Definition talk:Contour/Complex Plane

Might this work better if there was a definition for "piecewise smooth", then the latter can be used directly as the definition? In turn, the word "piecewise" can likewise be defined. In this way the concepts do not need to be reintroduced every time they are used. --prime mover (talk) 20:19, 5 December 2012 (UTC)
 * A definition for "piecewise smooth" would be nice to have. A definition of "piecewise" for a mapping might possibly be too general to be of much use, since "pieces" both can mean a components of a partition and, in this case, a not-quite partition of a closed interval. --Anghel (talk) 21:22, 5 December 2012 (UTC)


 * Piecewise: as in piecewise smooth, piecewise continuous, etc. and linked with Definition:Subdivision (Real Analysis), should be straightforward to craft a general definition. --prime mover (talk) 21:28, 5 December 2012 (UTC)

Suggestion of complete change of definition
I'm suggesting that we change this definition of contour to the following:

Let $C_1, \ldots, C_n$ be directed smooth curves.

For all $i \in \left\{ {1, \ldots, n}\right\}$, $C_i$ is parameterized by $\gamma_i: \left[{a_i\,.\,.\,b_i}\right] \to \C$.

Suppose that for all $i \in \left\{ {1, \ldots, n}\right\}$: $\gamma\left({b_i}\right) = \gamma\left({a_{i + 1} }\right)$.

Then the finite sequence $C_1, \ldots, C_n$ is called a contour.

The present content of this page should then be incorporated in a new page: Definition:Parameterization of Contour (Complex Plane).

The reason for this proposed change is threefold: First, it seems to be more common among the notes on complex analysis that I've read on the net. Second, it makes it clear that a contour is not just a function, or an image of a function. Third, it makes it possible to reparameterize just one $C_i$, instead of having to reparameterize the whole contour. (And fourth, it means that we can avoid all those restriction signs $\restriction$ in the proofs about contour integrals.)

This page is linked to in only 10 pages, and all have been started by myself. I'm willing to spend my christmas holidays changing all these results so they utilize the proposed definition. --Anghel (talk) 16:41, 26 December 2012 (UTC)


 * It seems you have put considerable thought into this. I don't object as it appears to be an improvement. TIA for the considerable effort. --Lord_Farin (talk) 16:45, 26 December 2012 (UTC)


 * Endorsed. You are an asset, Anghel. --prime mover (talk) 17:16, 26 December 2012 (UTC)