106 Samuel Karp, Jack Kotik, and Jerome Lurye 
1/2 F 
| H(B') ¥,(Flsin 8, - sin 6'|) dB' 
0 
fe F 
H(B') v,(Zlsin By 7 sin 6'|) dg' 
Bist F 
+f H(B') v,($ sin Bo San 6 |) dp’ 
Biiy 
Bnei 
ml H(B') ¥,($ |sin 8, - sin 6'|) dp' (i = 2,3,...,m). (B7) 
i+1 
Since the total number of subintervals, n, is even, the first and third integrals on the 
right of (B7) (these contain no singularity) either both extend over an even or both extend 
over an odd number of subintervals. In the even case, the integrals are evaluated by 
Simpson’s rule. In the odd case, they are evaluated by the trapezoidal rule over the first 
subinterval and by Simpson’s rule over the rest. (Note that when i = 2, the first integral on 
the right of (B7) vanishes, while when i = n, the third integral vanishes.) 
The second integral on the right of (B7) is approximated by first writing 
Biay F 
H(B') vy, (F |sin 8, - sin e't) dB' 
Bi. 
Bi+y F 
© H, I Y ( |sin 8, - sin 6 |) dB’. (Ba) 
i-1 
This approximation is sufficiently accurate for our purpose when n> 6. The integral on 
the right of (B8) is evaluated by expanding the Bessel function about the point B'=B8. The 
result is 
Biks 2F2 2 
ie ae 2 my! F il 
(elfen ane =n tana 
2p2 2 
2: 2 1 - , )- 1 cos B, , B 
oe Wan pied) CLE BY Awe s : 9 
= ie + s(t + tan“6, ein (1 at ,n) (B9) 
where log y’ = y = Euler’s constant = 0.577... . 
