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Thread: Higher Level Integration Marathon & Questions

  1. #51
    Supreme Member seanieg89's Avatar
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    Re: Extracurricular Integration Marathon

    Quote Originally Posted by Drsoccerball View Post




    The integrand has its modulus squared equal to (x^2 + (1-x^2)cos^2(theta))^n =< (x^2 + (1-x^2))^n=1.

    The triangle inequality completes the proof.
    Drsoccerball likes this.

  2. #52
    -insert title here- Paradoxica's Avatar
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    Re: Extracurricular Integration Marathon

    Quote Originally Posted by omegadot View Post


    The Laurent series for arctan(z) looks unpromising.... We need a different approach.
    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. #53
    Loquacious One Drsoccerball's Avatar
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    Re: Extracurricular Integration Marathon

    When we do Cauchy's Integrals what are we supposed to picture in our heads?

  4. #54
    -insert title here- Paradoxica's Avatar
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    Re: Extracurricular Integration Marathon

    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. #55
    Supreme Member seanieg89's Avatar
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    Re: Extracurricular Integration Marathon

    Quote Originally Posted by Paradoxica View Post


    This integral is clearly a prime target for contour integration. (We could factorise into its real quadratic factors, use partial fractions, and integrate the simpler summands, but contour integration seems faster).

    Note that since the quartic denominator is both even and real, the poles of the integrand occur at where is an arbitrarily chosen root, say the one in the first quadrant of the the complex plane.

    Now if we take a semicircular contour (radius R) with diameter on the real axis centred at the origin, and semicircular arc in the upper half-plane, then the integral around this contour (with positive orientation) is just equal to I, because the integrand decays as 1/R^2, and so the contribution from the curved segment of length O(R) tends to zero.

    On the other hand, for sufficiently large R this integral is just equal to



    It remains to compute , which is just the principal square root of .

    Writing , we get

    The resulting biquadratic yields , where is the golden ratio.

    Hence

    Last edited by seanieg89; 1 Feb 2016 at 11:57 AM.

  6. #56
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    Re: Extracurricular Integration Marathon

    Quote Originally Posted by omegadot View Post




    Last edited by Paradoxica; 17 Feb 2016 at 11:08 AM.
    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. #57
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    Re: Extracurricular Integration Marathon

    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.

  8. #58
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    Re: Extracurricular Integration Marathon

    Quote Originally Posted by Paradoxica View Post



  9. #59
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    Re: Extracurricular Integration Marathon

    Quote Originally Posted by omegadot View Post
    #ISC+MasterRace
    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.

  10. #60
    Loquacious One Drsoccerball's Avatar
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    Re: Extracurricular Integration Marathon


  11. #61
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    Re: Extracurricular Integration Marathon

    Quote Originally Posted by Drsoccerball View Post


    Well, I mean, Sophomore's Dream is nice and all, but there isn't a closed form...
    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.

  12. #62
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    Re: Extracurricular Integration Marathon

    Quote Originally Posted by Paradoxica View Post


    Well, I mean, Sophomore's Dream is nice and all, but there isn't a closed form...
    Yes that's the answer how did you do it though O.o ? Did you create a taylor series and just integrate each term or?

  13. #63
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    Re: Extracurricular Integration Marathon

    Quote Originally Posted by Drsoccerball View Post
    Yes that's the answer how did you do it though O.o ? Did you create a taylor series and just integrate each term or?
    Inspection
    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.

  14. #64
    Rambling Spirit
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    Re: Extracurricular Integration Marathon

    Quote Originally Posted by Drsoccerball View Post
    Yes that's the answer how did you do it though O.o ? Did you create a taylor series and just integrate each term or?
    Yeah, it can be proved that way: https://en.wikipedia.org/wiki/Sophomore%27s_dream#Proof .

  15. #65
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    Re: Extracurricular Integration Marathon

    Quote Originally Posted by Paradoxica View Post
    I can provide a hint if anyone is attempting this.
    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.

  16. #66
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    Re: Extracurricular Integration Marathon

    Quote Originally Posted by omegadot View Post
    I have not found a complete solution to this integral, only generalised forms. Do you happen to have a solution that specifically deals with this integral?
    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. #67
    Retired Carrotsticks's Avatar
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    Re: Extracurricular Integration Marathon

    Quote Originally Posted by Paradoxica View Post


    Quote Originally Posted by Paradoxica View Post
    Inspection
    And by 'inspection' you mean 'Google' or 'Wolfram'?

    Nobody's buying it, mate.

  18. #68
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    Re: Extracurricular Integration Marathon

    Quote Originally Posted by Carrotsticks View Post
    And by 'inspection' you mean 'Google' or 'Wolfram'?

    Nobody's buying it, mate.
    It was a joke -_-

    I've already encountered Sophomore's Dream before, on one of my many followed blogs.

    As for Ahmed's Integral, I didn't know the answer was exactly that until I put the numerical answer into ISC+.
    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. #69
    Retired Carrotsticks's Avatar
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    Re: Extracurricular Integration Marathon

    Quote Originally Posted by Paradoxica View Post
    It was a joke -_-

    I've already encountered Sophomore's Dream before, on one of my many followed blogs.
    Didn't look like a joke to me. Looked like taking credit where credit was not due.

    But I'll humour you. Explain the your first "wild guess" then.

    I'm sure we're all fascinated to see how you guessed such an answer, whilst refusing to provide at least any sort of outline of a method.

  20. #70
    -insert title here- Paradoxica's Avatar
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    Re: Extracurricular Integration Marathon

    Quote Originally Posted by Carrotsticks View Post
    Didn't look like a joke to me. Looked like taking credit where credit was not due.

    But I'll humour you. Explain the your first "wild guess" then.

    I'm sure we're all fascinated to see how you guessed such an answer, whilst refusing to provide at least any sort of outline of a method.
    From above: As for Ahmed's Integral, I didn't know the answer was exactly that until I put the numerical answer into ISC+.

    Also, putting the integral into Wolfram Alpha doesn't give you the closed form. Google... well you can't exactly search tex code easily. So I resorted to the above, taking the numerical value of the integral to 50 decimal places.
    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.

  21. #71
    -insert title here- Paradoxica's Avatar
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    Re: Extracurricular Integration Marathon

    Quote Originally Posted by Carrotsticks View Post
    I'm sure we're all fascinated to see how you guessed such an answer, whilst refusing to provide at least any sort of outline of a method.
    Well I would post my solution which I figured out yesterday, but I'm not sure it's worth anything at this point.
    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.

  22. #72
    Retired Carrotsticks's Avatar
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    Re: Extracurricular Integration Marathon

    Quote Originally Posted by Paradoxica View Post
    Well I would post my solution which I figured out yesterday, but I'm not sure it's worth anything at this point.
    No, I must insist.

    Just a brief and general outline will do.

  23. #73
    -insert title here- Paradoxica's Avatar
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    Re: Extracurricular Integration Marathon

    Quote Originally Posted by Carrotsticks View Post
    Just a brief and general outline will do.


    omegadot likes this.
    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.

  24. #74
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    Re: Extracurricular Integration Marathon

    Thank you. The spirit of this thread exists in the solution methods, not the answers.

    This is still not fully consistent with the below post, but I'll drop the case as an answer has now been given, regardless of its true origins.

    Quote Originally Posted by Paradoxica View Post
    I have not found a complete solution to this integral, only generalised forms. Do you happen to have a solution that specifically deals with this integral?
    Last edited by Carrotsticks; 21 Feb 2016 at 1:49 PM.

  25. #75
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    Re: Extracurricular Integration Marathon

    Quote Originally Posted by Carrotsticks View Post
    Thank you. The spirit of this thread exists in the solution methods, not the answers.

    This is still not fully consistent with the below post, but I'll drop the case as an answer has now been given, regardless of its true origins.
    In saying so, I can't do anything that can only be done through contour integration, so my box of tools is much smaller than yours, or most people who are on this thread.
    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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