DIFFERENTIAL EQUATIONS 2OQ 



# 



2. 



dS n (x) n 



dx 



S n (x) - 





9.30 The hypergeometric differential equation: 



9.31 The equation 9.30 is satisfied by the hypergeometric series: 



(a + 2) /3(ff + i) (/3 + 2) 





The series converges absolutely when x<i and diverges when x>i. When 

 ^ = +i it converges only when a + /3 - 7<o, and then absolutely. When 

 x = i it converges only when a-{-/3 Y" i<o, and absolutely if 

 a + ]8 - 7<o. 



9.32 



r(7)r(7--0) 



9.33 Representation of various functions by hypergeometric series. 



(i + xY = F(-n, ft ft -*), 

 log (i + x) = afF(i, i, 2, -j), 



Limit / o x 



