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: 9. Bessel Functions of Integer Order Mathematical Properties Not ation The tables in t is chapter are for Bessel func- tions of integer order; the text treats general orders. The conventions used are: z=x+iy; x, y real. n is a positive integer or zero. V, p are unrestricted except where otherwise indicated; v is supposed real in the sections devoted to Kelvin functions 9.9, 9.10, and 9.11. The notation used for the Bessel functions is that of Watson [9.15] and the British Association and Royal Society Mathematical Tables. The function Yv(z) is often denoted N,(z) by physicists and European workers. Aldis, Airey: Other notations are those of: G,(z) for -+?rY,(z),K,(z) for (-)"K,(z). Clifford : C,(z) for s+"Jn(2&). Gray, Mathews and MacRobert [9.9]: Y,(z) for +?rYn(z>+(ln 2--Y)Jn(z), - Yy(z) for ?rev"' sec(v?r)Y,(z), Gv(z) for +riH;l)(z). Jahnke, Emde and Losch [9.32]: L ( Z ) for r(v+i)(&)-"J,(z). Jeffreys: Hsv(z) for HL1)(z), Hiv(z) for H!')(z), Kh,(z) for (2/r)Kv(z). Heine: K,( z ) for - +rY,(z). Neumann: Y"(z) for +?rY,(z>+ On 2--y)J,(z). Whittaker and Watson t9.181: Kv(z) for cos(v?r)KY(~). 358 Bessel Functions Jand Y 9.1. Definitions and Elementary Properties Differential Equation 22--+2 d2w dw -+(22-v2)w=O 9.1.1 dz2 dz Solutions are the Bessel functions of the first kind J*y(z), of the second kind Yy(z) (also called Weber's function) and of the third kindH;"(z), Hp)(z) (also called the Hankel functions). Each is a regular (holomorphic) function of z throughout the z-plane cut along the negative real axis, and for fixed z ( Z 0 ) each is an entire (integral) func- tion of v. When v= f n , Jv(z) has no branch point and is an entire (integral) function of z. Important features of the various solutions are as follows: Jv(z)(,B'v20) is bounded as z-+O in any bounded range of arg z. J y ( z ) and J-,(z) are linearly independent except when v is an integer. Jy(z) and Yv(z) are linearly independent for all values of v. Hjl)(z) tends to zero as jzI+m in the sector O