INFINITE SERIES OP ANALYTIC FUNCTIONa 487 



For the serin 



the circle of convergence is a cut, and the function defined by the series is regular 

 within the circle, but has a line of singularities round the circle that are everywhere 

 dense. 



Hence the series 



) ........ (13) 



Ml 



defines a function that is regular within the two loops of the oval of < ' \-i\i 



where x = f+nj, which has singularities everywhere dense round the t\vo loops, and 

 which consequently cannot be continued across the boundary of either of the loojw. 



Hence we have constructed an analytic function that exists within two separate 

 regions, is the same function in Initli regions, but cannot be analytically continued 

 from one region into the other. 



6. Consider the equation 



g= jP Vy 2 *- .......... (14), 



which is satisfied by the Bessel functions 



n is a positive integer, p is real, and the cases /> = and (2/>)~' an integer are 

 excluded. 



Take as the standard solutions 



and 



From the asymptotic expansions of the Bessel functions, or from Theorem I., we 

 find that when n is large 



and 



'/(*) = A/ - sin - 



V 717T -/> 



approximately. 



Hence the series of Bessel functions 



