The Cartesian coordinates (X, Y) of the transition curves are calculated via using Fresnel integral (Jeffrey, Dai 2008).
These integrals are known as the Fresnel Integral (Jeffrey, Dai 2008).
In either case, the diffraction pattern due to the principal aperture A can be evaluated using a generalized Fresnel integral derived by this author from the general equations for the propagation of cross-spectral density in a partially coherent optical field (3).
Therefore, it may be desirable to avoid the need for these computations in the first place by designing diffraction experiments so that the aperture illumination will be "almost" coherent and the Fresnel integral (2) can still be used, in spite of the finite size of a given source.
It is well known and easy to show that in the mid zone these expressions are all reduced to the Fresnel integral (2) so that
Formula (5) and (6) only existing numerical solution, and when the integral upper limit tends to infinity, the two Fresnel integrals
all tend to [square root of (2[pi]/4)], i.
The necessary integrals can be expressed in terms of Fresnel integrals, and he goes on to explain how to evaluate these special functions.
Matlab does not include software for Fresnel integrals, so in Section 4.
Using MuPAD we found the expressions for the integrals in terms of Fresnel integrals that we provide in Section 4.
In the Fresnel limit on either side of the aperture plane (|z| [much greater than] [lambda]) the forward and reverse fields are unidirectional, and the forward field is reduced to the standard expressions in terms of Fresnel integrals
for slits and Lommel functions for circular apertures  for z > 0, and to the unperturbed geometrical field for z < 0.
While this integral has an analytic form in terms of the Fresnel integrals
, a simple approximation is more useful in this context.
Error Functions, Dawson's Integral, Fresnel Integrals