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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 | MEDIUM | LARGE Below is the OCR-scanned text from this page: 8. Legendre Functions Mathematical Properties Notation The conventions used are z=r+iy, x, y real, and in particular, z always means a real number in the interval -1 5x5 +l with cos e=x where e is likewise a real number; n and m are positive integers or zero; v and p fire unrestricted except where otherwise indicated. Other notations are : Various other definitions of the functions occur as well as mixing of definitions. 8.1. Differential Equation 8.1.1 (1-2) d2w --2 z dw -+[v(v+l)--]w=O EL2 dzZ dz 1- 2 2 Solutions (Degree v and order p with singularities at as ordinary branch points-p, v arbi- z = & l , trary complex constants.) P : ( z ) , Qf( Z ) -Associated Legendre Functions (Spher- ical Harmonics) of the First and Second Kinds larg (zk l>l (For F(a, b; c ; z ) see chapter 15.) r(v+p+l) z-'-r-1(22-1)t~F(1+Y+P, A+:+!; v+?; ') 8.1.3 @(z) = e2p*2--v-1~f (121>1) 2 2 2 2 2 2 2 2 r(v++) Alternate Forms (Additional forms may be obtained by means of the transformation formulas of the hyper- geometric function, see [8.1].) The functions YF(8, P:(cos 8) called surface harmonics of the first kind, tesseral for m