:: If the closed Jordan curve ? separates the surface, it is homotopic to a smooth Jordan curve ? ( with non-vanishing derivative ) that separates the surface into two halves.
42.
:: If the closed Jordan curve ? separates the surface, it is homotopic to a smooth Jordan curve ? ( with non-vanishing derivative ) that separates the surface into two halves.
43.
Hence, for the definition of the functional calculus, indeed a suitable family of Jordan curves can be found for each " f " that is holomorphic on some " D ".
44.
Thus to check exactness of a closed form it suffices to show that the vanishing of the integral around any regular closed curve, i . e . a simple smooth Jordan curve with nowhere vanishing derivative.
45.
The Jordan curve theorem was independently generalized to higher dimensions by H . Lebesgue and L . E . J . Brouwer in 1911, resulting in the "'Jordan Brouwer separation theorem " '.
46.
Here A is the area of the region bounded by a closed Jordan curve of length ( perimeter ) L in the plane, R is the circumradius of the bounded region, and r is its inradius.
47.
So, roughly speaking, in the case of one function the image is a slit region in the complex plane; and in the case of two functions the two regions are separated by a closed Jordan curve.
48.
The previous discussion has shown that the integral makes sense, i . e . a suitable collection ? of Jordan curves does exist for each " f " and the integral does converge in the appropriate sense.
49.
A fundamental domain for the action of a Schottky group " G " on its regular points ? ( " G " ) in the Riemann sphere is given by the exterior of the Jordan curves defining it.
50.
Both the Mizar and the HOL Light proof rely on libraries of previously proved theorems, so these two sizes are not comparable . showed that the Jordan curve theorem is equivalent in proof-theoretic strength to the weak K�nig's lemma.
