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Thread: Higher Level Integration Marathon & Questions

  1. #101
    Ancient Orator leehuan's Avatar
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    Re: Extracurricular Integration Marathon


  2. #102
    -insert title here- Paradoxica's Avatar
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    Re: Extracurricular Integration Marathon

    Quote Originally Posted by leehuan View Post
    Done earlier on this thread.

    We all know the answer is
    If I am a conic section, then my e = ∞

    Just so we don't have this discussion in the future, my definition of the natural numbers includes 0.

  3. #103
    Senior Member KingOfActing's Avatar
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    Re: Extracurricular Integration Marathon

    Paradoxica likes this.
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  4. #104
    -insert title here- Paradoxica's Avatar
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    Re: Extracurricular Integration Marathon

    Quote Originally Posted by leehuan View Post
    Not sure if this is doable

    Then why post it?
    If I am a conic section, then my e = ∞

    Just so we don't have this discussion in the future, my definition of the natural numbers includes 0.

  5. #105
    Ancient Orator leehuan's Avatar
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    Re: Extracurricular Integration Marathon

    Nvm deleted it. I was mindlessly experimenting with Wolfram when I realised what the f**** I actually did

  6. #106
    -insert title here- Paradoxica's Avatar
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    Re: Extracurricular Integration Marathon

    Quote Originally Posted by leehuan View Post
    Not sure if this is doable

    Since this is a series of rectangles across the real line, convert it into a sum.

    Unsing the standard trigo-geometric summation identity for sin(x), the answer is

    If I am a conic section, then my e = ∞

    Just so we don't have this discussion in the future, my definition of the natural numbers includes 0.

  7. #107
    Ancient Orator leehuan's Avatar
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    Re: Extracurricular Integration Marathon


  8. #108
    -insert title here- Paradoxica's Avatar
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    Re: Extracurricular Integration Marathon

    Quote Originally Posted by leehuan View Post
    Take the Imaginary part of the following integral:



    The imaginary component of the integral is absolutely convergent, and so is the Taylor expansion, so (insert reasoning here) it is valid to exchange summation, integration and complex extraction.

    Take the Taylor series of ex and replace x with eix

    Swap the order of integration and summation. Integrate termwise.

    final answer:

    π(e-1)/2
    If I am a conic section, then my e = ∞

    Just so we don't have this discussion in the future, my definition of the natural numbers includes 0.

  9. #109
    Supreme Member seanieg89's Avatar
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    Re: Extracurricular Integration Marathon

    Quote Originally Posted by Paradoxica View Post
    Take the Imaginary part of the following integral:



    The imaginary component of the integral is absolutely convergent, and so is the Taylor expansion, so (insert reasoning here) it is valid to exchange summation, integration and complex extraction.

    Take the Taylor series of ex and replace x with eix

    Swap the order of integration and summation. Integrate termwise.

    final answer:

    π(e-1)/2
    I don't think that integral is absolutely convergent...there might be a nice way of justifying that interchange, but its not a trivial matter.

  10. #110
    -insert title here- Paradoxica's Avatar
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    Re: Extracurricular Integration Marathon

    Quote Originally Posted by seanieg89 View Post
    I don't think that integral is absolutely convergent...there might be a nice way of justifying that interchange, but its not a trivial matter.
    The Imaginary part is absolutely convergent. The real part is divergent.

    IDK about the justification of interchange.
    If I am a conic section, then my e = ∞

    Just so we don't have this discussion in the future, my definition of the natural numbers includes 0.

  11. #111
    Rambling Spirit
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    Re: Extracurricular Integration Marathon

    Quote Originally Posted by Paradoxica View Post
    The Imaginary part is absolutely convergent. The real part is divergent.

    IDK about the justification of interchange.
    Are you absolutely (lol) sure about this?

  12. #112
    -insert title here- Paradoxica's Avatar
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    Re: Extracurricular Integration Marathon

    Quote Originally Posted by InteGrand View Post
    Are you absolutely (lol) sure about this?
    Dirichlet's test.
    If I am a conic section, then my e = ∞

    Just so we don't have this discussion in the future, my definition of the natural numbers includes 0.

  13. #113
    Supreme Member seanieg89's Avatar
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    Re: Extracurricular Integration Marathon

    The imaginary part of the integrand is the original integrand, which is



    This is not absolutely convergent. We have something continuous and periodic divided by x. Upon taking absolute values, integrating this is like summing a harmonic series.

    The integral converges in the sense of an improper Riemann integral because it oscillates as well as decays (things like the Dirichlet test or alternating series test pin this notion down), but its rate of decay is too slow to give us absolute convergence.

    The most common ways of justifying an interchange rely on our limit function being absolutely integrable (the monotone/dominated/vitali convergence theorems), or on us having an absolutely integrable error term which we can bound and show tends to zero, so this is not as routine as you might think, even if it is probably justifiable somehow.
    Last edited by seanieg89; 29 Jun 2016 at 10:11 AM.

  14. #114
    -insert title here- Paradoxica's Avatar
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    Re: Extracurricular Integration Marathon

    Quote Originally Posted by seanieg89 View Post
    The imaginary part of the integrand is the original integrand, which is



    This is not absolutely convergent. We have something continuous and periodic divided by x. Upon taking absolute values, integrating this is like summing a harmonic series.

    The integral converges in the sense of an improper Riemann integral because it oscillates as well as decays (things like the Dirichlet test or alternating series test pin this notion down), but its rate of decay is too slow to give us absolute convergence.

    The most common ways of justifying an interchange rely on our limit function being absolutely integrable (the monotone/dominated convergence theorems), or on us having an absolutely integrable error term which we can bound and show tends to zero, so this is not as routine as you might think, even if it is probably justifiable somehow.
    I don't even know, Leehuan says this is from an IB textbook. Advanced much?
    If I am a conic section, then my e = ∞

    Just so we don't have this discussion in the future, my definition of the natural numbers includes 0.

  15. #115
    Supreme Member seanieg89's Avatar
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    Re: Extracurricular Integration Marathon

    Quote Originally Posted by Paradoxica View Post
    I don't even know, Leehuan says this is from an IB textbook. Advanced much?
    There has to be a less potentially dodgy way of doing it, I also thought of expanding out a series in e^(ix) to start with, but I couldn't see any way of justifying that so I left it.

    Gotta be careful with these things, because functions/sequences that oscillate and decay but are not absolutely integrable are a common source of counterexamples to otherwise believable claims.
    Paradoxica likes this.

  16. #116
    -insert title here- Paradoxica's Avatar
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    Re: Extracurricular Integration Marathon

    Quote Originally Posted by seanieg89 View Post
    There has to be a less potentially dodgy way of doing it, I also thought of expanding out a series in e^(ix) to start with, but I couldn't see any way of justifying that so I left it.

    Gotta be careful with these things, because functions/sequences that oscillate and decay but are not absolutely integrable are a common source of counterexamples to otherwise believable claims.
    Well the only other way I see is by contouring the first quadrant but like I said.

    Too advanced.
    If I am a conic section, then my e = ∞

    Just so we don't have this discussion in the future, my definition of the natural numbers includes 0.

  17. #117
    Ancient Orator leehuan's Avatar
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    Re: Extracurricular Integration Marathon

    I don't even know at all. When I first saw this integral I had absolutely no idea on what to do obviously.

  18. #118
    -insert title here- Paradoxica's Avatar
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    Re: Extracurricular Integration Marathon

    Quote Originally Posted by leehuan View Post
    I don't even know at all. When I first saw this integral I had absolutely no idea on what to do obviously.
    I only saw the imaginary extraction during an epiphany on the train last week.

    My first thought was contouring due to the singularity but I have little experience with that.

    Taylor expansion didn't occur to me until a lot later.
    If I am a conic section, then my e = ∞

    Just so we don't have this discussion in the future, my definition of the natural numbers includes 0.

  19. #119
    Supreme Member seanieg89's Avatar
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    Re: Extracurricular Integration Marathon

    Quote Originally Posted by leehuan View Post
    I don't even know at all. When I first saw this integral I had absolutely no idea on what to do obviously.
    Could you clarify the source and the typical integration techniques used in this book? I will have another look at it later today if I have time.
    Paradoxica likes this.

  20. #120
    Supreme Member seanieg89's Avatar
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    Re: Extracurricular Integration Marathon

    Quote Originally Posted by Paradoxica View Post
    Well the only other way I see is by contouring the first quadrant but like I said.

    Too advanced.
    We are in the extracurricular marathon, so post a contour integration solution if you do find one.

    The slow decay of the integrand again makes things nontrivial (what contour do you suggest?), and the singularity is removable, not a pole so there is no benefit in contouring about it (for the original function, it is a pole of the complexified function, but the problem of slow decay remains).
    Last edited by seanieg89; 29 Jun 2016 at 2:10 PM.

  21. #121
    Rambling Spirit
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    Re: Extracurricular Integration Marathon

    So is this integral from an IB maths textbook?

  22. #122
    -insert title here- Paradoxica's Avatar
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    Re: Extracurricular Integration Marathon

    Quote Originally Posted by InteGrand View Post
    So is this integral from an IB maths textbook?
    According to Leehuan
    If I am a conic section, then my e = ∞

    Just so we don't have this discussion in the future, my definition of the natural numbers includes 0.

  23. #123
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    Re: Extracurricular Integration Marathon

    Quote Originally Posted by Paradoxica View Post
    According to Leehuan
    What techniques of integration do they learn in IB? Do they learn Complex Analysis methods? I'm guessing they'd do double integrals?
    kawaiipotato likes this.

  24. #124
    Ancient Orator leehuan's Avatar
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    Re: Extracurricular Integration Marathon

    Still waiting for my source to reply...

  25. #125
    -insert title here- Paradoxica's Avatar
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    Re: Extracurricular Integration Marathon

    Quote Originally Posted by seanieg89 View Post
    We are in the extracurricular marathon, so post a contour integration solution if you do find one.

    The slow decay of the integrand again makes things nontrivial (what contour do you suggest?), and the singularity is removable, not a pole so there is no benefit in contouring about it (for the original function, it is a pole of the complexified function, but the problem of slow decay remains).
    The classical semicircular contour indented at the origin works for every power series term (Jordan's Lemma) but the problem still remains at bay: How do we justify interchange of summation and integration?
    If I am a conic section, then my e = ∞

    Just so we don't have this discussion in the future, my definition of the natural numbers includes 0.

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