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