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 :
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 .
The Fresnel integrals and do not have periodicity.
Parity and symmetry
The Fresnel integrals and are odd functions and have mirror symmetry:
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:
The Fresnel integrals and have the following simple integral representations through sine or cosine that directly follow from the definition of these integrals:
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: