Accurate calculation of the modified Mathieu functions of integer order

Several different expressions involving infinite series are available for calculating the radial (i.e., modified) Mathieu functions of integer order. Mathieu functions depend on three parameters: order, radial coordinate, and a size parameter that is chosen here to be either real or imaginary. Expressions traditionally used to calculate the radial functions result in inaccurate function values over some parameter ranges due to unavoidable subtraction errors that occur in the series evaluation. For many of the expressions the errors for small orders increase without bound as the size parameter increases. In the present paper, the subtraction error obtained using traditional expressions is explored with regard to parameter values. 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 here 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. This effort was guided by recent advancements in the calculation of prolate spheroidal functions. The Mathieu functions are a special case of spheroidal functions, resulting in a similarity of behavior in their evaluation.