of Bessel Functions of Imaginary Argument. 911 



Thenif k = m\n + ^ v 2 = //, 2 -l, 

 T=(v 2 /2n-flV[(v 2 -fF)-^ + X 3 (^ + P)-f + X5(v 2 + /c 2 )-t + ...} 



x [log { , Wt JT^i\-in og ^^^±^ 



+ ^ lo sr^|J' • • • (9) 



where k is less than unity, and the coefficients of types X r , 

 /jir are given by 



X x = l, x 3 = -i(U*-l)/(n + iy-l\ 



4*{(n^-2 2 }X 5 =-6£ 2 (2-3^^ 



and in general 



^(P-l)(r-2)(r-4)(r-6)X r _ 6 + F(2-3P)(r-2)0—3)(r-4)X r _ 4 



+ (r + 2){P + 3# 2 (r-2) 2 -(r-2) 2 }x r _ 2 



+ (r-l){l(^ + l) 2 -(r-l) 2 ^X r =0, . (10) 



whereas the jjl's are defined by the identical relation 



l + /x 1 o- + ^ 2 o- 2 4-.. = (l4-X 3 <r + X 50 - 2 4-...) _1 - C 11 ) 



We proceed to the limit when n is infinite and m finite, 

 so that k = 0. In this case, 



Lt n 2 X 3 = -, 



and so on. In fact, these limits are the coefficients which in 

 the notation o£ previous papers dealing with the Bessel 

 functions of real argument, were denoted by 



X 2 , X 4 , — X 6 , . . . 



with m taking the place of n. Similarly, in the formula 

 for £, n 2 fjui, n*/jt 2 ) . . . must be replaced in the limit by 



-/*2, /*4, -/* 6? • • • 

 where the fis are now the coefficients of earlier papers *. 

 * Vide e. g. Phil. Mag. Feb. 1910, p. 240. 



