The Fresnel integrals S(x) and C(x) are two transcendental functions named after Augustin-Jean Fresnel that are used in optics and are closely related to the error function (erf). They arise in the description of near-field Fresnel diffraction phenomena and are defined through the following integral representations:
The Fresnel integrals admit the following power series expansions that converge for all x:
Some widely used tables use π/2t2 instead of t2 for the argument of the integrals defining S(x) and C(x). This changes their limits at infinity from 1/2·√ to 1/2 and the arc length for the first spiral turn from √ to 2 (at t = 2). These alternative functions are usually known as normalized Fresnel integrals.
The Euler spiral, also known as Cornu spiral or clothoid, is the curve generated by a parametric plot of S(t) against C(t). The Cornu spiral was created by Marie Alfred Cornu as a nomogram for diffraction computations in science and engineering.
From the definitions of Fresnel integrals, the infinitesimals dx and dy are thus:
Thus the length of the spiral measured from the origin can be expressed as
That is, the parameter t is the curve length measured from the origin (0, 0), and the Euler spiral has infinite length. The vector (cos(t2), sin(t2)) also expresses the unit tangent vector along the spiral, giving θ = t2. Since t is the curve length, the curvature κ can be expressed as
Thus the rate of change of curvature with respect to the curve length is
An Euler spiral has the property that its curvature at any point is proportional to the distance along the spiral, measured from the origin. This property makes it useful as a transition curve in highway and railway engineering: if a vehicle follows the spiral at unit speed, the parameter t in the above derivatives also represents the time. Consequently, a vehicle following the spiral at constant speed will have a constant rate of angular acceleration.
- C(x) and S(x) are odd functions of x.
- Asymptotics of the Fresnel integrals as x → ∞ are given by the formulas:
- Using the power series expansions above, the Fresnel integrals can be extended to the domain of complex numbers, where they become analytic functions of a complex variable.
- C(z) and S(z) are entire functions of the complex variable z.
- The Fresnel integrals can be expressed using the error function as follows:
Limits as x approaches infinity
around the boundary of the sector-shaped region in the complex plane formed by the positive x-axis, the bisector of the first quadrant y = x with x ≥ 0, and a circular arc of radius R centered at the origin.
As R goes to infinity, the integral along the circular arc γ2 tends to 0
Note too that because the integrand is an entire function on the complex plane, its integral along the whole contour is zero. Overall, we must have
where γ3 denotes the bisector of the first quadrant, as in the diagram. To evaluate the left hand side, parametrize the bisector as
where t ranges from 0 to +∞. Note that the square of this expression is just +it2. Therefore, substitution gives the left hand side as
Using Euler's formula to take real and imaginary parts of e−it2 gives this as
where we have written 0i to emphasize that the original Gaussian integral's value is completely real with zero imaginary part. Letting
and then equating real and imaginary parts produces the following system of two equations in the two unknowns IC and IS:
Solving this for IC and IS gives the desired result.
which reduces to Fresnel integrals if real or imaginary parts are taken:
The leading term in the asymptotic expansion is
For m = 0, the imaginary part of this equation in particular is
with the left-hand side converging for a > 1 and the right-hand side being its analytical extension to the whole plane less where lie the poles of Γ(a−1).
The Kummer transformation of the confluent hypergeometric function is
For computation to particular target precision, other approximations have been developed. Cody developed a set of efficient approximations based on rational functions that give relative errors down to 2×10−19. A FORTRAN implementation of the Cody approximation that includes the values of the coefficients needed for implementation in other languages was published by van Snyder. Boersma developed an approximation with error less than 1.6×10−9.
The Fresnel integrals were originally used in the calculation of the electromagnetic field intensity in an environment where light bends around opaque objects. More recently, they have been used in the design of highways and railways, specifically their curvature transition zones, see track transition curve. Other applications are rollercoasters or calculating the transitions on a velodrome track to allow rapid entry to the bends and gradual exit.
- Abramowitz & Stegun 1983, eqn 7.3.1 – 7.3.2. sfn error: no target: CITEREFAbramowitzStegun1983 (help)
- Temme 2010. sfn error: no target: CITEREFTemme2010 (help)
- Abramowitz & Stegun 1983, eqn 7.3.20. sfn error: no target: CITEREFAbramowitzStegun1983 (help)
- functions.wolfram.com, Fresnel integral S: Representations through equivalent functions and Fresnel integral C: Representations through equivalent functions. Note: Wolfram uses the Abramowitz & Stegun convention, which differs from the one in this article by factors of √.
- Another method based on parametric integration is described for example in Zajta & Goel 1989.
- Mathar 2012.
- Temme 2010, §7.12(ii). sfn error: no target: CITEREFTemme2010 (help)
- Press et al. 2007.
- Cody 1968.
- van Snyder 1993.
- Boersma 1960.
- Beatty 2013.
- Stewart 2008, p. 383.
- Abramowitz, Milton; Stegun, Irene Ann, eds. (1983) [June 1964]. "Chapter 7". Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Applied Mathematics Series. 55 (Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first ed.). Washington D.C.; New York: United States Department of Commerce, National Bureau of Standards; Dover Publications. ISBN 978-0-486-61272-0. LCCN 64-60036. MR 0167642. LCCN 65-12253.
- Alazah, Mohammad (2012). "Computing Fresnel integrals via modified trapezium rules". Numerische Mathematik. 128 (4): 635–661. arXiv:1209.3451. Bibcode:2012arXiv1209.3451A. doi:10.1007/s00211-014-0627-z. S2CID 13934493.
- Beatty, Thomas (2013). "How to evaluate Fresnel Integrals" (PDF). FGCU Math - Summer 2013. Retrieved 27 July 2013.
- Boersma, J. (1960). "Computation of Fresnel Integrals". Math. Comp. 14 (72): 380. doi:10.1090/S0025-5718-1960-0121973-3. MR 0121973.
- Bulirsch, Roland (1967). "Numerical calculation of the sine, cosine and Fresnel integrals". Numer. Math. 9 (5): 380–385. doi:10.1007/BF02162153. S2CID 121794086.
- Cody, William J. (1968). "Chebyshev approximations for the Fresnel integrals" (PDF). Math. Comp. 22 (102): 450–453. doi:10.1090/S0025-5718-68-99871-2.
- Hangelbroek, R. J. (1967). "Numerical approximation of Fresnel integrals by means of Chebyshev polynomials". J. Eng. Math. 1 (1): 37–50. Bibcode:1967JEnMa...1...37H. doi:10.1007/BF01793638. S2CID 122271446.
- Mathar, R. J. (2012). "Series Expansion of Generalized Fresnel Integrals". arXiv:1211.3963 [math.CA].
- Nave, R. (2002). "The Cornu spiral". (Uses π/2t2 instead of t2.)
- Press, W. H.; Teukolsky, S. A.; Vetterling, W. T.; Flannery, B. P. (2007). "Section 6.8.1. Fresnel Integrals". Numerical Recipes: The Art of Scientific Computing (3rd ed.). New York: Cambridge University Press. ISBN 978-0-521-88068-8.
- van Snyder, W. (1993). "Algorithm 723: Fresnel integrals". ACM Trans. Math. Softw. 19 (4): 452–456. doi:10.1145/168173.168193. S2CID 12346795.
- Stewart, James (2008). Calculus Early Transcendentals. Cengage Learning EMEA. ISBN 978-0-495-38273-7.
- Temme, N. M. (2010), "Error Functions, Dawson's and Fresnel Integrals", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, MR 2723248
- van Wijngaarden, A.; Scheen, W. L. (1949). Table of Fresnel Integrals. Verhandl. Konink. Ned. Akad. Wetenschapen. 19.
- Zajta, Aurel J.; Goel, Sudhir K. (1989). "Parametric Integration Techniques". Mathematics Magazine. 62 (5): 318–322. doi:10.1080/0025570X.1989.11977462.
- Cephes, free/open-source C++/C code to compute Fresnel integrals among other special functions. Used in SciPy and ALGLIB.
- Faddeeva Package, free/open-source C++/C code to compute complex error functions (from which the Fresnel integrals can be obtained), with wrappers for Matlab, Python, and other languages.
- "Fresnel integrals", Encyclopedia of Mathematics, EMS Press, 2001 
- "Roller Coaster Loop Shapes". Archived from the original on September 23, 2008. Retrieved 2008-08-13.
- Weisstein, Eric W. "Fresnel Integrals". MathWorld.
- Weisstein, Eric W. "Cornu Spiral". MathWorld.