522 Dr. J. R. Airey on Bessel and Neumann 



do not differ widely : 



J»W = -3 X /^ £ f !i) (J i + J- i ), ... (6) 



the J functions of fractional order having the argument 



" , where a= —p. 



From a consideration of Sommerfeld's Integral, Debye * 

 has given the following formula for J ?l (Y) when n and z are 



nearly equal. When - = 1 — e, 



J -«-sS«-wg) ¥ r(!±i).i»(. + i,J, (7) 



where 



B„(«) = 1, 





B t («) = ee, 





B 2 (e,)=f - 



1 



"20' 





~I5' 



-D / N <*V 



€ 2 £ 2 



TAe Bessel Function J n (z) . 



These results can be obtained very easily by evaluating 

 the integral 



J»(s) = — 1 cos (.srsin w— nw) die. . . . (8) 

 In the case where n and z are large and equal |, 

 J.W = -J o oos»(8m»-«)& =-^ cos ^ T -120 + 5^0 •)^- 



Changing the variable by writing 



x iv' s iv 5 to 7 6% w 5 w 7 



^ = ¥~l2d + 5040'" ° r Ti Z=ZW ~2"0 + 84b""' 





+ 



60 ' 8400 "■• 



* jlf^/i. Annalen, lxvii. p. 557 (1909). 

 f Lord Rayleigh, Phil. Mag. Dec. 1910. 



