The best-known properties and formulas for Fresnel integrals

Real values for real arguments

For real values of argument , the values of the Fresnel integrals and are real.

Simple values at zero and infinity

The Fresnel integrals and have simple values for arguments and :

Analyticity

The Fresnel integrals and are defined for all complex values of , and they are analytical functions of over the whole complex ‐plane and do not have branch cuts or branch points. They are entire functions with an essential singular point at .

Periodicity

The Fresnel integrals and do not have periodicity.

Parity and symmetry

The Fresnel integrals and are odd functions and have mirror symmetry:

Series representations

The Fresnel integrals and have rather simple series representations at the origin:

These series converge at the whole ‐plane and their symbolic forms are the following:

Interestingly, closed-form expressions for the truncated version of the Taylor series at the origin can be expressed through the generalized hypergeometric function , for example:

Asymptotic series expansions

The asymptotic behavior of the Fresnel integrals and can be described by the following formulas (only the main terms of asymptotic expansion are given):

The previous formulas are valid in any directions of approaching point to infinity (). In particular cases when and , the formulas can be simplified to the following relations:

Integral representations

The Fresnel integrals and have the following simple integral representations through sine or cosine that directly follow from the definition of these integrals:

Transformations

The argument of the Fresnel integrals and with square root arguments can sometimes be simplified:

Simple representations of derivatives

The derivatives of the Fresnel integrals and are the sine or cosine functions with simple arguments:

The symbolic derivatives of the order have the following representations:

Simple of differential equations

The Fresnel integrals and satisfy the following third-order linear ordinary differential equation:

They can be represented as partial solutions of the previous equation under the following corresponding initial conditions: