Fresnel Integral Calculator S(x) and C(x)

Compute the Fresnel integrals S(x) and C(x) and plot the Euler spiral (clothoid).

Input

Enter x to compute the Fresnel integrals S(x) and C(x).

Upper limit x

Enter the real number used as the upper limit of integration.

S(1)

0.4382591474

C(1)

0.7798934004

Input x

Euler spiral (clothoid)

Horizontal axis C(t), vertical axis S(t). The orange dot marks the point for your input x.

The Fresnel integrals are defined as S(x)=∫_0^x sin(πt²/2)dt and C(x)=∫_0^x cos(πt²/2)dt (normalized form).
This calculator evaluates them with a power series for small x and an asymptotic expansion of the auxiliary functions for large x.
As x→∞ both S(x) and C(x) approach 1/2, and for negative x they are odd functions, so the sign flips.
Plotting the point (C(t), S(t)) traces the Euler spiral (clothoid); because its curvature is proportional to arc length, it is used for transition curves on roads and railways.
Fresnel integrals also appear in the analysis of the diffraction of light (Fresnel diffraction).

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