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Re: Problem 101, amended

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  • adh_math
    Suppose f is analytic and bounded in the open unit disk D, admits a continuous extension to the closed disk minus {1}, and has norm <= 1 on the boundary
    Message 1 of 4238 , Nov 1, 1999
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      Suppose f is analytic and bounded in the open
      unit disk D, admits a continuous extension to the
      closed disk minus {1}, and has norm <= 1 on the
      boundary (minus {1}). Show |f| <= 1 in D.
      <br><br>First, it suffices to show |f(0)| <= 1 since every
      point of D can be carried to 0 by a disk automorphism,
      and this normalization does not change the other
      hypotheses (but makes the Cauchy integral formula much
      easier to use :).<br>Next, consider a contour \gamma of
      the following form: \gamma traces out `most' of the
      unit circle, but makes a small detour near 1 (so the
      enclosed region is a `notched' unit disk). Standard
      estimates show that the contribution from the detour can be
      made arbitrarily small (by taking the detour to be
      short, since f is bounded), while the contribution from
      the portion of \gamma on dD is no larger than 1 in
      absolute value. `Formally'<br>|f(0)| = 1/(2\pi) |int(0,
      2\pi, f(e^{i\theta}), d\theta)| (Cauchy integral
      formula)<br><= 1/(2\pi) int(0, 2\pi, |f(e^{i\theta})|, d\theta)
      (triangle inequality)<br><= 1 (|f| bounded above by 1 on
      dD);<br>the gymnastics are required to control the integral
      near 1.<br><br>Don't remember the agreed-upon notation
      for integrals is in this club, but have used a
      modification of what I recall to be the notation for sums. As
      an aside, Jason asked about TeX (notation I used in
      an earlier post), and I forgot to answer. <br>TeX
      (pron: tech, with a soft k) is a typesetting language
      for mathematics, developed by Donald Knuth and now
      (in various incarnations) widely used in mathematics
      and physics publishing. A document is prepared with a
      text editor, then `compiled' with TeX into a printable
      binary file (with the extension dvi, for DeVice
      Independent). Mathematical symbols (sums, integrals, Greek
      letters, exotic binary operators, etc) are produced by
      short descriptive commands beginning with a backslash
      (examples: \sum, \int, \alpha, \beta,..., \times, \otimes,
      \oplus, \leq, \geq...dozens of them). Subscripts are
      produced with an underscore, superscripts with a caret. On
      sums and integrals, sub- and superscripts become lower
      and upper limits. Thus<br>\sum_{i=1}^n (x_i)^2 = 1 is
      the equation of the unit sphere in R^n. In actual
      use, mathematical symbols are delimited (so that TeX
      knows the scope of the mathematics in the input file),
      usually with dollar signs (e.g. $R^n$).
    • bqllpd
      ((c/2 + x)^2 + y^2)^(n/2) + ((c/2 - x)^2 + y^2)^(n/2) = c^n where c/2 = x = -c/2 x^2+y^2=r^2 y/r=sin(T) Parametric form x= (+or- (-c^2 * r^2
      Message 4238 of 4238 , May 1, 2011
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        ((c/2 + x)^2 + y^2)^(n/2) + ((c/2 - x)^2 + y^2)^(n/2) = c^n where c/2 >= x >= -c/2
        x^2+y^2=r^2
        y/r=sin(T)

        Parametric form
        x= (+or- (-c^2 * r^2 *(sin^2(T)-1))^(1/2))/c
        y=r*sin(T)
        T=0 to 2*pi Radians (but T=0 to pi/2 is sufficient).

        I had to wait this long for the solving software to catch up.

        Show that for all n>2 and c=rational, and for all x xor y rational, then x and y are never rational.

        Doing so will give a "simple" proof of Fermat's Last Theorem.


        --- In mathforfun@yahoogroups.com, jeshields_98 wrote:
        >
        > I've just joined this club, and I have theorem
        > for<br>you. It's also been posted on Spherical Cow but here
        > seemed more appropriate, so I'll just throw it out
        > here.<br><br>Theorem:<br><br>[y^2 + (c/2 - x)^2]^(n/2) +<br>[y^2 + (c/2 +
        > x)^2]^(n/2) + c^n<br><br>Proposition 1:<br><br>If c can be
        > solved in terms of x,y,n, the result is the equation for
        > the set of curves that applies to triangles of an
        > n-dimensional Pythagorean Theorem. (including the
        > circle)<br><br>Proposition 2:<br><br>If c,y are rational, y not equal to
        > zero, then x is never rational.<br><br>I have no way of
        > knowing if proposition 2 is correct, but I HAVE derived
        > the theorem and proposition 1. I will post the link
        > to this if you would like some time in the future.
        >
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