then 15.5.3, 15.5.4 are replaced by 15.5.16 ZD~(O)=F(~, b; 1; Z ) 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. (800) 430-7532 | info@maps.com | Online Privacy Policy©2000 Maps.com. All rights reserved.Portions ©2000 ConvertIt.com, Inc. All rights reserved. Terms of Use.