Fresnel Integrals and Cornu Spiral

Worked Example: Series Expansion for Small x

We use the explicit π/2 normalization for the Fresnel integrals:
C(x) = ∫₀ˣ cos(πt²/2) dt
S(x) = ∫₀ˣ sin(πt²/2) dt

Both functions can be expanded into Maclaurin series that converge for all x. For x = 1, substituting into the series up to the first few terms provides a highly accurate approximation.

1. C(x) = x - (π²/40)x⁵ + (π⁴/3456)x⁹ - ...
2. For x = 1: C(1) ≈ 1 - 0.24674 + 0.02818 ≈ 0.78144. (The precise value is 0.7798934...).
3. S(x) = (π/6)x³ - (π³/336)x⁷ + (π⁵/42240)x¹¹ - ...
4. For x = 1: S(1) ≈ 0.52360 - 0.09228 + 0.00724 ≈ 0.43856. (The precise value is 0.4382591...).

The unit-speed tangent to the spiral is given by (C'(x), S'(x)) = (cos(πx²/2), sin(πx²/2)). At x = 1, the tangent is (cos(π/2), sin(π/2)) = (0, 1).

Experiment

Explore the path of the Cornu spiral by controlling the upper bound of the integration, x.

1. Open the Fresnel Integrals and Cornu Spiral module.
2. Select a value from the "Probe x" dropdown, for example 0.5 or 2.
3. Observe the calculated values for C(x) and S(x) along with the derivatives C'(x) and S'(x).
4. Watch the parametric plot on the right update to highlight the portion of the Cornu spiral from 0 to the chosen x, complete with the tangent vector indicating the instantaneous direction.

Limitations

The power series used for evaluation converges everywhere, but for very large values of x, calculating it directly leads to severe numerical instability and cancellation errors. In a full implementation, asymptotic expansions or rational approximations would be used for large |x|. Our visualization strictly focuses on the domain x ∈ [-2, 2].

References

The properties and series expansions for Fresnel Integrals are formally established in standard mathematical literature.

• NIST Digital Library of Mathematical Functions: Section 7.2 Fresnel Integrals