From Eq. (5-63) we may derive
2f+ P(cos7) = Y*'ll ) l (, ) (5-93)
2 1m
2 + sin0 (cos7 sin0 y 0')) (5-94)
where cos 7 = cos 0 cos 0' + sin 0 sin 0' cos( Q'). In the coincidence limit x' xa ,
we have cos7 sinO'cos(4' Q*) cos where = Jo JrA/f (r2 + J2),
with the definition of the rotated angles (0, K) given in Eq. (4-151). Using the
chain rule along with Eq. (4-151), we may rewrite Eqs. (5-93) and (5-94) in the
coincidence limit as
o [0 47r dX
Y (K om, B] o 4(o--)sin 8 cos X P (X), (5-95)
L 2a+1 d
sin 0 Y*(00, ) 2 sin sin s d Pd(X), (5-96)
ao 0 47 wX
where X = cos 0. Using Eqs. (5-95) and (5-96) for Eqs. (5-91) and (5-92),
respectively, in the coincidence limit x' x4, we have
hs4ev 2f+1 ddX
h- 4) d+F(1, J) sin cos 1 P(X), (5-97)
hs od 4 ( + ) dQG(0, ) sin 8sin d P (X), (5-98)
where the angles were rotated in the integrals, from dR' = sin O'dO'dq' to dQ =
sin OdOdK, and accordingly the integrands changed their variables,
F(0, ) = [cot O'hs, + ho, + csc2 0'01h,] (0, ), (5-99)
G(0, ) L [csc0' (0o, hs, a00ho,)] (0, ). (5-100)
One should note that Eqs. (5-97) and (5-98) contain dPe(X)/dX rather
than Pe(X) unlike Eq. (5-66) in the case of scalars. These first derivatives of the
Legendre polynomials are orthogonal over (-1, 1) with weighting function 1 X2