Rational-fraction approximations and asymptotic series for functions which arise in skin-effect and allied problems

The first two functions are denoted by the symbols ÿ(x) and ¿(x) and are defined by the relation ÿ(x) ¿ j¿(x) = I0(x¿j)/¿jI1(x¿j) where In(x) is the modified Bessel function of order n. They can be expressed in terms of Kelvin functions and their derivatives as follows: ÿ(x) = ber x ber¿x + bei x bei¿x/(ber¿ x)2 + (bei¿ x)2 ¿(x) = ber x bei¿x + bei x ber¿x/(ber¿ x)2 + (bei¿ x)2 The other two functions are Butterworth´s functions ÿn(x) and ¿n(x) defined by the relation ÿn(x) + j¿n(x) = In+1(x¿j)/In-1(x¿j) A sequence of up to seven progressively-more-accurate rational-fraction approximations is obtained for each of the four functions by taking real and imaginary parts of successive convergents of a continued fraction in the complex variable x¿j. In addition there are, for each function, asymptotic series in which the general coefficients can be computed from simple recurrence relations. A detailed description of an Algol 60 procedure for calculating values of ÿ(x) and ¿(x) to an accuracy of at least five significant decimal digits is given in an appendix.