C = ½ − G cos u + H sin u, S = ½ − G sin u − H cos u (14).
The constant parts in (14), viz. ½, may be determined by direct integration of (12), or from the observation that by their constitution G and H vanish when u = ∞, coupled with the fact that C and S then assume the value ½.
Comparing the expressions for C, S in terms of M, N, and in terms of G, H, we find that
G = ½ (cos u + sin u) − M, H = ½ (cos u − sin u) + N (15),
formulae which may be utilized for the calculation of G, H when u (or v) is small. For example, when u = 0, M = 0, N = 0, and consequently G = H = ½.
Descending series of the semi-convergent class, available for numerical calculation when u is moderately large, can be obtained from (12) by writing x = uy, and expanding the denominator in powers of y. The integration of the several terms may then be effected by the formula
| ∫ | ∞ | e−y yq−½dy = Γ(q + ½) = (q − ½)(q − 3⁄2) ... ½ √π; |
| 0 |
and we get in terms of v
| G = | 1 | − | 1·3·5 | + | 1·3·5·9 | − ... (16), |
| π²v³ | π4v7 | π6v11 |
| H = | 1 | − | 1·3 | + | 1·3·5·7 | − ... (17). |
| πv | π³v5 | π5v9 |