119? 



«ft „2 OO „2 



te * sin .vu sin uvdii = i | ^ * \cos (x — v) u — cos {x -f ?') m] du 



which may be reduced by means of the relation 



~2p 



I g—p-u- COS Xpudii = — ^4 ^^) 



In this way we get 



00 ,.2 , 



C-- ^^ 



\e ^ COS XU COS Uvdu = -^^ g-x'^—v^ (g2.rr _|_ g— 2.r(>) 



o 







V-: 



r-- . y^ , 



I e ^ sm xu sin uvdu z= g—x^-v'^ ^gSxr _ g-2xy) 







and 



GO s„2 



1 r-~ 



P = e'^' I e "^ cos av (e^^^ + g-^^^i') dv 







00 5, .2 



. 1 r-— 







To evaluate these integrals we may remark that the relation {a) 

 holds for complex values of ?.. Putting therefore X = a -]- ih and 

 equating the real and imaginary part in both members of the equation, 

 we obtain 



a2 62 



e— /''«- cos apu {e'^P"' -\- e-'^P'^) du =z e '^ ^ cos - 



P 







00 



e 4 4 s^>j _ 



P 2 







(^) 



which reduce the values of P and Q to 



1 ^^^ 4«.. » . i/2^.(..)//2^(«) 



i"^ =r - — e ^ cos = 2: (1)'^ . . . (19) 



(J = e •> S171 =rr ^ ( — 1)'^ (JO) 



1/5 5 0^ ^ (2/.-+1)/ ^ 



Investigating in the same way a second series 



