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Calculating integrals involving floor function

  1. Apr 3, 2013 #1
    Is it possible to claculate ∫[cot(x)]dx from x=0 to x=[itex]\pi[/itex]where [.] 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?
     
  2. jcsd
  3. Apr 3, 2013 #2

    mfb

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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.
     
  4. Apr 3, 2013 #3
    Impossible because this defined integral is not converging.
     
  5. Apr 3, 2013 #4

    mfb

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    Well, we can consider
    $$\lim_{\delta \to 0} \int_\delta^{\pi-\delta} [\cot(x)] dx$$
    This should be well-defined and finite.
     
  6. Apr 3, 2013 #5
    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.
     
  7. Apr 3, 2013 #6

    mfb

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    Right, we can use the symmetry of the function. Nice.
     
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