[delta] = (a + b) s^2/2ab (1),
so that, if we write
2[pi][delta] = [pi](a + b)s^2 [pi]v^2
------------ --------------- = ------ (2),
[lambda] ab[lambda] 2
the effect at B is
_ _
/ab[lambda]\1/2 / 2[pi]t / 2[pi]t / \
( ---------- ) ( cos ------ | cos 1/2[pi]v^2.dv + sin ------ | sin 1/2[pi]v^2.dv ) (3),
\2(a + b) / \ [tau] _/ [tau] _/ /
the limits of integration depending upon the disposition of the
diffracting edges. When a, b, [lambda] are regarded as constant, the
first factor may be omitted,--as indeed should be done for
consistency's sake, inasmuch as other factors of the same nature have
been omitted already.
The intensity I^2, the quantity with which we are principally
concerned, may thus be expressed
_ _
/ / \^2 / / \^2
I^2= ( | cos 1/2[pi]v^2.dv ) + ( | sin 1/2[pi]v^2.dv ) (4).
\ _/ / \ _/ /
These integrals, taken from v = 0, are known as Fresnel's integrals;
we will denote them by C and S, so that
_ _
/ v / v
C = | cos 1/2[pi]v^2.dv, S = | cos 1/2[pi]v^2.dv (5).
_/0 _/0
When the upper limit is infinity, so that the limits correspond to the
inclusion of half the primary wave, C and S are both equal to 1/2, by a
known formula; and on account of the rapid fluctuation of sign the
parts of the range beyond very moderate values of v contribute but
little to the result.
Ascending series for C and S were given by K. W. Knockenhauer, and are
readily investigated. Integrating by parts, we find
_v _v
/ i.1/2[pi]v^2 i.1/2[pi]v^2 1 / i.1/2[pi]v^2
C + iS = | e dv = e . v - - i[pi] | e dv^3;
_/0 3 _/0
and, by continuing this process,
i.1/2[pi]v^2 / i[pi] i[pi] i[pi] i[pi] i[pi] i[pi] \
C + iS = e ( v - ----- v^3 + ----- ----- v^5 - ----- ----- ----- v^7 + ... ).
\ 3 3 5 3 5 7 /
By separation of real and imaginary parts,
C = M cos 1/2[pi]v^2 - N sin 1/2[pi]v^2 \
S = M sin 1/2[pi]v^2 - N cos 1/2[pi]v^2 / (6)
where
v [pi]^2v^5 [pi]^4v^9
M = - - --------- + --------- - ... (7)
1 3.5 3.5.7.9
[pi]v^3 [pi]^3v^7 [pi]^5v^11
N = ------ - --------- + ------------ ... (8)
1.3 1.3.5.7 1.3.5.7.9.11
These series are convergent for all values of v, but are practically
useful only when v is small.
Expressions suitable for discussion when v is large were obtained by
L. P. Gilbert (_Mem. cour. de l'Acad. de Bruxelles_, 31, p. 1). Taking
1/2[pi]v^2 = u (9),
we may write
_
1 /u e^iu du
C + iS = ------------- | -------- (10).
[sqrt](2[pi]) _/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