Integration in the Problem of Diffraction. 377 



the approximate value of J- m (z) is given by the equation 



J_ m (z) = [a) — • V ^^ • e" n lo ° m ~ m , 



\&i mir 



and this does not vanish at infinity, when the real part of m 

 is positive. 



However, since 



U »W=2si^( J --^-- i "' J ^> 



the expression for J m (z')V m {z) simplifies, and we find its 

 approximate value to be given by 



J, 



1 ft' 



■W D -W-5ii© 



Hence this product vanishes at infinity, when the real part 

 of m is positive, provided that z>z' . 

 Further, it is readily shown that 



cosw ( tt — + 0') 

 sin mir 



vanishes at infinity, when < 9 — 6 f < 2ir. Thus 



1 f cosm(7r- 0+0') T h >\tt n \i ( v>v '\ 

 2i^} c ' ; sW - J -^) L ^(^-'^>W' 



over the path (C) of fig. 1 is equal to the sum of the residues 

 of this function. Therefore 



1 C cos«i(7r — 6 + 0') r , 7 , TT ' ; , Lr>r'\ 



over the path (C) of fig. 2 is equal to 



i [j (kr')XJ (kr ) + 22 B=1 J„ (W) Vjkr) cos n (0- 0')] , 



the path ((7) differing from the path (C) by the removal of 

 the small semicircle at the origin. 



We have therefore obtained the following expression For 

 U (£T1) as a contour integral: — 



1 f GO *m('n—0 + ff)j n a'tt ii w (r>i J \ 

 I l, siu^vr J ^ kr ^>>>^>»\0>0>h 



c C 



