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I was browsing a book in complex analysis which said 'the contour is defined by admissible parametrization'.Unfortunately,no definition followed.

I presumed it was just z(t) defined on a closed interval-the position at time t on the contour?

Why the fancy name and when and why was it originally coined?

Thanks!

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    I use the term "admissible" out of lazyness sometimes (I say "$f$ is an admissible function" instead of saying $f$" is a function satisfying condition 1, condition 2, ...", if it clear what the conditions should be). I surmise in your case "admissible" means a parameterization that has the properties it should have. The term was likely used because the required parameterization clearly exists but would be somewhat messy or lengthy to explicitly write down (or that the explicit form of the parameterization is irrelevant to the discussion at hand).2012-03-24

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As seen in the comments the adjective admissible can mean everything in this context, and is not standard.

A sufficient condition for Cauchy's integral formula and all consequences to hold is that the contour is rectifiable, see

http://en.wikipedia.org/wiki/Cauchy%27s_integral_theorem