Fresnel integrals

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The special functions

Figure: f041720a

The Fresnel integrals can be represented in the form of the series

An asymptotic representation for large is:

In a rectangular coordinate system the projections of the curve

where is a real parameter, onto the coordinate planes are the Cornu spiral and the curves , (see Fig. b).

Figure: f041720b

The generalized Fresnel integrals (see [1]) are functions of the form

The Fresnel integrals are related to the generalized Fresnel integrals as follows:


[1] H. Bateman (ed.) A. Erdélyi (ed.) et al. (ed.) , Higher transcendental functions , 2. Bessel functions, parabolic cylinder functions, orthogonal polynomials , McGraw-Hill (1953)
[2] E. Jahnke, F. Emde, "Tables of functions with formulae and curves" , Dover, reprint (1945) (Translated from German)


A word of warning. There are different normalizations in use for the Fresnel integrals. E.g., in [a3] they are defined as

so that

The Fresnel integrals defined in the article are related to the probability integral for a complex argument ,

(integration along the line ), by


[a1] A. Segun, M. Abramowitz, "Handbook of mathematical functions" , Appl. Math. Ser. , 55 , Nat. Bur. Standards (1970)
[a2] J. Spanier, K.B. Oldham, "An atlas of functions" , Hemisphere & Springer (1987) pp. Chapt. 39
[a3] N.N. Lebedev, "Special functions and their applications" , Dover, reprint (1972) pp. 21–33 (Translated from Russian)
How to Cite This Entry:
Fresnel integrals. A.B. Ivanov (originator), Encyclopedia of Mathematics. URL:
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098