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Conformal Mapping

  1. Nov 13, 2007 #1
    1. The problem statement, all variables and given/known data

    "Study the infinitesimal behavior of f at the point c. (In other words, use the conformal mapping theorem to describe what is happening to the tangent vector of a smooth curve passing through c.)"

    f(z) = 1/(z-1), c=i

    2. Relevant equations

    |f'(c)| and arg f'(c)


    3. The attempt at a solution

    I know what |f'(c)| is, d/dz is -1/(z-1)^2, and evaluates out to 1/2i. However, I'm not sure what exactly that's saying about the behavior. Does it mean it's shrinking on the imaginary axis by 1/2 ?

    Also, about the argument... this is something I can't quite wrap my head around. I've read in this math text, and the wiki entry on arguments, but I'm not quite sure I get it. The equation in this book is, the argument of z = |z|(cos(theta)+ i*sin(theta)) where |z| = sqrt(x^2+y^2).

    Thanks.
     
  2. jcsd
  3. Nov 13, 2007 #2

    HallsofIvy

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    Since the problem specifically says "use the conformal mapping theorem", what is the conformal mapping theorem and how does it apply to this problem?

    I very much doubt that your book says that the "argument" of x. If z= a+ bi, in polar form is [itex]r (cos(\theta)+ i sin(\theta))= re^{i\theta}[/itex] then the "modulus" of z is [itex]r= \sqrt{a^2+ b^2}[/itex] and the "argument" of z is [itex]\theta= arctan(b/a)[/itex]. By the time you are working with "conformal mapping", that should be old stuff.
     
  4. Nov 13, 2007 #3
    It is old stuff, but I didn't quite understand it then. The conformal mapping theorem, according to the book, is, "If f is analytic in the disc |z-zo|<r and if f'(zo) != 0, then f is conformal at zo."

    Where zo is z with subscript 0.
     
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