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The work of Friedman and Ko and Müller guarantee the polynomial time computability of the integrals of analytic functions inside the circle of convergence. But do algorithms have practical value? Is there any implementation? Edit: the function should be also polynomial - time computable

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    $\begingroup$ I think you are misstating the linked results. In particular, in the abstract of Müller's paper, we see "As central result, the function is computable in polynomial time if and only if the coefficients of the [Taylor] series are uniformly computable in polynomial time." So, in general there is no "guarantee [of] the polynomial time computability of the integrals of analytic functions inside the circle of convergence": for such computability of the function, the coefficients of the Taylor series must be uniformly computable in polynomial time. $\endgroup$
    – Iosif Pinelis
    Commented Feb 18 at 17:14
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    $\begingroup$ Previous comment continued: On the other hand, if the latter condition on the Taylor series holds, then we clearly have the polynomial-time algorithm to compute the values of the function (and its integral). $\endgroup$
    – Iosif Pinelis
    Commented Feb 18 at 17:14
  • $\begingroup$ @Iosif Pinelis, thank you for your remark, I have edited the statement $\endgroup$
    – poeaqnwgo
    Commented Feb 18 at 18:21
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    $\begingroup$ Well, then, as noted in my comment, the Taylor series clearly provides the polynomial-time algorithm to compute the values of the function (and its integral). $\endgroup$
    – Iosif Pinelis
    Commented Feb 18 at 18:41

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