9-10 Dr. J. W. Nicholson on the Approximate Calculati 



i on 



With these definitions, a well-known formula due to Heine 

 shows that 



I„,(.r) = Lt n~ m PjYcosh-), . . . (5) 



n = » \ 11/ 



and a companion formula may be readily derived as follows : — 

 When fi is greater than unity, and 0i+i, n— m + 1 are 

 positive, Hobson * has shown that 



VUW- 2« «r(n-m)«r(m-±)^ J> ' J ^ + ^/ M «-l.oo»h »)•+-« 



. . . (6) 



where ot(.v) is Gauss' ['unction, identical with r(s + l), or if 

 s be an integer, with s ! 



\\ rite /a = cosh a //, where n tends towards infinity. Then 

 {u 2 — !)*"■ tends to the value (xjn) m 9 and v(n-f-m)/v(n— m) 

 to the value »**. 



Thus 





sinh 2m wrfw 



"■o Lt (l+-coshuA' 



lint 



n + tn + l 



p-x cosh rc 



( 7 , \ 7i + m+1 



1 + - cosh IV I 

 » / 



and therefore we deduce by the definition (3), 



K m (x) = Uj-™n^Q%(cosh~y . . (7) 



which is the required companion formula to (5). 



Asymptotic expansions. 



It is now possible to derive the asymptotic expansions of 

 the Bessel functions \ m (x) and K TO (V) from those of the 

 Legendre functions. The latter will be quoted from the 

 writer's paper f, for the case of argument greater than unity. 



Writing 



* L. c. ante, p. 500. 



t Quarterly Journal, April 1910, pp. 250-252. 



