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Accurate calculation of the modified Mathieu functions of integer order. (English) Zbl 1124.33019
The authors explore the ability of traditional expressions to calculate accurate values for Mathieu functions of integer order. Emphasis is given on the subtraction errors that occur in some parameter ranges for all of the expressions. They determine how to calculate accurate values of radial Mathieu functions and their first derivatives. Included is a discussion of the Bessel function product series, which has an integer offset for the order of the Bessel functions that is traditionally chosen to be zero (or one). It is shown in the paper that the use of larger offset values that tend to increase with increasing radial function order usually eliminates the subtraction errors. This paper identifies the expressions and evaluation procedures that provide accurate radial Mathieu function values. A brief discussion of the calculation of the angular functions of the first kind that appear in many of these expressions is included. The paper also gives a description of a Fortran computer program that provides accurate values of radial Mathieu functions together with the associated angular functions over extremely wide parameter ranges.
MSC:
33E10Lamé, Mathieu, and spheroidal wave functions
33F05Numerical approximation and evaluation of special functions