Fresnel integrals

Fresnel integrals, S(x) and C(x), are two transcendental functions named after Augustin-Jean Fresnel that are used in optics, which are closely related to the error function (erf). They arise in the description of near-field Fresnel diffraction phenomena and are defined through the following integral representations:

The simultaneous parametric plot of S(x) and C(x) is the Euler spiral (also known as the Cornu spiral or clothoid). Recently, they have been used in the design of highways and other engineering projects.

The Fresnel integrals admit the following power series expansions that converge for all x:

Some authors, including Abramowitz and Stegun, (eqs 7.3.1 – 7.3.2) use ${\displaystyle {\frac {\pi }{2}}t^{2}}$ for the argument of the integrals defining S(x) and C(x). To get these functions, multiply the above integrals by ${\displaystyle {\sqrt {\frac {2}{\pi }}}}$ and multiply the argument x by ${\displaystyle {\sqrt {\frac {\pi }{2}}}}$.

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