354 
CALCULUS 
(a) Bessel’s Equation : 
(Py 1 dy 
dx 2 x dx 
(6) Legendre’s Equation : 
— j (1 — X 2 ) ] + Wl (m + 1) y = 0. 
dx | ; dx j 
The first of these cannot be solved in terms of the elementary 
functions. It defines a new class of functions, the Bessel’s Func 
tions of the First Kind, denoted by J n (x), and those of the Second 
Kind, denoted by K n (x). 
The Series for J Q (x). On setting n = 0, Bessel’s Equation be 
comes : 
(1) d! 2'+i't +v = 0. 
v dx 2 xdx 3 
Let us see if we can obtain a solution of this differential equation 
in the form of a power series, 
(2) y = «„ -f oqx + a 2 x 2 + .... 
Writing (1) in the form 
we compute the left-hand side of (3) by means of (2) : * 
— 2 • 1 a 2 x -f- 3 • 2 a 3 x 2 -f- ••• -f- (a -f- 2) (n + l)a n+2 x n+1 -f- ••• 
j = ffli + 2a 2 x -f 3a 3 æ 2 + ••• + 
( n + 2)a„ +2 x n+1 + ... 
dx 1 81 
xy a 0 x —j— oqx 2 -|— ... -|- 
a n x n+1 + ... 
On adding these three equations together we obtain a single 
power series in x, whose constant term is a 1 . The coefficients of all 
subsequent terms are given by the formula : 
(n + 2) (n -f-1) a n+2 -f- (?i -f- 2) a„ +2 a n = (n -f- 2) 2 a n+2 + a n . 
Set each of these coefficients equal to 0 ; thus 
* We assume here without proof that a power series can be differentiated term- 
by-term, as if it were a polynomial. —The object in writing the general term as 
the one in x« +1 , rather than as the one in x n , is to obtain a somewhat simpler form 
of the relation between the coefficients of (2).