Abstract
It is well known that interpolation by polynomials of high degree is not well behaved. In this paper we give a theoretical basis for this by examining C. Runge's famous example of a continuous function f(x)over a closed interval for which the Lagrange interpolating polynomials fail to converge. Runge showed that there is a convergence curvedefined in the complex plane whose interior contains the (real) closed interval on which the function f(x)is defined. If f(x) has no (complex) singularities within this convergence curve, then the Lagrange interpolating polynomials will converge uniformly to f(x)over the interval. This discussion uses only elementary topics from advanced calculus and beginning complex variable theory, and thus provides for a capstone series of one or two lectures to show the applicability of these courses on an applied problem of considerable interest. The advent of computer algebra programs such as Mathematicaand Maple V make this discussion all the more tractable.
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