32 



REPORTS ON THE STATE OF SCIENCE. — 1915. 



Tables of the Bessel Functions J„(a;) — continued. 



Table II. 



The Neumann Functions 6o(a;), Gi(a;), Yo(a;), and Y|(x). 



The Neumann Functions or Logarithmic Bessel Functions, G(,{x) 

 and Gi\{x), were calculated to twelve or more places of decimals for values 

 of the argument 6-5 to 15-5 from their asymptotic expansions. 



Go(x) = 





ix) sin 



- ^j + Qo(a;) cos (a; — J 



X — 



and G,(x) = + a/'^ \'P\{x) cos L-''')- Q,(x) sin 



If the calculation be restricted to the convergent part of Po(a;), Qo(a;), 

 &c., the value of Go(65) can be found to about seven places of decimals, 

 but it is possible to transform the divergent part of any one of these 

 series and express it as a fraction of the least term of the convergent part 

 of the series.' The fraction or 'converging factor' is a function of x 

 in descending powers of the variable, the absolute term being |. By 

 this means, Gn(6'5), &c..can be calculated to about twelve or more places 

 of decimals. The functions Yu(x) and Y,(a;) were then found from the 

 relations Yo(x) = (log 2 - -y) ,Jo(x) — Go(x), &c. 



' Annates de I'Ecole Normale Supirieure, 1886; ArcMv der Math, und Phyaik, 

 1914. 



