Complex integration and the homotopy version of Cauchy’s theorem

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are , (with endpoints fixed),
in the fixed endpoint case:
;
or in the closed case:
.

, if is closed in which is homotopic to a point, then

This is the case if is the boundary of a subregion such that
is to a closed disk.

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We first introduce some embedding and approximation theorems.

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Let be a and let be a function. Given any positive function , there exists a function that is close to .

Let be , and let be a map. Then is homotopic to a map .

If are homotopic smooth maps, then they are If is homotopic to relative to some closed subset , then they are smoothly homotopic relative to .

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By virtue of these properties of homotopy, one can apply Stoke’s Theorem which demands both , the oriented dimensional manifold with boundary, and , the compactly supported form on , are smooth.

: such that .
: .

: a closed covector field on a smooth manifold . Then every has a neighborhood on which is exact.

Now follows naturally the proof in page 21.

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