# Fresnel integrals

The special functions

$$C ( x) = { \frac{1}{\sqrt {2 \pi } } } \int\limits _ { 0 } ^ { {x ^ {2}} } \frac{\cos t }{\sqrt t } dt = \ \sqrt { \frac{2} \pi } \int\limits _ { 0 } ^ { x } \cos t ^ {2} dt,$$

$$S ( x) = { \frac{1}{\sqrt {2 \pi } } } \int\limits _ { 0 } ^ { {x } ^ {2} } \frac{\sin t }{\sqrt t } dt = \sqrt { \frac{2} \pi } \int\limits _ { 0 } ^ { x } \sin t ^ {2} dt;$$

$$\lim\limits _ {x \rightarrow + \infty } C ( x) = \lim\limits _ {x \rightarrow + \infty } S ( x) = { \frac{1}{2} } .$$

Figure: f041720a

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

$$C ( x) = \sqrt { \frac{2} \pi } x \sum _ {k = 0 } ^ \infty \frac{(- 1) ^ {k} x ^ {4k} }{( 2k)! ( 4k + 1) } ,$$

$$S ( x) = \sqrt { \frac{2} \pi } x \sum _ {k = 0 } ^ \infty \frac{(- 1) ^ {k} x ^ {2 ( 2 k + 1) } }{( 2k + 1)! ( 4k + 3) } .$$

An asymptotic representation for large $x$ is:

$$C ( x) = { \frac{1}{2} } + \frac{1}{\sqrt {2 \pi } x } \ \sin x ^ {2} + O \left ( \frac{1}{x ^ {2} } \right ) ,$$

$$S ( x) = { \frac{1}{2} } - \frac{1}{\sqrt {2 \pi } x } \cos x ^ {2} + O \left ( \frac{1}{x ^ {2} } \right ) .$$

In a rectangular coordinate system $( x, y)$ the projections of the curve

$$x = t,\ \ y = C \left ( { \frac \pi {2} } t ^ {2} \right ) ,\ \ z = S \left ( { \frac \pi {2} } t ^ {2} \right ) ,$$

where $t$ is a real parameter, onto the coordinate planes are the Cornu spiral and the curves $y = C ( \pi t ^ {2} /2)$, $z = S ( \pi t ^ {2} /2)$( see Fig. b). $x$ $S( x)$ $C( x)$

Figure: f041720b

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

$$C ( x, \alpha ) = \ \int\limits _ { x } ^ \infty t ^ {\alpha - 1 } \ \cos t dt,$$

$$S ( x, \alpha ) = \int\limits _ { x } ^ \infty t ^ {\alpha - 1 } \sin t dt.$$

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

$$C ( x) = { \frac{1}{2} } - \frac{1}{\sqrt {2 \pi } } C \left ( x ^ {2} , { \frac{1}{2} } \right ) ;$$

$$S ( x) = { \frac{1}{2} } - \frac{1}{\sqrt {2 \pi } } S \left ( x ^ {2} , { \frac{1}{2} } \right ) .$$

#### References

 [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

$$C _ {1} ( z) = \int\limits _ { 0 } ^ { z } \cos \frac{\pi t ^ {2} }{2} d t ,\ S _ {1} ( z) = \ \int\limits _ { 0 } ^ { z } \sin \frac{\pi t ^ {2} }{2} d t ,$$

so that

$$C _ {1} ( z) = C \left ( \sqrt { \frac \pi {2} } z \right ) \ \ \textrm{ and } \ \ S _ {1} ( z) = S \left ( \sqrt { \frac \pi {2} } z \right ) .$$

The Fresnel integrals defined in the article are related to the probability integral for a complex argument $z = \sqrt i x$,

$$\Phi ( z) = \frac{2}{\sqrt \pi } \int\limits _ { 0 } ^ { z } e ^ {- t ^ {2} } d t$$

(integration along the line $\mathop{\rm Re} z = \mathop{\rm Im} z$), by

$$\frac{\Phi ( \sqrt i x ) }{\sqrt i } = \ \sqrt 2 C ( x) - i \sqrt 2 S ( x) .$$

#### References

 [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. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fresnel_integrals&oldid=46990
This article was adapted from an original article by A.B. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article