Let us describe more precisely the open problems on the example of oscillatory and Fresnel integrals. Let [PHI] [member of] [C.sup.[infinity]] ([R.sup.n], R) be a fixed function.
Mazzucchi, "Generalized infinite-dimensional Fresnel integrals," Comptes Rendus Mathematique, vol.
The choice [PHI] (x) = (i/2h)[[absolute value of (x)].sup.2] is of particular interest and is known under the name of Fresnel integral. This choice gives us a mean, up to normalization by a factor [(2i[pi]h).sup.-d/2], and can be generalized to a Hilbert space H the following way.
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.e.
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 4.2 we consider how to evaluate them efficiently in this computing environment.
He pays special attention to formulas of derivatives of nth-order (with respect to the argument) and of the first derivatives (with respect to the parameter) for most elementary and special functions, covering the derivatives (including the Hurwitz zeta function and Fresnel integrals
) limits (including special functions), indefinite integrals (including elementary and special functions), definite integrals (including Bessel, Mcdonald, Struve, Kelvin, Legendre, Chebyshev, Hermite, Laguerre and Jacobi functions and polynomials), finite sums, infinite series, the connection formulas and representations of hypergeometric functions and of the Meijer G function.
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
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.