Functions of Equal Order and Argument. 527 



The following values calculated from these formulae may 

 be compared with those * found from Gr (n) and Gi(n) by 



means of the recurrence formula, G n+1 = — G n — G n _i. 



n. 



».(»)- 



<*.-!<«)- 



7 



0-636755 



0-313070 



8 



0-608951 



0-314261 



9 



0-585445 



0-314077 



10 



0-565195 



0-313060 



11 



0-547485 



0-311526 



12 



0-531806 



0*309676 



13 



0-517782 



0-307632 



Gr»_i(n) has a maximum value near rc = 8'3.16. 



To find the first root of G ]0 o(^), the values of G 104 (104) 

 and G 103 (104) are calculated, and from these results functions- 

 of higher and lower orders are readily obtained. 



n. 



(*,(104). 



n. 



GJ104). 



100 



0-025594 



104 



0-258794 



101 



0-091802 



105 



0-309836 



102 



0-152713 



106 



0-366837 



103 



0-207751 



107 



0-437957 



Bessel's formula (15) is equally applicable to the G and 

 Y functions, giving finally 104-380, as the first root of 

 GiooOO- 



Since the Neumann function f Y n (V) is given by 



Y n (z)=(log2-y)J n (z)-G n (z), . . (20> 



expressions for Y n (n) and Y n _i(n) can be found by substi- 

 tuting (11) and (17) or (14) and (19) in (20). The first 

 root of YxooCz) calculated as in the previous examples is. 

 104-133. 



It can be shown that p m , the mth. root of J n (z), is given 

 very approximately by the formula 



p m = n sec (f>, (21), 



where tan 6 — 6= ^ m ~" ^ (22) 



4:11 V ' 



* Britisli Association Report: Calculation of Mathematical Tables, 

 1914. 



t Gray & Mathews, ' Bessel Functions/ pp. 14, 242. 



