Series Expansions Immune to Round-Off Errors for the Transition Function Used in High Frequency Electromagnetic Methods

A series expansion is given for efficient computation of an integral that is called transition function F(x) in the high frequency electromagnetics literature or Fresnel integrals in the optics literature. It is obtained as a limiting case of some series expansions that appear in a solution of an electromagnetic problem in a recent letter. As an interesting property, it involves an arbitrary parameter r that may be used for specifying an interval in which the series becomes somewhat immune to the round-off errors and so the function F(x) can be calculated with any accuracy in that interval. We also give a computational scheme whose speed and accuracy performances fulfill the needs of high frequency electromagnetic methods.