194 



so tliat we can write for (9) 



/2« 







Substituted in (8) we get the double integral 



00 „2 00 



gx2 r -— r 



. r/)„(.f'') =: I e ^ ii'-'-+'^ COS xudn \(:—'''<f„{ — f-)(U 







if we introduce /y = ut. 



By substitution of u =: 2// it passes into 



00 00 



ff,.{w^) = \ e-y'' y^>'+^ cos 2,vydy i e-K>/hf ,,{—f)dt . . (10) 



nf \/:tJ J 







Now we make use of the relation ') also deduced bj Prof. Kaptein 



l-,^^^M=l->'''^M (II) 



c . u 



In (10) we substitute x = Vt and then multiply i)0th members by 



1 



e-' dt 



and integrate between and x. then we get by making use of (11) : 

 (1+^f+i n!\/:x.J ''J ' ' ' J 1-i-t 



Q u 



According to a well-known integral in the theory of the integral- 

 logarithm, is-) 



00 00 



Jcos 2y[/t rxcos2yx ^ „...„^ o;•/o^ 

 ^^ dt = 2 — dx = — e-2" /«. (g2//)_e2^ /i (g-2^/) 



U 



consequent!} 



00 00 I 



J (1 + 0"+' n![/jrj j 



U 00 / > 



. (e-2.'/«^„( — ?<-)(:??< I 



n ) 



By summation from 7i = to y^ :^ x we find: 



i) Lc. XV, p. 1250 (14). 



~) See for instance "Theorie des liitegrallogarithmus Dr. Nielsem page 24. 



(12) 



