396 ON LEG-ENDRE'S FUNCTION Pn(0), [404
But whether n be even or odd, (12) fails when 6 is so small that n6 is not moderately large. For this case our original equation (1) takes approximately the form
dzu 1 du
where a2 is written for n(n + l); and of this the solution is
u = J0(a8) .............................. .(14)
It is evident that the Bessel's function of the second kind, infinite when 6 — 0, does not enter, and that no constant multiplier is required, since u is to be unity when 6 = 0. For a second approximation we replace (13) by
dht, I du „ _ du /I cos 6\ _ 6 du _ ad T, , ...
w + dM + a~u-de\d~^re)~3dd~-^'Jo(a°)'
or, if ad = .z,
d?u I du z T K \
5?+JS+tf!5S3*'/'(*)
In order to solve (15) we assume as usual
u = v.J0(z) ............................... (16)
This substitution gives
_ Za? J0'
a linear equation of the first order in dv/dz. In this
, i , dv A
80that -
Here
Thus ^-^--L^:, .......
t>
which has now to be integrated again.
. = ~~jf +
___zjj _ f 7 _ zJ<>' z-
regard being paid to the differential equation satisfied by /„3] • ' _]