The Location of Zeros
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We investigate in this chapter real functions of the real variable x. In particular we assume that the coefficients a 0, a 1, a 2,... of the polynomials a 0 + a 1 x +a 2 x 2 + ... + a n x n and of the power series a 0 + a 1 x + a 2 x 2 + ... which we shall be considering are real. We assume further, unless the contrary is stated, that all functions are analytic in the corresponding intervals. The theorems, however, are changed only slightly or not at all if we introduce more general assumptions, e.g. the existence of derivatives up to some order. The zeros in the following are always to be counted according to their multiplicity.