The constant parts in (14), viz. 1/2, may be determined by direct
integration of (12), or from the observation that by their
constitution G and H vanish when u = [oo], coupled with the fact that
C and S then assume the value 1/2.
Comparing the expressions for C, S in terms of M, N, and in terms of
G, H, we find that
G = 1/2 (cos u + sin u) - M, H = 1/2 (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 = 1/2.
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
_ [oo]
/ -y q-1/2
| e y dy = [Gamma](q + 1/2) = (q - 1/2)(q - 3/2) ... 1/2[sqrt][pi];
_/0
and we get in terms of v
1 1.3.5 1.3.5.9
G = --------- - ---------- + ----------- - (16),
[pi]^2v^3 [pi]^4 v^7 [pi]^6 v^11
1 1.3 1.3.5.7
H = ----- - ---------- + ---------- - (17).
[pi]v [pi]^3 v^5 [pi]^5 v^9
The corresponding values of C and S were originally derived by A. L.
Cauchy, without the use of Gilbert's integrals, by direct integration
by parts.