238 
POWER SERIES 
We shall show in the next chapter that 
lim II (n, x) 
71 = 00 
exists for all x different from a negative integer. Gauss denotes 
it by II(V); as we shall see, 
r (x) = II (x — 1) , for x > 0. 
Letting n = oo, 6) gives 
F(a, /3, 7) 
11(7 — 1)11(7— a ~~ fi — 1) 
n (7 — « — 1) TI (7 — /3 — 1) 
We must of course suppose that 
7, 7 - 7 “ A 7 - a - £, 
are not negative integers or zero, as otherwise the corresponding 
n or F function are not defined. 
Bessel Functions 
193. Ï. The infinite series 
J n (x) = X n ï(- 1)‘ 
X 2 ' 
n = 0, 1, 2 ••• 
(1 
s=o 2 n+2s s! (n + s') ! 
converges for every x. For the ratio of two successive terms of 
the adjoint series is | x 12 
2 2 (s + l)(w + s + 1) 
which = 0 as s = 00 for any given x. 
The series 1) thus define functions of x which are everywhere 
continuous. They are called Bessel functions of order 
n— 0, 1, 2 ••• 
In particular we have 
4,0*0 = 1 
+ 
xi 6 
2 • 2 2 2 • 4 2 2 2 • 4 2 • 6 2 
x? ¡r 6 x 
+ 
J ( 1 ^ 
1 W ~ 2 _ 2 2 • 4 ' 2 2 • 4 2 .6 2 2 • 4 2 • 6 2 • 8 
+ 
(2 
(3 
Since 1) is a power series, we may differentiate it termwise and 
get J'(x) = V , 
) -72"+^«!(» + «)! ■ *