Fresnel integrals

Fresnel integrals are functions formed by integrating the sine or cosine of a quadratic phase. They describe wave propagation in near-field diffraction and trace the Cornu spiral.

Fresnel diffraction: A wave-optics approximation that describes diffraction when source, aperture, and observation distances are finite. Its field amplitudes are expressed through differences of Fresnel integrals.
Oscillatory integral: An integral whose integrand oscillates rapidly, often requiring special methods to evaluate or approximate it. Fresnel integrals are a standard example of an oscillatory integral.
Knife-edge diffraction: Diffraction of a wave around a straight boundary that blocks part of the wavefront. The field near a straight edge is computed from Fresnel integrals.
Augustin-Jean Fresnel: A French physicist whose wave theory of light established the mathematical foundations of diffraction. His diffraction theory gave the integrals their name and physical setting.
Cornu spiral: A plane curve traced by the complex Fresnel integral as its argument varies. It converts an integral between two limits into a geometrical displacement.
Gaussian integral: The integral of an exponential of a negative quadratic, with value linked to the square root of pi. Analytic continuation connects Gaussian integrals to quadratic-phase exponentials.
Rectangular aperture diffraction: The diffraction pattern produced when a wave passes through a rectangular opening. Its one-dimensional edge contributions can be evaluated with Fresnel integrals.
Cornu method: A graphical method for evaluating diffraction by measuring displacements along the Cornu spiral. It turns Fresnel’s integrals into a construction using a drawn curve.
Quadratic phase: A phase whose value varies as the square of a spatial or temporal coordinate. The squared coordinate in this phase generates the Fresnel functions.
Euler's formula: The identity relating complex exponentials to sine and cosine: e^(ix) = cos(x) + i sin(x). It combines the sine and cosine integrals into one complex-valued function.

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Integration by partsRelated: Integration by parts relates their tails to oscillatory expressions and supports asymptotic estimates.