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Fresnel integral
Special function defined by an integral

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:

S ( x ) = ∫ 0 x sin ⁡ ( t 2 ) d t , C ( x ) = ∫ 0 x cos ⁡ ( t 2 ) d t . {\displaystyle S(x)=\int _{0}^{x}\sin \left(t^{2}\right)\,dt,\quad C(x)=\int _{0}^{x}\cos \left(t^{2}\right)\,dt.}

The parametric curve ⁠ ( S ( t ) , C ( t ) ) {\displaystyle {\bigl (}S(t),C(t){\bigr )}} ⁠ is the Euler spiral or clothoid, a curve whose curvature varies linearly with arclength.

The term Fresnel integral may also refer to the complex definite integral

∫ − ∞ ∞ e ± i a x 2 d x = π a e ± i π / 4 {\displaystyle \int _{-\infty }^{\infty }e^{\pm iax^{2}}dx={\sqrt {\frac {\pi }{a}}}e^{\pm i\pi /4}}

where a is real and positive; this can be evaluated by closing a contour in the complex plane and applying Cauchy's integral theorem.

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Definition

The Fresnel integrals admit the following power series expansions that converge for all x: S ( x ) = ∫ 0 x sin ⁡ ( t 2 ) d t = ∑ n = 0 ∞ ( − 1 ) n x 4 n + 3 ( 2 n + 1 ) ! ( 4 n + 3 ) , C ( x ) = ∫ 0 x cos ⁡ ( t 2 ) d t = ∑ n = 0 ∞ ( − 1 ) n x 4 n + 1 ( 2 n ) ! ( 4 n + 1 ) . {\displaystyle {\begin{aligned}S(x)&=\int _{0}^{x}\sin \left(t^{2}\right)\,dt=\sum _{n=0}^{\infty }(-1)^{n}{\frac {x^{4n+3}}{(2n+1)!(4n+3)}},\\C(x)&=\int _{0}^{x}\cos \left(t^{2}\right)\,dt=\sum _{n=0}^{\infty }(-1)^{n}{\frac {x^{4n+1}}{(2n)!(4n+1)}}.\end{aligned}}}

Some widely used tables12 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·√⁠π/2⁠ to ⁠1/2⁠3 and the arc length for the first spiral turn from √2π to 2 (at t = 2). These alternative functions are usually known as normalized Fresnel integrals.

Euler spiral

Main article: Euler spiral

The Euler spiral, also known as a Cornu spiral or clothoid, is the curve generated by a parametric plot of S(t) against C(t). The Euler spiral was first studied in the mid 18th century by Leonhard Euler in the context of Euler–Bernoulli beam theory. A century later, Marie Alfred Cornu constructed the same spiral as a nomogram for diffraction computations.

From the definitions of Fresnel integrals, the infinitesimals dx and dy are thus: d x = C ′ ( t ) d t = cos ⁡ ( t 2 ) d t , d y = S ′ ( t ) d t = sin ⁡ ( t 2 ) d t . {\displaystyle {\begin{aligned}dx&=C'(t)\,dt=\cos \left(t^{2}\right)\,dt,\\dy&=S'(t)\,dt=\sin \left(t^{2}\right)\,dt.\end{aligned}}}

Thus the length of the spiral measured from the origin can be expressed as L = ∫ 0 t 0 d x 2 + d y 2 = ∫ 0 t 0 d t = t 0 . {\displaystyle L=\int _{0}^{t_{0}}{\sqrt {dx^{2}+dy^{2}}}=\int _{0}^{t_{0}}dt=t_{0}.}

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)), where θ = t2, also expresses the unit tangent vector along the spiral. Since t is the curve length, the curvature κ can be expressed as κ = 1 R = d θ d t = 2 t . {\displaystyle \kappa ={\frac {1}{R}}={\frac {d\theta }{dt}}=2t.}

Thus the rate of change of curvature with respect to the curve length is d κ d t = d 2 θ d t 2 = 2. {\displaystyle {\frac {d\kappa }{dt}}={\frac {d^{2}\theta }{dt^{2}}}=2.}

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.

Sections from Euler spirals are commonly incorporated into the shape of rollercoaster loops to make what are known as clothoid loops.

Properties

C(x) and S(x) are odd functions of x,

C ( − x ) = − C ( x ) , S ( − x ) = − S ( x ) . {\displaystyle C(-x)=-C(x),\quad S(-x)=-S(x).}

which can be readily seen from the fact that their power series expansions have only odd-degree terms, or alternatively because they are antiderivatives of even functions that also are zero at the origin.

Asymptotics of the Fresnel integrals as x → ∞ are given by the formulas:

S ( x ) = 1 8 π sgn ⁡ x − [ 1 + O ( x − 4 ) ] ( cos ⁡ ( x 2 ) 2 x + sin ⁡ ( x 2 ) 4 x 3 ) , C ( x ) = 1 8 π sgn ⁡ x + [ 1 + O ( x − 4 ) ] ( sin ⁡ ( x 2 ) 2 x − cos ⁡ ( x 2 ) 4 x 3 ) . {\displaystyle {\begin{aligned}S(x)&={\sqrt {{\tfrac {1}{8}}\pi }}\operatorname {sgn} x-\left[1+O\left(x^{-4}\right)\right]\left({\frac {\cos \left(x^{2}\right)}{2x}}+{\frac {\sin \left(x^{2}\right)}{4x^{3}}}\right),\\[6px]C(x)&={\sqrt {{\tfrac {1}{8}}\pi }}\operatorname {sgn} x+\left[1+O\left(x^{-4}\right)\right]\left({\frac {\sin \left(x^{2}\right)}{2x}}-{\frac {\cos \left(x^{2}\right)}{4x^{3}}}\right).\end{aligned}}}

Using the power series expansions above, the Fresnel integrals can be extended to the domain of complex numbers, where they become entire functions of the complex variable z.

The Fresnel integrals can be expressed using the error function as follows:4

S ( z ) = π 2 ⋅ 1 + i 4 [ erf ⁡ ( 1 + i 2 z ) − i erf ⁡ ( 1 − i 2 z ) ] , C ( z ) = π 2 ⋅ 1 − i 4 [ erf ⁡ ( 1 + i 2 z ) + i erf ⁡ ( 1 − i 2 z ) ] . {\displaystyle {\begin{aligned}S(z)&={\sqrt {\frac {\pi }{2}}}\cdot {\frac {1+i}{4}}\left[\operatorname {erf} \left({\frac {1+i}{\sqrt {2}}}z\right)-i\operatorname {erf} \left({\frac {1-i}{\sqrt {2}}}z\right)\right],\\[6px]C(z)&={\sqrt {\frac {\pi }{2}}}\cdot {\frac {1-i}{4}}\left[\operatorname {erf} \left({\frac {1+i}{\sqrt {2}}}z\right)+i\operatorname {erf} \left({\frac {1-i}{\sqrt {2}}}z\right)\right].\end{aligned}}}

or

C ( z ) + i S ( z ) = π 2 ⋅ 1 + i 2 erf ⁡ ( 1 − i 2 z ) , S ( z ) + i C ( z ) = π 2 ⋅ 1 + i 2 erf ⁡ ( 1 + i 2 z ) . {\displaystyle {\begin{aligned}C(z)+iS(z)&={\sqrt {\frac {\pi }{2}}}\cdot {\frac {1+i}{2}}\operatorname {erf} \left({\frac {1-i}{\sqrt {2}}}z\right),\\[6px]S(z)+iC(z)&={\sqrt {\frac {\pi }{2}}}\cdot {\frac {1+i}{2}}\operatorname {erf} \left({\frac {1+i}{\sqrt {2}}}z\right).\end{aligned}}}

Limits as x approaches infinity

The integrals defining C(x) and S(x) cannot be evaluated in the closed form in terms of elementary functions, except in special cases. The limits of these functions as x goes to infinity are known: ∫ 0 ∞ cos ⁡ ( t 2 ) d t = ∫ 0 ∞ sin ⁡ ( t 2 ) d t = 2 π 4 = π 8 ≈ 0.6267. {\displaystyle \int _{0}^{\infty }\cos \left(t^{2}\right)\,dt=\int _{0}^{\infty }\sin \left(t^{2}\right)\,dt={\frac {\sqrt {2\pi }}{4}}={\sqrt {\frac {\pi }{8}}}\approx 0.6267.}

Proof of the formula

This can be derived with any one of several methods. One of them5 uses a contour integral of the function e − z 2 {\displaystyle e^{-z^{2}}} 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 | ∫ γ 2 e − z 2 d z | = | ∫ 0 π 4 e − R 2 ( cos ⁡ t + i sin ⁡ t ) 2 R e i t d t | ≤ R ∫ 0 π 4 e − R 2 cos ⁡ 2 t d t ≤ R ∫ 0 π 4 e − R 2 ( 1 − 4 π t ) d t = π 4 R ( 1 − e − R 2 ) , {\displaystyle \left|\int _{\gamma _{2}}e^{-z^{2}}\,dz\right|=\left|\int _{0}^{\frac {\pi }{4}}e^{-R^{2}(\cos t+i\sin t)^{2}}\,Re^{it}dt\right|\leq R\int _{0}^{\frac {\pi }{4}}e^{-R^{2}\cos 2t}\,dt\leq R\int _{0}^{\frac {\pi }{4}}e^{-R^{2}\left(1-{\frac {4}{\pi }}t\right)}\,dt={\frac {\pi }{4R}}\left(1-e^{-R^{2}}\right),} where polar coordinates z = Reit were used and Jordan's inequality was utilised for the second inequality. The integral along the real axis γ1 tends to the half Gaussian integral ∫ γ 1 e − z 2 d z = ∫ 0 ∞ e − t 2 d t = π 2 . {\displaystyle \int _{\gamma _{1}}e^{-z^{2}}\,dz=\int _{0}^{\infty }e^{-t^{2}}\,dt={\frac {\sqrt {\pi }}{2}}.}

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 ∫ γ 3 e − z 2 d z = ∫ γ 1 e − z 2 d z = ∫ 0 ∞ e − t 2 d t , {\displaystyle \int _{\gamma _{3}}e^{-z^{2}}\,dz=\int _{\gamma _{1}}e^{-z^{2}}\,dz=\int _{0}^{\infty }e^{-t^{2}}\,dt,} where γ3 denotes the bisector of the first quadrant, as in the diagram. To evaluate the left hand side, parametrize the bisector as z = t e i π 4 = 2 2 ( 1 + i ) t {\displaystyle z=te^{i{\frac {\pi }{4}}}={\frac {\sqrt {2}}{2}}(1+i)t} where t ranges from 0 to +∞. Note that the square of this expression is just +it2. Therefore, substitution gives the left hand side as ∫ 0 ∞ e − i t 2 2 2 ( 1 + i ) d t . {\displaystyle \int _{0}^{\infty }e^{-it^{2}}{\frac {\sqrt {2}}{2}}(1+i)\,dt.}

Using Euler's formula to take real and imaginary parts of e−it2 gives this as ∫ 0 ∞ ( cos ⁡ ( t 2 ) − i sin ⁡ ( t 2 ) ) 2 2 ( 1 + i ) d t = 2 2 ∫ 0 ∞ [ cos ⁡ ( t 2 ) + sin ⁡ ( t 2 ) + i ( cos ⁡ ( t 2 ) − sin ⁡ ( t 2 ) ) ] d t = π 2 + 0 i , {\displaystyle {\begin{aligned}&\int _{0}^{\infty }\left(\cos \left(t^{2}\right)-i\sin \left(t^{2}\right)\right){\frac {\sqrt {2}}{2}}(1+i)\,dt\\[6px]&\quad ={\frac {\sqrt {2}}{2}}\int _{0}^{\infty }\left[\cos \left(t^{2}\right)+\sin \left(t^{2}\right)+i\left(\cos \left(t^{2}\right)-\sin \left(t^{2}\right)\right)\right]\,dt\\[6px]&\quad ={\frac {\sqrt {\pi }}{2}}+0i,\end{aligned}}} where we have written 0i to emphasize that the original Gaussian integral's value is completely real with zero imaginary part. Letting I C = ∫ 0 ∞ cos ⁡ ( t 2 ) d t , I S = ∫ 0 ∞ sin ⁡ ( t 2 ) d t {\displaystyle I_{C}=\int _{0}^{\infty }\cos \left(t^{2}\right)\,dt,\quad I_{S}=\int _{0}^{\infty }\sin \left(t^{2}\right)\,dt} and then equating real and imaginary parts produces the following system of two equations in the two unknowns IC and IS: I C + I S = π 2 , I C − I S = 0. {\displaystyle {\begin{aligned}I_{C}+I_{S}&={\sqrt {\frac {\pi }{2}}},\\I_{C}-I_{S}&=0.\end{aligned}}}

Solving this for IC and IS gives the desired result.

Generalization

The integral ∫ x m e i x n d x = ∫ ∑ l = 0 ∞ i l x m + n l l ! d x = ∑ l = 0 ∞ i l ( m + n l + 1 ) x m + n l + 1 l ! {\displaystyle \int x^{m}e^{ix^{n}}\,dx=\int \sum _{l=0}^{\infty }{\frac {i^{l}x^{m+nl}}{l!}}\,dx=\sum _{l=0}^{\infty }{\frac {i^{l}}{(m+nl+1)}}{\frac {x^{m+nl+1}}{l!}}} is a confluent hypergeometric function and also an incomplete gamma function6 ∫ x m e i x n d x = x m + 1 m + 1 1 F 1 ( m + 1 n 1 + m + 1 n ∣ i x n ) = 1 n i m + 1 n γ ( m + 1 n , − i x n ) , {\displaystyle {\begin{aligned}\int x^{m}e^{ix^{n}}\,dx&={\frac {x^{m+1}}{m+1}}\,_{1}F_{1}\left({\begin{array}{c}{\frac {m+1}{n}}\\1+{\frac {m+1}{n}}\end{array}}\mid ix^{n}\right)\\[6px]&={\frac {1}{n}}i^{\frac {m+1}{n}}\gamma \left({\frac {m+1}{n}},-ix^{n}\right),\end{aligned}}} which reduces to Fresnel integrals if real or imaginary parts are taken: ∫ x m sin ⁡ ( x n ) d x = x m + n + 1 m + n + 1 1 F 2 ( 1 2 + m + 1 2 n 3 2 + m + 1 2 n , 3 2 ∣ − x 2 n 4 ) . {\displaystyle \int x^{m}\sin(x^{n})\,dx={\frac {x^{m+n+1}}{m+n+1}}\,_{1}F_{2}\left({\begin{array}{c}{\frac {1}{2}}+{\frac {m+1}{2n}}\\{\frac {3}{2}}+{\frac {m+1}{2n}},{\frac {3}{2}}\end{array}}\mid -{\frac {x^{2n}}{4}}\right).} The leading term in the asymptotic expansion is 1 F 1 ( m + 1 n 1 + m + 1 n ∣ i x n ) ∼ m + 1 n Γ ( m + 1 n ) e i π m + 1 2 n x − m − 1 , {\displaystyle _{1}F_{1}\left({\begin{array}{c}{\frac {m+1}{n}}\\1+{\frac {m+1}{n}}\end{array}}\mid ix^{n}\right)\sim {\frac {m+1}{n}}\,\Gamma \left({\frac {m+1}{n}}\right)e^{i\pi {\frac {m+1}{2n}}}x^{-m-1},} and therefore ∫ 0 ∞ x m e i x n d x = 1 n Γ ( m + 1 n ) e i π m + 1 2 n . {\displaystyle \int _{0}^{\infty }x^{m}e^{ix^{n}}\,dx={\frac {1}{n}}\,\Gamma \left({\frac {m+1}{n}}\right)e^{i\pi {\frac {m+1}{2n}}}.}

For m = 0, the imaginary part of this equation in particular is ∫ 0 ∞ sin ⁡ ( x a ) d x = Γ ( 1 + 1 a ) sin ⁡ ( π 2 a ) , {\displaystyle \int _{0}^{\infty }\sin \left(x^{a}\right)\,dx=\Gamma \left(1+{\frac {1}{a}}\right)\sin \left({\frac {\pi }{2a}}\right),} 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 ∫ x m e i x n d x = V n , m ( x ) e i x n , {\displaystyle \int x^{m}e^{ix^{n}}\,dx=V_{n,m}(x)e^{ix^{n}},} with V n , m := x m + 1 m + 1 1 F 1 ( 1 1 + m + 1 n ∣ − i x n ) . {\displaystyle V_{n,m}:={\frac {x^{m+1}}{m+1}}\,_{1}F_{1}\left({\begin{array}{c}1\\1+{\frac {m+1}{n}}\end{array}}\mid -ix^{n}\right).}

Numerical approximation

For computation to arbitrary precision, the power series is suitable for small argument. For large argument, asymptotic expansions converge faster.7 Continued fraction methods may also be used.8

For computation to particular target precision, other approximations have been developed. Cody9 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.10 Boersma developed an approximation with error less than 1.6×10−9.11

Applications

The Fresnel integrals were originally used in the calculation of the electromagnetic field intensity in an environment where light bends around opaque objects.12 More recently, they have been used in the design of highways and railways, specifically their curvature transition zones, see track transition curve.13 Other applications are rollercoasters14 or calculating the transitions on a velodrome track to allow rapid entry to the bends and gradual exit.

See also

  • Mathematics portal

Notes

References

  1. Abramowitz & Stegun 1983, eqn 7.3.1–7.3.2. - Abramowitz, Milton; Stegun, Irene Ann, eds. (1983) [June 1964]. "Chapter 7". Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Applied Mathematics Series. Vol. 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. https://lccn.loc.gov/64-60036 ↩

  2. Temme 2010. - 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 http://dlmf.nist.gov/7 ↩

  3. Abramowitz & Stegun 1983, eqn 7.3.20. - Abramowitz, Milton; Stegun, Irene Ann, eds. (1983) [June 1964]. "Chapter 7". Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Applied Mathematics Series. Vol. 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. https://lccn.loc.gov/64-60036 ↩

  4. 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 √π⁄2. http://functions.wolfram.com/GammaBetaErf/FresnelS/27/01/ ↩

  5. Another method based on parametric integration is described for example in Zajta & Goel 1989. /wiki/Integration_using_parametric_derivatives ↩

  6. Mathar 2012. - Mathar, R. J. (2012). "Series Expansion of Generalized Fresnel Integrals". arXiv:1211.3963 [math.CA]. https://arxiv.org/abs/1211.3963 ↩

  7. Temme 2010, §7.12(ii). - 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 http://dlmf.nist.gov/7 ↩

  8. Press et al. 2007. - 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. Archived from the original on 2011-08-11. Retrieved 2011-08-09. https://web.archive.org/web/20110811154417/http://apps.nrbook.com/empanel/index.html#pg=297 ↩

  9. Cody 1968. - 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. https://www.ams.org/journals/mcom/1968-22-102/S0025-5718-68-99871-2/S0025-5718-68-99871-2.pdf ↩

  10. van Snyder 1993. - van Snyder, W. (1993). "Algorithm 723: Fresnel integrals". ACM Trans. Math. Softw. 19 (4): 452–456. doi:10.1145/168173.168193. S2CID 12346795. https://doi.org/10.1145%2F168173.168193 ↩

  11. Boersma 1960. - Boersma, J. (1960). "Computation of Fresnel Integrals". Math. Comp. 14 (72): 380. doi:10.1090/S0025-5718-1960-0121973-3. MR 0121973. https://doi.org/10.1090%2FS0025-5718-1960-0121973-3 ↩

  12. Beatty 2013. - Beatty, Thomas (2013). "How to evaluate Fresnel Integrals" (PDF). FGCU Math - Summer 2013. Retrieved 27 July 2013. https://web.archive.org/web/20230409062958/https://www.thomasbeatty.com/MATH%20PAGES/ARCHIVES%20-%20NOTES/Complex%20Variables/How%20to%20evaluate%20Fresnel%20Integrals.pdf ↩

  13. Stewart 2008, p. 383. - Stewart, James (2008). Calculus Early Transcendentals. Cengage Learning EMEA. ISBN 978-0-495-38273-7. https://books.google.com/books?id=v4-1Rk7uaEcC&q=%22design+of+highways%22&pg=PA383 ↩

  14. Beatty 2013. - Beatty, Thomas (2013). "How to evaluate Fresnel Integrals" (PDF). FGCU Math - Summer 2013. Retrieved 27 July 2013. https://web.archive.org/web/20230409062958/https://www.thomasbeatty.com/MATH%20PAGES/ARCHIVES%20-%20NOTES/Complex%20Variables/How%20to%20evaluate%20Fresnel%20Integrals.pdf ↩