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There are ways to approximate a sum through integration like the Euler–Maclaurin formula, which requires the function $f(x)$ to be continuous, but there are several ways to extend the formula to functions with singularities:

  1. Buchheit and Keßler - Singular Euler–Maclaurin expansion (See also their post at the Wolfram Blog: The singular Euler–Maclaurin expansion)
  2. Waterman, Yos, and Abodeely - Numerical Integration of Non-analytic Functions.

For me, it's not clear if the mentioned rules are acceptable and if not, what happens with the approximation after the application of the formula to the following functions, if one (or more) of the points in the sum are points of discontinuity for a given function (assuming $f$ is continuous):

  1. $\Gamma(x)$
  2. $\ln\Gamma(x)$
  3. $\ln(f(x))$
  4. $\dfrac{1}{f(x)^2}$

The other question is: is it possible to determine if there is a point of singularity in a sum, for example, $\sum_{a}^{b}\ln(f(x))$, without computing the sum componentwise?

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    $\begingroup$ MathOverflow users are advised to avoid answering posts with multiple questions. Your post seems to contain at least 5 (!) questions. $\endgroup$
    – Iosif Pinelis
    Commented Jul 24, 2023 at 2:21
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    $\begingroup$ you might be interested in the answers here: mathoverflow.net/questions/239558/… (disclaimer, I have written one of the answers there) $\endgroup$
    – Sidharth Ghoshal
    Commented Jul 24, 2023 at 16:01
  • $\begingroup$ @IosifPinelis, sure, I took a note. The main question is the last, which concerns the detection of a discontinuity. $\endgroup$
    – roignoirewg
    Commented Jul 24, 2023 at 16:51

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