( 348 ) 



infinity, circulating around the origin in the positive direction, and 

 returning to negative infinity again ; thus 



2jti r* r w 



= e 1 ": I fi-'t-- dt — e~ mz I e-H-- dt 



r(z) J J 







and if z =z a -f- iv 



2m / . . \ f / , \ 



= r- TU) \ cos (mt)-\-i sin (mi) I I c~ <t~ u l cos (dq t) — / sin (vlq t) <lt 



u + iv) V )J \ J 



o 



— <•""( cos (mi) — i sin (mi) J f e~' t' u \ cos (vlg t) — i sin (vlg t) I 



n 



dt. 



Writiiu 



J e-H-" cos (vlg t) dt = M 



o 



I e~ { t— u sin (vlg t) dt = N 

 o 



2üT 



I\u-\~iv) 



a -\- ift 



we obtain 



a = (e 71 » -f e— nr ) sin (mi) M -\- (e" r — e-~") cos (jtu) N 

 ft = (e ne — «- 71 ") cos (mi) M — (e-<' -\- e-' 1 ') sin (mt) N 



and 



«* -f f = f/™ — 2 cos 2mt -f e~^") (M i + A 72 ). 

 Now we have 



J/ 2 = I c' — x arr~ u cos (v g x) dm . I £~~ V y ~ u cos (vlg //) dy 

 o 



A 72 = I «-* «— « sin (vlo x) dx . J e—y y— u sin (vlg y) dy 







so 



J/ 2 -f A' 2 =| / ,---'+.'/ {.>■,,)-» cos f vlg !l - j ctefy 

 ii ii 

 or in polar coordinates, putting 



# z=r ;■ cos 0, // = r .iin 6 



M 2 -|- 2V S = f C e -r(cose+m,e) (,.* s j„ ö ,.,,, ffy-u ,.„,. (,./„ ^ #) ,,/,.,/# . 



This double integral may be reduced to a single one, for 



