Chebyshev series approximation on complex domains

This thesis is an account of work carried out at the Department of Mathematics, Durham University, between October 1979 and September 1982. A method of approximating functions in regions of the complex plane is given. Although it is not, in general, a near minimax approximation it is shown that it can give good results. A review of approximation in the complex plane is given in Chapter 1.Chapter 2 contains the basic properties of Chebyshev polynomials and the Chebyshev series, together with methods for calculating the coefficients in the series. The maximum error, over a complex domain, of a truncated Chebyshev series is investigated in Chapter 3 and Chapter 4 shows how the Bessel functions of the first and second kinds of integer order could be approximated over the entire complex plane. Numerical calculations were performed on the NUMAC IBM370.168 computer.