GV

GeezerVison

The World Wide Web, restored to 800 pixels of glory.

Home|About|Gaming cabinet|Archives|How it works|Starter sites

On this site

  • Algebra
  • Applied Mathematics
  • Calculus and Analysis
  • Discrete Mathematics
  • Foundations of Mathematics
  • Geometry
  • History and Terminology
  • Number Theory
  • Probability and Statistics
  • Recreational Mathematics
  • Topology
  • Alphabetical Index
  • Random Entry
  • New in MathWorld

This view

  • Original page
  • New address

GeezerVison is showing mathworld.wolfram.com — Fresnel Integrals -- from Wolfram MathWorld

Search

Fresnel Integrals

DOWNLOAD Mathematica NotebookDownload Wolfram Notebook
Fresnel

The term "Fresnel integral" has a number of slightly different definitions. In physics, the Fresnel integrals denoted C(u) and S(u) are most often defined by

C(u)+iS(u)=int_0^ue^(ipix^2/2)dx
(1)
=int_0^ucos(1/2pix^2)dx+iint_0^usin(1/2pix^2)dx,
(2)

so

C(u)=int_0^ucos(1/2pix^2)dx
(3)
S(u)=int_0^usin(1/2pix^2)dx.
(4)

These Fresnel integrals are implemented in the Wolfram Language as FresnelC[z] and FresnelS[z].

C(u) and S(u) are entire functions.

FresnelCReIm
FresnelCContours
FresnelSReIm
FresnelSContours

The C(u) and S(u) integrals are illustrated above in the complex plane.

They have the special values

C(-infty)=-1/2
(5)
C(0)=0
(6)
C(infty)=1/2
(7)

and

S(-infty)=-1/2
(8)
S(0)=0
(9)
S(infty)=1/2.
(10)

An asymptotic expansion for u>>1 gives

C(u) approx 1/2+1/(piu)sin(1/2piu^2)
(11)
S(u) approx 1/2-1/(piu)cos(1/2piu^2).
(12)

Therefore, as u->infty, C(u)=1/2 and S(u)=1/2. The Fresnel integrals are sometimes alternatively defined as

x(t)=int_0^tcos(v^2)dv
(13)
y(t)=int_0^tsin(v^2)dv.
(14)

Letting x=v^2 so dx=2vdv=2sqrt(x)dv, and dv=x^(-1/2)dx/2

x(t)=1/2int_0^(sqrt(t))x^(-1/2)cosxdx
(15)
y(t)=1/2int_0^(sqrt(t))x^(-1/2)sinxdx.
(16)

In this form, they have a particularly simple expansion in terms of spherical Bessel functions of the first kind. Using

j_0(x)=(sinx)/x
(17)
n_1(x)=-j_(-1)(x)
(18)
=-(cosx)/x,
(19)

where n_1(x) is a spherical Bessel function of the second kind

x(t^2)=-1/2int_0^tn_1(x)x^(1/2)dx
(20)
=1/2int_0^tj_(-1)(x)x^(1/2)dx
(21)
=x^(1/2)sum_(n=0)^(infty)j_(2n)(x)
(22)
y(t^2)=1/2int_0^tj_0(x)x^(1/2)dx
(23)
=x^(1/2)sum_(n=0)^(infty)j_(2n+1)(x).
(24)

Related functions C_1(z), C_2(z), S_1(z), and S_2(z) are defined by

C_1(z)=C(sqrt(2/pi)z)=sqrt(2/pi)int_0^zcost^2dt
(25)
S_1(z)=S(sqrt(2/pi)z)=sqrt(2/pi)int_0^zsint^2dt
(26)
C_2(z)=C(sqrt((2z)/pi))=1/(sqrt(2pi))int_0^z(cost)/(sqrt(t))dt
(27)
S_2(z)=S(sqrt((2z)/pi))=1/(sqrt(2pi))int_0^z(sint)/(sqrt(t))dt.
(28)

See also

Cornu Spiral

Related Wolfram sites

https://functions.wolfram.com/GammaBetaErf/FresnelC/, https://functions.wolfram.com/GammaBetaErf/FresnelS/

Explore with Wolfram|Alpha

WolframAlpha

More things to try:

  • representations C(x)
  • representations S(x)
  • identities C(x)

References

Abramowitz, M. and Stegun, I. A. (Eds.). "Fresnel Integrals." §7.3 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 300-302, 1972.Leonard, I. E. "More on Fresnel Integrals." Amer. Math. Monthly 95, 431-433, 1988.Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. "Fresnel Integrals, Cosine and Sine Integrals." §6.79 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 248-252, 1992.Prudnikov, A. P.; Marichev, O. I.; and Brychkov, Yu. A. "The Generalized Fresnel Integrals S(x,nu) and C(x,nu)." §1.3 in Integrals and Series, Vol. 3: More Special Functions. Newark, NJ: Gordon and Breach, p. 24, 1990.Spanier, J. and Oldham, K. B. "The Fresnel Integrals S(x) and C(x)." Ch. 39 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 373-383, 1987.

Referenced on Wolfram|Alpha

Fresnel Integrals

Cite this as:

Weisstein, Eric W. "Fresnel Integrals." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/FresnelIntegrals.html

Subject classifications

You are visitor 01402503 — GeezerVison does not store the pages you look up.
Best viewed with Netscape Navigator 4.7 or Internet Explorer 6 at 800×600. No cookies required.