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Thread: Higher Level Integration Marathon & Questions

  1. #226
    -insert title here- Paradoxica's Avatar
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    Re: Higher Level Integration Marathon & Questions

    Quote Originally Posted by seanieg89 View Post
    Eh I wouldn't say that necessarily, I did the integral in a not particularly imaginative way...there could be a clever shortcut.

    To me it does seem like computing this integral is roughly the same mathematical "depth" as computing some particular values of the di/trilogarithm though.

    Anyway, is there a reason why you were seeking to avoid them?
    because I like a nice balance of low tech and efficient solutions to integrals
    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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    Re: Higher Level Integration Marathon & Questions

    Quote Originally Posted by Paradoxica View Post
    because I like a nice balance of low tech and efficient solutions to integrals
    Fair enough, seems a rather subjective matter...to me polylog and polygamma manipulations are okay, but I also have no qualms about using contour integration etc.

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    Re: Higher Level Integration Marathon & Questions

    Quote Originally Posted by Paradoxica View Post
    because I like a nice balance of low tech and efficient solutions to integrals





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    Re: Higher Level Integration Marathon & Questions

    I think you guys should enrol in Extreme Integration (MATH1052), or if you're feeling ambitious try Insane Integration (MATH3025)!

    PM me for details.

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    Ancient Orator leehuan's Avatar
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    Re: Higher Level Integration Marathon & Questions



    (Try ignoring the lengthy 4U way in this thread :P)
    Last edited by leehuan; 15 Nov 2017 at 8:38 PM.
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    Supreme Member seanieg89's Avatar
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    Re: Higher Level Integration Marathon & Questions

    Quote Originally Posted by leehuan View Post


    (Try ignoring the lengthy 4U way in this thread :P)
    Last edited by seanieg89; 15 Nov 2017 at 8:50 PM.
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    -insert title here- Paradoxica's Avatar
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    Re: Higher Level Integration Marathon & Questions





    Prove the integral diverges if the argument of the exponential does not describe an ellipse.

    Challenge: Formulate a generalisation to the above integral in higher dimensions. (Experience with Linear Algebra will be very handy)
    Last edited by Paradoxica; 15 Nov 2017 at 11:30 PM.
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    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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    Re: Higher Level Integration Marathon & Questions

    Quote Originally Posted by Paradoxica View Post




    Prove the integral diverges if the argument of the exponential does not describe an ellipse.

    Challenge: Formulate a generalisation to the above integral in higher dimensions. (Experience with Linear Algebra will be very handy)


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    -insert title here- Paradoxica's Avatar
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    Re: Higher Level Integration Marathon & Questions

    Quote Originally Posted by InteGrand View Post


    An answer that isn't an answer....

    (As in you technically didn't do the computation, not that you're being cryptic)
    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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    Supreme Member seanieg89's Avatar
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    Re: Higher Level Integration Marathon & Questions

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    Re: Higher Level Integration Marathon & Questions

    Quote Originally Posted by Paradoxica View Post
    An answer that isn't an answer....

    (As in you technically didn't do the computation, not that you're being cryptic)
    Well I did say I was sketching a method (i.e. not giving the full answer).

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    Supreme Member seanieg89's Avatar
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    Re: Higher Level Integration Marathon & Questions

    Oh didn't see you edited it, was the divergence thing always there and I just blind before?

    In any case, we can still represent the quadratic form by a symmetric, hence diagonalisable matrix Q. If the quadratic form does not represent an ellipse, then at least one eigenvalue is non-positive, which implies that at least one of the integrals in the above product of single-variable integrals is infinite and we are done.

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    Re: Higher Level Integration Marathon & Questions

    Quote Originally Posted by seanieg89 View Post
    Oh didn't see you edited it, was the divergence thing always there and I just blind before?

    In any case, we can still represent the quadratic form by a symmetric, hence diagonalisable matrix Q. If the quadratic form does not represent an ellipse, then at least one eigenvalue is non-positive, which implies that at least one of the integrals in the above product of single-variable integrals is infinite and we are done.
    I think it was there before. I remember answering it in my head but then forgetting to comment on it in my post. Haha

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    Re: Higher Level Integration Marathon & Questions

    Quote Originally Posted by leehuan View Post


    (Try ignoring the lengthy 4U way in this thread :P)



























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    Re: Higher Level Integration Marathon & Questions

    Easy difficulty


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    Re: Higher Level Integration Marathon & Questions

    Quote Originally Posted by leehuan View Post
    Easy difficulty


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    Re: Higher Level Integration Marathon & Questions

    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.

  18. #243
    -insert title here- Paradoxica's Avatar
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    Re: Higher Level Integration Marathon & Questions

    Quote Originally Posted by He-Mann View Post
    I think you guys should enrol in Extreme Integration (MATH1052), or if you're feeling ambitious try Insane Integration (MATH3025)!

    PM me for details.
    this feels like a meme
    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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    Ancient Orator leehuan's Avatar
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    Re: Higher Level Integration Marathon & Questions

    Quote Originally Posted by Paradoxica View Post
    At least quick partial fractions is possible this time round lol



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    Supreme Member seanieg89's Avatar
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    Re: Higher Level Integration Marathon & Questions



    Last edited by seanieg89; 27 Nov 2017 at 10:20 AM.

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    Ancient Orator leehuan's Avatar
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    Re: Higher Level Integration Marathon & Questions



    (Not sure if already asked)

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    Re: Higher Level Integration Marathon & Questions

    Really hope I'm right coz this took me forever. Also sorry for bad LaTeX.







    Let



    where

    This is easily evaluated depending on the different cases:

    If or , we have , if then and , and if then and .

    Case 1:

    hence for some constant .

    Evaluating the integral for , we obtain that:

    when .

    Case or .

    (which follows from Case 1).

    when or .

    Case 3:



    Let

    By symmetry,

    Let



    By symmetry,

    Therefore , so .

    Subbing the values back in, we obtain .

    Case 4:



    This can be done by the same method as case 3, and so .

    Hence, when , , and otherwise.
    Last edited by calamebe; 4 Dec 2017 at 11:43 PM.
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  23. #248
    -insert title here- Paradoxica's Avatar
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    Re: Higher Level Integration Marathon & Questions

    Quote Originally Posted by calamebe View Post
    Really hope I'm right coz this took me forever. Also sorry for bad LaTeX.
    Fourier Expansion :P
    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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    Re: Higher Level Integration Marathon & Questions

    Quote Originally Posted by Paradoxica View Post
    Fourier Expansion :P
    No idea what that is, I'm not all that far beyond HSC maths
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