Bessel Fwictions of Fractional Order. 1021 



Accordingly 



\/c -\-kn 



- (3) 



2 

 If we put 



l~n_ 



then 



l + 2 5 



71= — . 



1+5 



Substituting this expression for n in (3), we obtain 



z 1 = 2 s / / (l + s)[c ^k+(c + 2k)s]. . . . (4) 



This formula was already given in my previous paper 

 mentioned in the footnote. 



The accuracy of the approximate value of z l given by (4) 

 depends of course on the proper choice of the .constants 

 c and k. For example, take c = 2'471 and &=— 1*032; 

 then 



Zl = 2y/(1 + *)(l-439 + 0-4075), ... (5) 



and the values of Z\ given by (5) are shown in the third 

 column of the following table, the difference between the 

 correct and approximate values being given in the last 

 column. 



Values of z v 



s. , * N Difference. 



Correct. Approximate. Correct— approximate. 



-| 1-571 1-572 -0001 



-f 1-700 1-700 0-000 



-| 1-866 1-864 0-002 



2-405 2-399 0-006 



| 3-142 3-139 0003 



1 3-832 3-843 -0-011 



By actual calculation it can be further shown that (5) 

 maybe used even in the case of .?=— § probably with a 

 similar degree of accuracy to the case of 5 = 1. Thus the 

 approximate equation (4) or (5) may be very conveniently 

 used for calculating the first root of Bessel functions of 

 fractional order, if minute accuracy is not wanted as in the 

 problem referred to at the beginning of the present note. 



Yours faithfully, 

 Kyushu Imperial University, AKIMASA ONO, 



Fukuoka, Japan, 

 April 2, 1921. 



