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Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables (AMS55)
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8. Legendre Functions
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
(1-2) d2w --2 z dw -+[v(v+l)--]w=O EL2
dzZ dz 1- 2 2
(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<T, 18% Zl<T
(22- l)tp= (2- 1)tr(z+ l)*P
(For P:(z), p=O, Legendre polynomials, see
(I 1 - 4 <2>
(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++)
(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<n and sectoral
unit sphere z2+y2+ z2= 1 where x=sin 8 cos 9, y=sin e sin 9 and z=cos 0.
With 0 < 0 < ~ , 0 < 9 < 2 ~ , they are everywhere one valued and continuous functions on the surface of the
'See page n.
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