...Prove Their Worth...

"Problems worthy of attack
prove their worth
by hitting back." - Piet Hein

A kind of running diary and rambling pieces on my struggles with assorted books, classes, and other things, as they happen. You must be pretty bored to be reading this...

Wednesday, June 05, 2002

Say f(z) = z^m. f' = m*z^(m-1). f'' = m*(m-1)*z^(m-2). f''/f' = (m-1)*z^(m-2-m+1) = (m-1)*z^(-1). Call K the complex curvature of f(z). -i*Conj[K] = (m-1)/(z*Length[z]). Ok. So that pile of crap right there is an intrinsic property of the mapping f: z -> z^m. It tells us that even if we were to apply the mapping to a straight line, with zero curvature, the image curve would have non-zero curvature. For m=1, a linear mapping, it produces zero, as expected.

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