448 



as we might expect since equation (3) may be established bj 

 integrating- repeatedly both sides of the eqnatioii 



«=00 



e-'!i . . , , -c-i (— 1)"S» ■" 



y'::i = -iog,/ + iv + 5:^^.'^.(o<y<2^). 



By substituting 2 = ^ in the formulae (1) and (2) we find 



<f^ (0 = Ï log» 2 -i C(2) log2 + ^ £(3), 

 and then, remembering tiiat 



log log 



2 = _i,og2+"vti)!^".*'°i^, 



we deduce from (3) by taking yz=i\o^1 



I log 2 '^ ( - 1)"-^ ^in (log2)2"+i 



«=i 



>i= CO 



I log' 2 ^-. (-IV'-i^,, (og2p+2 



C (3) = I log» 2 + 4 J -°— + \ — ^' . ^— ^-^ . 



- V ; 7 '^ë ^ I 2 ^ ^ 2n / 2n 4- 2 



The index of these expansions is about ^^V a'x' the error involved 

 in neglecting terms beyond the second does not affect tlie fifth 

 decimal. 



It is possible to connect these expansions with the equation 



— Coth — = 1 + > 



(_l)n-li^„ 



2 2 ^6—' 2n! 



and to deduce in tiiis way definite integrals representing ?(2) and $(3). 

 Thus we arrive in the first place at the well-known formula 



1 



,,2) = 2j'°i<L±i)^. 







but we get also the less familiar result 



2 3 n — 1 



"=' ' ' " 



Further, we may prove Eulku's result 



1 





