But by choosing j = N in eq (82) we see that terms 1 to N for C2N equal terms 2 to N+1 in 

 E, so 



E = -^N,N^2N"^^2N • 



However, since B^ N' ^^^ coefficient of y'^ in Bj^, is always 1 by eq (80), E = 0. Therefore 

 bjsj+i does not exist in C2N+1. 



Number of Terms 



The number of terms or stages to use in the continued fraction was arrived at by a 

 trial and error process. For a given number of terms, a real positive argument was decreased 

 until the accuracy began to drop. The magnitude just before this drop was considered to be 

 the optimum point to increase the number of stages by one. Because the argument to the 

 continued fractions is z^/2, v/e took the larger of the magnitudes of the real and imaginary 

 parts of z3/2 as the test number. This number is then compared to the 3/2 power of the 

 points determined along the real axis by trial and error. 



The above method appears to work well although it involves no thorough under- 

 standing of the way complex numbers affect the successive convergents of a continued 

 fraction. Table 2 shows the points down to which a given number of stages gives full accu- 

 racy for positive real arguments and hsts the 3/2 power of these numbers as used in the 

 FORTRAN program list called ZMLA5. 



Division Lines 



The power series method is now to be used for small arguments and the continued 

 fraction method for large arguments. The exact dividing line between them is needed. The 

 division hne of figure 4 was arrived at by computing the functions along rays from the ori- 

 gin, using both power series and continued fractions. The number of decimal places to 

 which the functions determined by the two methods agree tends to reach a maximum at 

 some distance from the origin along each ray. At distances short of this maximum we can 

 assume that the continued fraction method is less accurate than the power series. At dis- 

 tances beyond the maximum, the power series is assumed to be less accurate. The maxi- 

 mum therefore indicates the ideal place to change from one method to the other if the 

 decision is to be based solely on accuracy. This method was used to determine figure 4. 



A compHcation arises, however. Along certain rays from the origin, hj and its deriva- 

 tive reach a maximum number of places at very different distances from ho and its deriva- 

 tive. The principal problem is at ±60° but persists from about 30° to 90°. At 60°, hj is 

 small in magnitude and h2 is large. The power series method cannot compute the small 

 values accurately due to loss in accuracy in subtraction in eq (48). The accuracy of the 

 continued fraction for h2 is poor at 60° because eq (69) becomes a nonalternating series and 

 continued fraction approximations are not known to improve the accuracy of nonalternat- 

 ing asymptotic series as they do for alternating series. 



A reasonable solution to this problem is to compute h| by continued fractions and 

 h2 by power series for arguments at these angles and magnitudes from 4 to 10. However, as 

 will be shown later, the above solution has not been employed at this time since this area is 

 not of great importance for normal mode computations. Instead, the argument was chosen 



32 



