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THEOREM 54. Let y be a closed smooth path in C and 2 = C\r*. The function du 1 2πi S 2 on is an analytic function assumi
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THEOREM 54. Let y be a closed smooth path in C and 2 = C\r*. The function du 1 2πi S 2 on is an analytic function assumi
THEOREM 54. Let y be a closed smooth path in C and 2 = C\r*. The function du 1 2πi S 2 on is an analytic function assuming only integer values. It is constant on each connected component of and, in particular, zero near infinity. Hints: Use Theorem 49 to show analyticity. Define F as in 53 and show first that F(1) = 1 and then, using part (7) of Theorem 52, that the values of Ind are integer. Why can they not change when z changes only little? What happens when z becomes very large?