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: 13. Confluent Hypergeometric Functions Mathematical Properties 13.1. Definitions of Kummer and Whittaker Functions Kummer's Equation 2 d2W -+(b-Z) dw --uw=O dz2 d z 13.1.1 It has a regular singularity at z=O and an irregular singularity at . Independent solutions are Kummer's Function 13.1.2 (a)nz"+ . . . a2 (a)zzZ M(u,b,z)=l+-+- b (b)22!+ * - +- (b)nn! where (a),=a(a+l)(a+2) . . . (a+n-l), (&=l, and 13.1.3 Parameters (m, n positive integers) M a , b, 4 all values of a, b and z in2 b # - n a#-m a convergent series for b # - n a=--m a polynomial of degree m b=-n a#-m b=-n a=-m, a simple pole at b= -n m>n b=-n a=-m, undefined U(a, b, z) is defined even when b + f n 13.1.4 m5n As IzI--+w, M(a, b, z ) = W e"~"-~[[l+O(lzl-~)] (%'z>O) (4 and 13.1.5 M(u, b, z)=- (- z)-"1 +O(lzl-')] ( 9 Z < O ) r(b--a) U(a, b, z) is a many-valued function. 1t.s princi- pal branch is given by - ?r< arg z _< a. 13.1.6 Logarithmic Solution (n-l)! - +- 2 nM(a-n, 1-n, z ) , I? (a) for n=O, 1, 2 , . . ., where the last function is the sum to n terms. It is to be interpreted as zero when n=O, and +(a)=r'(a)/r(a). 13.1.7 U(a, 1-n, z)=z"U(a+?t, l+n, z) As Wz+w 13.1.8 U(U, b, Z) = z - ~ [ 1 + O( I zl-')] Analytic Continuation 13.1.9 U(a, b, ze*rf)=- lr e-z M(b-a, b, Z) sin lrb { r (1 +a- b) r ( b ) e + r f ( l - b ) I-b 1 - z M ( ~ - c z , ~ - ~ , z ) r (a) r (2-b) where either upper or lower signs are to be taken throughout. 13.1.10 U(a, b, 2) + - 2ribn Alternative Notations IFI(a; b; z) or @(a; b; z ) for M(a, b, z) Z-"~F~(U, 1 +a-b; ;- l / z ) or *(a; b; z ) for U(a, b, z ) Complete Solution y=AM(a, b, z)+BU(a, b, Z ) 13.1.11 where A and B are arbitrary constants, b # -n. Eight Solutions 13.1.12 y l = M ( ~ , b, Z ) 13.1.13 y,=zl-bM(l+a-b, 2-b, Z ) 13.1.14 y,=e"M(b--a, b, -2) 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. ©2000 ConvertIt.com, Inc. All rights reserved. Terms of Use.