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A holonomic function is an analytic function, which satisfies a linear differential equation with polynomial coefficients. In particular, the elementary functions , etc. and many special functions like , , Bessel functions, etc. are holonomic functions.

Given a holonomic function (determined by the linear differential equation it satisfies and initial conditions in a non singular point ), we show how to perform arbitrary precision evaluations of at a non singular point on the Riemann surface of , while estimating the error.

Moreover, if the coefficients of the polynomials in the equation for are algebraic numbers, then our algorithm is asymptotically very fast: if is the time needed to multiply two digit numbers, then we need a time to compute digits of .

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**Note:** this paper also occurred as appendix C in my PhD. and as a 1996 preprint.