3 
ft-HhMW’HHNF-« 
is convergent since its argument is numerically < 1. Comparing 
3), 4) we see each term of 3) is numerically < the corresponding 
term of 4) for any \x \ < Gr and any a > /3. Thus the series 3) 
considered as a function of a is uniformly convergent in the 
interval (/3 + oo ) by 136, 2; and hereby x may have any value 
in (— G, G). Applying now 146, 4 to 3) and letting a = -f oo, 
we see 3) goes over into 2). 
3 x 2 
7. 
For 
lim xF[ a,«,-, 
a=4-oo \ 2 
= sin X 
xF[a, a, 2; ~-~)=x 
= G- > 0 and a = Gr. Then 
GF { a ' & ’l’ i) 
r , a 3 , f. , iv(? ± 
ff + 3T + l 1 + W5T + 
is convergent by 185. We may now reason as in 6. 
8. Similarly we may show : 
I "1 
lim F[a, a,, -; - —— ] = cos x. 
a=+ao \ 2 4 a 2 
f 3 a? 
lim F{ a, a, -, —- )= sinh x. 
a=+oo \ 2 4 a 2 , 
lim f(k, a, i, = cosh x. 
a=+ao V 2 4 a 2 / 
187. Contiguous Functions. Consider two F functions 
F(a, A 7 5 , F(a\ /3', 7'; z). 
If a differs from a' by unity, these two functions are said to be 
contiguous. The same holds for /3, and also for 7. Thus to 
F (ufi'yx) correspond 6 contiguous functions, 
F(a ± 1, /3 ± 1, 7 ± 1; x).