ensity 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
_