Fresnel Integrals \\\\ A Note

Fresnel Integrals A Note Johar M. Ashfaque The two Fresnel integrals are defined by   Z x π 2 C(x) = cos t dt, 2 0

Z S(x) =

x

 sin

0

 π 2 t dt 2

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 C(x) S(x)

 4 9  2 5 x π x π + − ... 2 5 · 2! 2 9 · 4!   3  3 7  5 11 π x π π x x = − + − ... 2 3 · 1! 2 7 · 3! 2 11 · 5! = x−

There is a complex continued fraction that yields both C(x) and S(x) simultaneously √ 1+i π erf z, z= (1 − i)x. C(x) + iS(x) = 2 2 Note. For large x C(x) ∼

1 1 + sin 2 πx



 π 2 x , 2

S(x) ∼

1

1 1 − cos 2 πx



 π 2 x . 2