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: 10. Bessel Functions of Fractional Order Mathematical Properties 10.1. Spherical Bessel Functions Definitions Differential Equation 10.1.1 22w"+2zw'+[ z2-n(n+ 1)]w=O (n=O,&l, f 2 , . . .) Particular solutions are the Spherical Bessel junctions oj thejirst kind j n ( 4 = r n J n + d z ) , Yn(2)=m2Yn+*iz), the Spherical Bessel junctions of the second kind and the Spherical Bessel junctions of the t h M kind h?(z)=j,(z) +~Y,(z)=I~&K~Y&), h?) ( Z> = j , ( 2 ) - iy, ( 2 ) = ~ z B A ~ + ( 2 ) . The pairs j,(z), y,(z) and h f ' ( z ) , h\$'(z) are For linearly independent solutions for every n. general properties see the remarks after 9.1.1. 10.1.2 Ascending Series (See 9.1.2, 9.1.10) 2" 3zz 'n(z)=1.3-5. . . (2n-I-1) { '-1!(2n+3) - 10.1.3 +2!(1--2n) (3z2>z (3-2n) - . . .} (n=O, 1, 2, . . .) Limiting Values as z - d 1 2-n'n(z)*1.3-5. . . (2n+l) 10.1.4 10.1.5 Z"+lyn(z)-+-1*3-5. . . (2n-1) (n=O, 1, 2 , . . .) Wronskians Representations by Elementary Functions 10.1.8 j,(z)=z-'[P(n++, z) sin (z-&) 10.1.9 +an+;, z> cos (z-h7r)] y,(z)=(-l)"+~Z-~[P(n++,z) cos (Z++n7r) -&(n++, z ) sin ( z + h r ) l =E (-l)k(n++, 2k)(2~)-~* 0 =[*F (-l)"n+\$, 2k+1)(22)-%-' 0 (n=O, 1,2, . . .) 437 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.