447 



As in general we have 



d dlogu 



— <pk^i (m) = <pk (u) —J— , 

 dz dz 



it is possible to extend partially the above relations to the function 

 (ft{z\, and indeed it follows that 



<Pt (^) - <f\ (t) = ~ ^ *°S' z + hm log' z + 2? (2) log z, i 



.f, {"-^^ -^ ^/'.(i-^) 4- r, (^) = i log' z - i log' z \og{\~z) 4- / (2) 



-fg(2)logz + i;(3), 



rz—1 



bor /(;^3 a linear relation between (fk{^),'l^k{^ — ■z)ix\m(^k 



no longer exists, there only remains, besides the relation 



1 

 Cpk {Z) + Cfk {—z) — -^^^^ (pk (^') 



an equation of the form 



r/, (z) + (-l)^- Cfk (^yj = - (2 :ri)>^gk (^-^^ . 



where ffki'^t) denotes the differential coefficient of Bkrnoulm's 

 polynomial /k{it). 



Another expansion valid for all positive and integer values of k 

 is the following: 



+ y ^=^?(A-n) (3) 



Here the right line (0, — oo ) must be considered as a barrier in 

 the complex //-plane and log y is real for positive values of y. The 

 accent in 2^' denotes that the term with the index Ji =z k — 1 is to 

 be excluded. As for the numerical values that the ^-function takes for 

 the value zero and for negative integer values of the argument, 

 we have 



?(0) = -i , C(-2A)^0 , 5(l_2A) = (-l)^^. 



Therefore after a certain stage the coefïicients in the expansion at 

 the right hand side of (3) are expressible by Bernoulli's numbers 

 and the radius of convergence of the expansion is evidently 2.t, 



29* 



