Specialized Mathematical Functions

This section contains specialized mathematical functions used in ATMTools.

 Name Description Computes the value of the Fresnel sine and cosine integrals \$C(x) - i S(x) = int_0^x dt e^{-ipi t^2/2}\$. Will either approximate or use quadl to compute, depending on the flag est. est = TRUE (default) for approximation, FALSE for quadl integration. Note that both \$C(x)\$ and \$S(x)\$ have odd symmetry, therefore, one need only compute for positive values of x. Returns the complex hypergeometric \$_aF_b({A};{B};{x})\$ function using a series approximation with N terms where a is the length of the vector A and b is the length of the vector B. Asymptotic approximation of \$_2F_3({B},{D},-{z})\$, z >> 1. This routine calculates the asymptotic expression for the generalized hypergeometric funcion for the case for which there are two numerator parameters and three denominator parameters. The asymptotic expression approximates \$_2F_3({B},{D},-{z})\$ when Z is large and positive. Computes values of the Lommel functions L and M using U and V Lommel functions to within an absolute tolerance of tol. If tol is an integer, TOL terms will be used in the series expansion. Function to calculate the ratio of gamma functions as defined in Sasiella's book. \$Gamma(a,b) = Gamma(a1)*Gamma(a2)*...*Gamma(ap)/ (Gamma(b1)*Gamma(b2)*...*Gamma(bq))\$ where a and b are vectors of length p and q respectively. Computes the radial and azimuthal order of Zernike polynomial using Noll's ordering scheme.