|
< _/0 [sqrt] u
Again, by a known formula,
_[oo]
1 1 / e^-ux dx
-------- = ---------- | -------- (11).
[sqrt] u [sqrt][pi] _/0 [sqrt]x
Substituting this in (10), and inverting the order of integration, we
get
_[oo] _u
1 / dx / e^u(i - x)
C + iS = ------- | -------- | ----------- dx
[sqrt]2 _/0 [sqrt] x _/0 [sqrt]x
_[oo]
1 / dx e^u(i - x) - 1
= ------- | -------- -------------- dx (12).
[sqrt]2 _/0 [sqrt] x i - x
Thus, if we take
_[oo]
1 / e^-ux [sqrt](x).dx
G = ----------- | ------------------,
[pi][sqrt]2 _/0 1 + x^2
_[oo]
1 / e^-ux dx
H = ----------- | ------------------ (13).
[pi][sqrt]2 _/ [sqrt]x . (1 + x^2)
0
C = 1/2 - G cos u + H sin u, S = 1/2 - G sin u - H cos u (14).
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 = ----- - ---------- + ---------- - ]]>
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