204 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS 



9.210 The differential equation: 



d 2 y i dy ( 



dx 2 x dx \ x 2 { 



with the substitution, 



x = iz, 

 becomes Bessel's equation. 



9.211 Two independent solutions of 9.210 are: 



IP (x) = i~ v J v (ix), 



V 7T I 



9.212 If v = n, an integer: 



'<*>- 2 in^fn 



k =o 



9.213 



/,(*) 



9.214 If ^ is large, to a first approximation: 



In (X) = (2TTX COSh #)- ^ ( cosh ^ - sinh ^) y 

 ^n (*) = TT(27TX COSh jS)-^- 1 (cosh ^ - ^ sinh /S) 



w = OP sinh j8. 



9.215 Ber and Bei Functions. 



ber jc + bei x'= I (xVt), 



ber x - i bei x = I (ix\/i), 



I X 



