Fresnel integral

The spiral is also closely related to the Fresnel integrals.

Cartesian coordinates of points along this spiral are given by the Fresnel integrals.

The Fresnel integrals can be expressed using the error function as follows:

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

In mathematics, a Böhmer integral is an integral introduced by generalizing the Fresnel integrals.

Using the power series expansions above, the Fresnel integrals can be extended to the domain of complex numbers, and they become analytic functions of a complex variable.

A similar argument can be made to derive continued fraction expansions for the Fresnel integrals, for the Dawson function, and for the incomplete gamma function.

Faddeeva Package, free/open-source C++/C code to compute complex error functions (from which the Fresnel integrals can be obtained), with wrappers for Matlab, Python, and other languages.