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 ø(x) and ψ(x) and are defined by the relationø(x) − jψ(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:ø(x) = ber x ber′x + bei x bei′x/(ber′ x)2 + (bei′ x)2ψ(x) = ber x bei′x + bei x ber′x/(ber′ x)2 + (bei′ x)2The other two functions are Butterworth's functions øn(x) and ψn(x) defined by the relationøn(x) + jψn(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 ø(x) and ψ(x) to an accuracy of at least five significant decimal digits is given in an appendix.
