Calculating integrals involving floor function

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Discussion Overview

The discussion revolves around the evaluation of the integral ∫[cot(x)]dx from x=0 to x=π, where [.] denotes the floor function. Participants explore whether this integral can be computed, considering both analytical and numerical approaches, as well as the implications of convergence and limits.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested, Mathematical reasoning

Main Points Raised

  • Some participants question the possibility of calculating the integral due to concerns about convergence, suggesting it may be impossible to evaluate directly.
  • Others propose converting the integral into a series involving arctan, though uncertainty remains about the series' analytical evaluability.
  • A participant suggests redefining the integral using limits to avoid convergence issues, specifically considering the limit as δ approaches 0.
  • Another participant agrees with the limit approach, comparing it to the Cauchy integral (Principal Value) and suggesting that the integral could yield finite values under certain conditions.
  • Symmetry of the function is noted as a potentially useful property in evaluating the integral.

Areas of Agreement / Disagreement

Participants express differing views on the convergence of the integral and its evaluability, with some proposing limits as a solution while others maintain that the integral is not converging. No consensus is reached regarding the overall feasibility of the integral's evaluation.

Contextual Notes

Participants highlight the importance of limits and the behavior of the floor function in the context of the integral, indicating that assumptions about convergence and the properties of the functions involved are critical to the discussion.

smatik
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Is it possible to claculate ∫[cot(x)]dx from x=0 to x=\piwhere [.] represents floor functon or the greatest integer function??
it seems impossible to me but can we use the properties of definite integrals to somehow evaluate the area?
 
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It is possible to convert this to a series (involving arctan), but I have no idea if this series can be evaluated analytically. It is possible to get a numeric approximation, of course.
 
smatik said:
Is it possible to claculate ∫[cot(x)]dx from x=0 to x=\piwhere [.] represents floor functon or the greatest integer function?
Impossible because this defined integral is not converging.
 
Well, we can consider
$$\lim_{\delta \to 0} \int_\delta^{\pi-\delta} [\cot(x)] dx$$
This should be well-defined and finite.
 
Yes, if both limits of the integral tend respectively to 0 and pi, with the same gap (delta), this is similar to the Cauchy integral (Principal Value). Then the limit of the value of the integral is finite : -pi/2 in case of the floor function and pi/2 in case of the ceiling function.
 
Right, we can use the symmetry of the function. Nice.
 

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