Numerical recipes

6.9 Fresnel Integrals, Cosine and Sine Integrals 255
6.9 Fresnel Integrals, Cosine and Sine Integrals
Fresnel Integrals
The two Fresnel integrals are defined by
� x
C(x) =
0
�
π
cos
2 t2� dt, S(x) =
� x
0
�
π
sin
2 t2� dt (6.9.1)
The most convenient way of evaluating these functions to arbitrary precision is
to use power series for small x and a continued fraction for large x. The series are
�
π
�2 x
C(x) =x −
2
5
5 · 2! +
�
π
�4 x
2
9
9 · 4! −···
�
π
� 3 x
S(x) =
2 3 · 1! −
�
π
�3 x
2
7
7 · 3! +
�
π
�5 x
2
11
11 · 5! −···
(6.9.2)
There is a complex continued fraction that yields both S(x) and C(x) simultaneously:
where
e z2
C(x)+iS(x) = 1+i
2
erfc z = 1
�
1
√
π z +
= 2z
√ π
�
1/2
z +
1
2z 2 +1−
√
π
erf z, z = (1 − i)x (6.9.3)
2
1
z +
3/2
z +
1 · 2
2z 2 +5−
2
z + ···
�
3 · 4
2z 2 +9− ···
� (6.9.4)
In the last line we have converted the “standard” form of the continued fraction to
its “even” form (see §5.2), which converges twice as fast. We must be careful not
to evaluate the alternating series (6.9.2) at too large a value of x; inspection of the
terms shows that x =1.5 is a good point to switch over to the continued fraction.
Note that for large x
C(x) ∼ 1 1
+
2 πx sin
�
π
2 x2� , S(x) ∼ 1 1
−
2 πx cos
�
π
2 x2�
(6.9.5)
Thus the precision of the routine frenel may be limited by the precision of the
library routines for sine and cosine for large x.
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