Summary. This paper deals with rational functions ~b(z) approximating the exponential function exp(z) related to numerical procedures for solving initial value problems. Motivated by positivity and contractivity requirements imposed on these numerical procedures we study the greatest nonnegative number R, denoted by R(qS), such that ~b is absolutely monotonic on (-R,0]. An algorithm for the computation of R(th) is presented. Application of this algorithm yields the value R(th) for the well-known Pad6 approximations to exp(z). For some specific values of m, n and p we determine the maximum of R(q~) when ~b varies over the class of all rational functions q~ with degree of the numerator
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