FresnelC[z]
gives the Fresnel integral
.
FresnelC
FresnelC[z]
gives the Fresnel integral
.
Details
- Mathematical function, suitable for both symbolic and numerical manipulation.
- FresnelC[z] is given by
. - FresnelC[z] is an entire function of z with no branch cut discontinuities.
- For certain special arguments, FresnelC automatically evaluates to exact values.
- FresnelC can be evaluated to arbitrary numerical precision.
- FresnelC automatically threads over lists.
- FresnelC can be used with Interval and CenteredInterval objects. »
Examples
open all close allBasic Examples (5)
FresnelC[1.8]Plot over a subset of the reals:
Plot[FresnelC[x], {x, -10, 10}]Plot over a subset of the complexes:
ComplexPlot3D[FresnelC[z], {z, -2 - 2I, 2 + 2I}, PlotLegends -> Automatic]Series expansion at the origin:
Series[FresnelC[x], {x, 0, 20}]Series expansion at Infinity:
Series[FresnelC[x], {x, ∞, 3}]//NormalScope (41)
Numerical Evaluation (5)
Evaluate numerically to high precision:
N[FresnelC[2], 50]The precision of the output tracks the precision of the input:
FresnelC[2.0000000000000000000000000000000]Evaluate for complex arguments:
FresnelC[2.5 + I]Evaluate FresnelC efficiently at high precision:
FresnelC[2`500]//TimingFresnelC[2`10000];//TimingCompute worst-case guaranteed intervals using Interval and CenteredInterval objects:
FresnelC[Interval[{1.7, 1.8}]]FresnelC[CenteredInterval[-1, 1 / 100]]Or compute average-case statistical intervals using Around:
FresnelC[ Around[1 / 2, 0.01]]Compute the elementwise values of an array:
FresnelC[{{1.1, .2}, {.2, 1.2}}]Or compute the matrix FresnelC function using MatrixFunction:
MatrixFunction[FresnelC, {{1.1, .2}, {.2, 1.2}}]Specific Values (3)
FresnelC[0]FresnelC[{-Infinity, Infinity, -I Infinity, I Infinity}]FresnelC[ComplexInfinity]Find a local maximum as a root of
:
xmax = Solve[D[FresnelC[x], x] == 0 && 0 < x < 1.5, x][[1, 1]]//QuietPlot[FresnelC[x], {x, -3, 3}, Epilog -> Style[Point[{xmax[[2]], FresnelC[xmax[[2]]]}], PointSize[Large], Red]]Visualization (2)
Plot the FresnelC function:
Plot[FresnelC[x], {x, -8, 8}]ComplexContourPlot[Re[FresnelC[z]], {z, -2 - 2I, 2 + 2 I}, Contours -> 20]ComplexContourPlot[Im[FresnelC[z]], {z, -2 - 2I, 2 + 2 I}, Contours -> 20]Function Properties (10)
FresnelC is defined for all real and complex values:
FunctionDomain[FresnelC[x], x]FunctionDomain[FresnelC[z], z, Complexes]Approximate function range of FresnelC:
FunctionRange[FresnelC[x], x, y]//NFresnelC is an odd function:
FresnelC[-x]FresnelC is an analytic function of x:
FunctionAnalytic[FresnelC[x], x]FresnelC is neither non-increasing nor non-decreasing:
FunctionMonotonicity[FresnelC[x], x]FresnelC is not injective:
FunctionInjective[FresnelC[x], x]Plot[{FresnelC[x], 1 / 2}, {x, -6, 6}]FresnelC is not surjective:
FunctionSurjective[FresnelC[x], x]Plot[{FresnelC[x], 2}, {x, -6, 6}]FresnelC is neither non-negative nor non-positive:
FunctionSign[FresnelC[x], x]FresnelC has no singularities or discontinuities:
FunctionSingularities[FresnelC[x], x]FunctionDiscontinuities[FresnelC[x], x]FunctionConvexity[FresnelC[x], x]Differentiation (3)
D[FresnelC[x], x]Table[D[FresnelC[x], {x, n}], {n, 1, 4}]Plot[Evaluate[%], {x, -2, 2}, PlotLegends -> {"First Derivative", "Second Derivative", "Third Derivative", "Fourth Derivative"}]D[FresnelC[x], {x, n}]Integration (3)
Indefinite integral of FresnelC:
Integrate[FresnelC[x], x]Definite integral of an odd integrand over an interval centered at the origin is 0:
Integrate[FresnelC[x], {x, -5, 5}]Integrate[z^αFresnelC[z^β], z]Integrate[(FresnelC[a Sqrt[z]]^2/Sqrt[z]), z]Integrate[t^α - 1FresnelC[t], {t, 0, Infinity}, Assumptions -> {-1 < α < 0}]Series Expansions (5)
Taylor expansion for FresnelC:
Series[FresnelC[x], {x, 0, 9}]Plot the first three approximations for FresnelC around
:
terms = Normal@Table[Series[FresnelC[x], {x, 0, m}], {m, 1, 9, 4}];
Plot[{FresnelC[x], terms}, {x, 0, 1.5}, PlotLegends -> "Expressions"]General term in the series expansion of FresnelC:
SeriesCoefficient[FresnelC[x], {x, 0, n}]Find series expansion at infinity:
Series[FresnelC[x], {x, Infinity, 3}]Give the result for an arbitrary symbolic direction
:
Series[FresnelC[x], {x, DirectedInfinity[z], 1}]FresnelC can be applied to power series:
FresnelC[x + (x^2/2) + (x^3/9) + O[x]^4]Integral Transforms (2)
Compute the Laplace transform using LaplaceTransform:
LaplaceTransform[FresnelC[t], t, s]MellinTransform[FresnelC[x], x, s]Function Identities and Simplifications (3)
Verify an identity relating HypergeometricPFQ to FresnelC:
(FresnelC[z]/z) == HypergeometricPFQ[{(1/4)}, {(1/2), (5/4)}, -(π^2 z^4/16)]//FullSimplifySimplify an integral to FresnelC:
(1/Sqrt[2π])Integrate[(Cos[t]/Sqrt[t]), {t, 0, (π z^2/2)}]//FullSimplifyFresnelC[I z]//FunctionExpandFresnelC[Sqrt[z^2]]//FunctionExpandFunction Representations (5)
Integrate[Cos[Pi / 2 t ^ 2], {t, 0, z}]Relation to the error function Erf:
(1 - I/4) (Erf[(1 + I/2) Sqrt[π] z] + I Erf[(1 - I/2)Sqrt[π] z])//FullSimplifyFresnelC can be represented as a DifferentialRoot:
DifferentialRootReduce[FresnelC[x], x]FresnelC can be represented in terms of MeijerG:
MeijerGReduce[FresnelC[x], x]Activate[%]//FullSimplifyTraditionalForm formatting:
FresnelC[x]//TraditionalFormApplications (5)
Intensity of a wave diffracted by a half‐plane:
Plot[(1/2)(FresnelS[d] + (1/2))^2 + (1/2)(FresnelC[d] + (1/2))^2, {d, -2, 4}]ParametricPlot3D[{FresnelC[x], FresnelS[x], x}, {x, -4, 4}]A solution of the time‐dependent 1D Schrödinger equation for a sudden opening of a shutter:
ψ[x_, t_] = (1/2) E^-(I(t - 2 x)/2) (1 + (1 + I) FresnelS[( t - x/Sqrt[π t])] + (1 - I) FresnelC[( t - x/Sqrt[π t])]);Check the Schrödinger equation:
I D[ψ[x, t], t] == -(1/2)D[ψ[x, t], x, x]//ExpandPlot the time‐dependent solution:
Plot3D[Abs[ψ[x, t]], {x, -5, 5}, {t, 0, 5}]Plot of FresnelC along a circle in the complex plane:
ParametricPlot[ReIm[FresnelC[2 Exp[I φ]]], {φ, 0, 2π}]Fractional derivative of Sin:
FractionalD[Sin[x], {x, m}]Derivative of order
of Sin:
FractionalD[Sin[x], {x, 1 / 2}]//FunctionExpand//FullSimplifyPlot a smooth transition between the derivative and integral of Sin:
Plot3D[Evaluate[FractionalD[Sin[x], {x, m}]], {x, 0, 2Pi}, {m, -1, 1}]Properties & Relations (6)
Use FullSimplify to simplify expressions containing Fresnel integrals:
FresnelC[z] - HypergeometricPFQ[{(1/4)}, {(1/2), (5/4)}, -(π^2 z^4/16)]//FullSimplifyFindRoot[FresnelC[z] == 1 / Pi, {z, 1}]Obtain FresnelC from integrals and sums:
Integrate[Cos[Pi / 2 t ^ 2], {t, 0, z}]Underoverscript[∑, k = 0, ∞](2^-2 k - 1 π^2 k + 1 (-z^4)^k/(4 k + 1) (2 k + 1)!)//FunctionExpandSolve a differential equation:
DSolve[z w'''[z] - w''[z] + π^2 z^3 w'[z] == 0, w, z]Det[Outer[D[#1, {x, #2}]&, {1, FresnelC[x], FresnelS[x]}, {0, 1, 2}]]Simplify[%]Compare with Wronskian:
Wronskian[{1, FresnelC[x], FresnelS[x]}, x]∫z^αFresnelC[z^β]ⅆzFourierTransform[FresnelC[t], t, s]LaplaceTransform[FresnelC[t], t, s]//FunctionExpand//SimplifyPossible Issues (3)
FresnelC can take large values for moderate‐size arguments:
FresnelC[1000.(1 + I)]A larger setting for $MaxExtraPrecision can be needed:
N[FresnelC[10 ^ 20] - 1 / 2, 20]Block[{$MaxExtraPrecision = 100}, N[FresnelC[10 ^ 20 ] - 1 / 2, 20]]Some references use a different convention for the Fresnel integrals:
FresnelC1[x_] = (1/Sqrt[2Pi])Integrate[(1/Sqrt[t])Cos[t], {t, 0, x}, Assumptions -> x > 0]See Also
Tech Notes
Related Guides
Related Links
History
Introduced in 1996 (3.0) | Updated in 2014 (10.0) ▪ 2021 (13.0) ▪ 2022 (13.1)
Text
Wolfram Research (1996), FresnelC, Wolfram Language function, https://reference.wolfram.com/language/ref/FresnelC.html (updated 2022).
CMS
Wolfram Language. 1996. "FresnelC." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2022. https://reference.wolfram.com/language/ref/FresnelC.html.
APA
Wolfram Language. (1996). FresnelC. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/FresnelC.html
BibTeX
@misc{reference.wolfram_2026_fresnelc, author="Wolfram Research", title="{FresnelC}", year="2022", howpublished="\url{https://reference.wolfram.com/language/ref/FresnelC.html}", note=[Accessed: 13-June-2026]}
BibLaTeX
@online{reference.wolfram_2026_fresnelc, organization={Wolfram Research}, title={FresnelC}, year={2022}, url={https://reference.wolfram.com/language/ref/FresnelC.html}, note=[Accessed: 13-June-2026]}