ÖFVBRSIGT AF K. VETENSK.-AKAD. FÖRHANDLINGAR 1901, N:0 9. 689 



If we liad difFerentiated the equation (6) directly in the 

 same mariner we sliould have obtained 



and 



^ = -^-+i^9(.™), (1&) 



where we have introduced the abridged notation 



fji\ nii. 



2^. '"' 



,e 



because this function will be frequently used in the following. 

 The points of infinity of the cotangent being all real, 



cot -—r — - cau only become infinite, when I,, is real — t being 

 2m 



a. real quantity. 



But when X.,, is real we have 



i^J:cot^=^ = 



i 2m 



and hence the sum in the right member of (16) ahoays has a 

 ånite value. 



When Ir is real the modulus of e "' is unity. On the con- 

 trary for 1.,. imaginary the modulus of the corresponding root is 

 greater or less than unity according as RiX,, is positive or 

 negative. 



In order to find a development of cot —p: — - when "k,, is 



2m 



imaginary we start from the relations 



_ t — I,. 



t — lr .1 + e'^' ~^^ 



cot —^ = ± I — 



2m 



1 — e 



_ t — X,, 



+ i 



