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Math Help - Complex analysis - mappings

  1. #1
    Super Member Bernhard's Avatar
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    Complex analysis - mappings

    I am a mathematical hobbyist teaching myself complex analysis from Gamelin's book. I have already struck a problem. Would appreciate help.

    On page 16 of Gamelin, "Complex Analysis" we find the following text:

    "Now we turn to the problem of finding an inverse function for w =  z^2. Every point w not equal to 0 is hit by exactly two values of z, the two square roots \pm \sqrt{w}. In order to define an inverse function, we must restrict the domain in the z-plane so the values of w are hit by only one z. There are many ways of doing this, and we proceed rather arbitrarily as follows.

    Note that as the rays sweep out the open right half of the z-plane, with the angle of the ray increasing from \frac{-\pi}{2} to \frac{\pi}{2}, the image rays under w =  z^2sweep out the entire w-plane except for the negative axis, with the angle of the ray increasing from  -\pi to +\pi . This leads us to draw a slit, or branch cut, in the w-plane along the negative axis from -\infty to 0 and to define the inverse function on the slit plane C\( -\infty,0]."


    Now Gamelin does say this is somewhat arbitrary - but he does not motivate this "branch cut" idea well. Why are we doing this - presumably because when arg(z) =  \frac{-\pi}{2} it repeats the same values as when arg(z) =  \frac{+\pi}{2} - and so we are getting rid of repeated values to achieve a single valued inverse function. Is this the reason for doing this?

    Presumably we could cut the plane at another ray? Is this right?

    Another question is why are we removing the point 0 - the function and the inverse seem to be defined there and there is no question of repeated or multiple values. Is the reason something I will discover [possibly related to continuity or differentiability?] later in the book
    Last edited by Bernhard; January 24th 2010 at 12:57 PM.
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  2. #2
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    One way to get a good understanding of multi-functions and branching is to draw the real and imaginary components of the function in the x-y plane for z=x+iy and note how these surfaces "interleave". For example, I've drawn the real and imaginary surfaces for f(z)=\sqrt{z} and note in each picture of the first row, the sheets overlap. This is a graphic illustration of multi-valuedness. Now, these surfaces are analytic but if we want a single-valued analytic function, then we excise (make a branch cut) a non-overlapping surface from the multi-sheets. Normally, we choose the maximal piece that doesn't overlap which in this case is any 2\pi contiguous part and since by definition, z^{1/2}=e^{1/2\log(z)} and the derivative of \log(z) is 1/z then we omit the origin since this derivative is not defined there and in fact if you look at the multi-sheets you'll see they have a vertical derivative at the origin. In the bottom set, I've cut out a part of the multi-sheet from -pi to pi and that is what is generally considered the principal branch. Those "cut" pieces are the principal branch functions and are analytic except at the branch cut.

    But I can cut out any piece that doesn't overlap to give me a single-valued function that is analytic all along the surface except at the cut since by definition, analyticity at a point means differentiable in a "region" surrounding the point and of course any point on the edge of the cut has no 'region' surrounding it.
    Attached Thumbnails Attached Thumbnails Complex analysis - mappings-squareroot-sheets.jpg  
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  3. #3
    Super Member Bernhard's Avatar
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    Thanks ... and by the way

    Thanks so much for that.

    By the way how did you include the diagrams - presumably you scanned the diagrams.

    My scanner only gives me A4 size. How did you get them to a good size for posting?

    Thanks again
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  4. #4
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    I used the following Mathematica code to generate the four separate plots then displayed it in a GraphicsGrid, selected it, then choose File/Save Selection As and saved it as a jpeg file on my disc. Then when I made the post here, at the bottom, I chose "Manage Attachments" and then just selected the file. The rest is magic


    Code:
    rpic = ParametricPlot3D[{{Re[z], Im[z], Re[Sqrt[z]]}, {Re[z], Im[z], 
         Re[-Sqrt[z]]}} /. z -> r Exp[I t], {r, 0, 2}, {t, -\[Pi], \[Pi]}]
    ipic = ParametricPlot3D[{{Re[z], Im[z], Im[Sqrt[z]]}, {Re[z], Im[z], 
         Im[-Sqrt[z]]}} /. z -> r Exp[I t], {r, 0, 2}, {t, -\[Pi], \[Pi]}]
    realpic = Show[rpic, PlotLabel -> Style["Real", 16]]
    imagpic = Show[ipic, PlotLabel -> Style["Imag", 16]]
    
    rprincp = 
     ParametricPlot3D[{Re[z], Im[z], Re[Sqrt[z]]} /. z -> r Exp[I t], {r, 
       0, 2}, {t, -\[Pi], \[Pi]}, 
      PlotLabel -> Style["Real principal branch", 16]]
    iprincp = 
     ParametricPlot3D[{Re[z], Im[z], Im[Sqrt[z]]} /. z -> r Exp[I t], {r, 
       0, 2}, {t, -\[Pi], \[Pi]}, 
      PlotLabel -> Style["Imag principal branch", 16]]
    GraphicsGrid[{{realpic, imagpic}, {rprincp, iprincp}}]
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