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Math Help - Smooth curve in complex space

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    Forum Admin topsquark's Avatar
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    Smooth curve in complex space

    I finally got my hands on a book on complex Analysis that is written for Mathematicians. (We really do need a "drooling" smiley.) Anyway, I have a brief question about a restriction made in the definition of a "smooth" curve. A curve isn't smooth at some value of the curve parameter t if z^{\prime}(t) = 0.

    Why the restriction here? Does it have something to do with the argument of the complex number 0 being undefined?

    -Dan
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    Quote Originally Posted by topsquark View Post
    I have a brief question about a restriction made in the definition of a "smooth" curve. A curve isn't smooth at some value of the curve parameter t if z^{\prime}(t) = 0.

    Why the restriction here? Does it have something to do with the argument of the complex number 0 being undefined?
    The reason is that a curve given by a differentiable function z(t) can fail to be "smooth" in the intuitive sense of the word at a point where z'(t)=0. For example, the curve given by z(t) = t^2+it^3 has a cusp at t=0.
    Attached Thumbnails Attached Thumbnails Smooth curve in complex space-cusp.png  
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