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Elliptic integral

  1. Mar 28, 2009 #1
    When replacing x with 1/kx then

    [tex]
    \int_{1/k}^\infty {\left[ {\left( {x^2 - 1} \right)\left( {k^2 x^2 - 1} \right)} \right]} ^{ - 1/2} dx = \int\limits_0^1 {\left[ {\left( {\frac{1}{{k^2 x^2 }} - 1} \right)\left( {\frac{1}{{x^2 }} - 1} \right)} \right]} ^{ - 1/2} \frac{{dx}}{{kx^2 }}
    [/tex]

    I do not see how. Why ranges the LHS integral over infinity, whereas the RHS from 0 to 1?

    Any help and hints very much appreciated.

    thanks
     
  2. jcsd
  3. Mar 28, 2009 #2

    HallsofIvy

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    As x goes from 1/k to infinity, kx goes from 1 to infinity, and 1/kx goes from 1 to 0. Make the change of variable u= 1/kx and then, since both u and x are "dummy variables", replace u with x.
     
  4. Mar 30, 2009 #3
    thank you
     
  5. Apr 7, 2009 #4
    From the same book, I have the following.

    Arcsin (which is given its integral form) maps the upper half plane 1:1 onto the shaded strip |x|<pi/2, y>0.

    Now the sentence I don't get. By reflection in the punctured plane (punctured at +1 and -1), it produces a full tiling of the target plane by congruent, nonoverlapping images of the upper and lower half-planes.

    So by this reflection we get many strips, that then completely cover the target plane.

    But how does that work?

    thank you
     
  6. Apr 14, 2009 #5
    I have attached a file.

    What goes on with 'By reflection in the punctured plane (punctured at +1 and -1), it produces a full tiling of the target plane by congruent, nonoverlapping images of the upper and lower half-planes'.

    I do not understand this.
     

    Attached Files:

  7. Apr 17, 2009 #6
    Hellllooooooooo!!!!!!!!!

    By the way, it is from the book 'Elliptic Curves' by McKean and Moll, p. 71. They call this example simple and a warm up.

    Anybody out there that can give a comment?
     
  8. Apr 19, 2009 #7
    don't be shy
     
  9. Apr 21, 2009 #8
    What is it what you people here don't like about my question?

    Please talk to me.
     
  10. Apr 25, 2009 #9
    Who do you find hotter, Scarlett Johanson or Jessica Alba?
     
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