Functions of Equal Order and Argument. 525 



In the important case where the order of the function is 

 n — 1 and the argument n, 



420 W W 8100 W \3/ 



23 /6\l_/2\ 9-1-7 /6\!;° ,1\ 1 ,,., 



+ il340oO r (3) + 7484i000U) r (3>"-J- " < U > 



The numerical values occurring in the above formulae are 

 logr(i)==0-42796 27493 1426 : r/|)=2'67893 85347 077 



Iogr(|) = 0-13165 64916 8402 : f/|)=1-35411 79394 264 



The formulae (11) and (14) give the values of J„(n) and 



2n 



*J n _!(n), and the recurrence formula J n -i J«4-j n+ i = 



enables one to find functions of higher or lower order. In 

 this way the behaviour of J„(#) in the neighbourhood of its 

 first root can be investigated. For example, in calculating 

 the first root of J\qo(z), Jio9(109) and J 108 (109) are first 

 found, and from these values the functions J 107 (109), &c, 

 •down to Jioo(109) as follows : — 



n. 



109 



J w (109). 

 0-093639 



n. 



104 



J„(109). 

 0-134966 



108 



0-111472 



103 



0-115672 



107 



0-127260 



102 



0-083643 



106 



0-138378 



101 



0040870 



105 



0-141879 



100 



-0-007901 



Bessel's addition theorem * for the J functions, viz., 



J.(*+*)Hp7^)" [ J "W- ^Jn + i(z) + ~ J*wW •-..], (15) 



, h(2z + h) 



where a= — -, 



then gives the value of A, ( — 0-164...), which makes 

 ■Jn{z + h) = when ?i = 100 and £ = 109. The first root of 

 JiooCO i s therefore 108*836. By the same method, and by 



* Abhandlungen der Berliner Akademie, 1824 ; Nielsen, ' Theorie der 

 CJylinderfunktionen,' S. 266. 



