Thread: Smooth curve in complex space

1. 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 $\displaystyle z^{\prime}(t) = 0$.

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

-Dan

2. Originally Posted by topsquark
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 $\displaystyle z^{\prime}(t) = 0$.

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 $\displaystyle z(t) = t^2+it^3$ has a cusp at t=0.