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Recent content by seanieg89

  1. seanieg89

    Higher Level Integration Marathon & Questions

    $Some high dimensional geometric weirdness that we can explore with integration:\\ \\ a) By analogy with the volumes by slices of MX2, find an expression for the volume of the n-dimensional unit ball. Express your answer in terms of factorials or the Gamma function. By using Stirling's formula...
  2. seanieg89

    Higher Level Integration Marathon & Questions

    \int \frac{e^x-e^{-x}}{e^{2x}+1}\, dx = \int \frac{u^2-1}{u^2(u^2+1)}\, du= \int \frac{2}{u^2+1}-\frac{1}{u^2}\, du\\ \\ = 2\arctan(e^x)+e^{-x}+C\\ \\ $(where $u:=e^x$)$
  3. seanieg89

    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...
  4. seanieg89

    Higher Level Integration Marathon & Questions

    $Let $Q$ be the symmetric matrix associated to the quadratic form defining the conic. $Q$ yields an ellipse iff it is a positive-definite matrix. Standard spectral theory shows that $Q$ is diagonalisable by orthogonal matrices. The resulting matrix is $\textrm{diag}(\lambda_j)$, so after our...
  5. seanieg89

    Higher Level Integration Marathon & Questions

    I=\int_0^{2\pi} (\frac{e^{it}+e^{-it}}{2})^{2n}\, dt \\ \\ = 2^{-2n}\binom{2n}{n}\int_0^{2\pi}\, dt\\ \\=\frac{\pi}{2^{2n-1}}\binom{2n}{n}.\\ \\ $(all nonconstant trig monomials integrate to zero.)$
  6. seanieg89

    Higher Level Integration Marathon & Questions

    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.
  7. seanieg89

    Higher Level Integration Marathon & Questions

    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...
  8. seanieg89

    Higher Level Integration Marathon & Questions

    ^ That was good practice! I need to go out for dinner v soon, so will supply gory details later. In short, the answer can be expressed solely in terms of Apery's constant zeta(3). 1. IBP to essentially reduce the integral to log^2(1+x)/x. (The bulk of the difficulty of the originally posted...
  9. seanieg89

    Chess thread

    Gg man, pretty well played. Shame about that pawn blunder.
  10. seanieg89

    Chess thread

    Someone has joined, but just isn't moving lol.
  11. seanieg89

    Chess thread

    Coming? Otherwise someone else join pls :).
  12. seanieg89

    Chess thread

    https://lichess.org/dhv78LH8
  13. seanieg89

    Chess thread

    Sweet, will host up in a sec. 15+10 or thereabouts seem decent? or would you prefer something else?
  14. seanieg89

    Chess thread

    Cool, should be a decent game then. Free now?
  15. seanieg89

    Chess thread

    Sure I'll be free later this arvo most likely for a few on lichess. How strong are you btw? Also theres some overlap between the part of the brain used in maths and chess respectively imo, but its hard to draw conclusions. The average mathematician who tries his hand at chess tends to do better...
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