Page 51 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Complex Analysis and Differential Geometry

Notes 5.3 Summary

Suppose D is a connected subset of the plane such that every point of D is an interior

point—we call such a set a region—and let C and C be oriented closed curves in D. We
2
1
say C is homotopic to C in D if there is a continuous function H : S  D, where S is the
2
1
square S = {(t, s) : 0  s, t  1}, such that H(t,0) describes C and H(t,1) describes C , and for
2
1
each fixed s, the function H(t, s) describes a closed curve C in D.
s
The function H is called a homotopy between C and C . Note that if C is homotopic to C
 1 2 1 2
in D, then C is homotopic to C in D. Just observe that the function K(t, s) = H(t,1 – s) is a
1
2
homotopy.
It is convenient to consider a point to be a closed curve. The point c is a described by a
constant function (t) = c. We, thus, speak of a closed curve C being homotopic to a
constant—we sometimes say C is contractible to a point.
Emotionally, the fact that two closed curves are homotopic in D means that one can be
continuously deformed into the other in D.
Suppose C and C are closed curves in a region D that are homotopic in D, and suppose f
 1 2
is a function analytic on D. Let H(t, s) be a homotopy between C and C . For each s, the
2
1
function  (t) describes a closed curve C in D. Let I(s) be given by I(s) =  f(z)dz.
s
s
C s
5.4 Keywords
Homotopy: The function H is called a homotopy between C and C . Note that if C is homotopic
2
1
1
to C in D, then C is homotopic to C in D. Just observe that the function K(t, s) = H(t,1 – s) is a
2
2
1
homotopy.
Contractible: It is convenient to consider a point to be a closed curve. The point c is a described
by a constant function (t) = c. We thus speak of a closed curve C being homotopic to a constant—
we sometimes say C is contractible to a point.
Cauchy’s Theorem: Suppose C and C are closed curves in a region D that are homotopic in D,
2
1
and suppose f is a function analytic on D. Let H(t, s) be a homotopy between C and C . For each
1
2
s, the function  (t) describes a closed curve C in D. Let I(s) be given by I(s) =  f(z)dz.
s
s
C s
5.5 Self Assessment
1. Suppose D is a connected subset of the plane such that every point of D is an interior
point—we call such a set a region—and let C and C be oriented closed .................
1 2
2. It is convenient to consider a point to be a closed curve. The point c is a described by a
constant function (t) = c. We thus speak of a closed curve C being homotopic to a constant—
we sometimes say C is ................. to a point.
3. Emotionally, the fact that two closed curves are ................. in D means that one can be
continuously deformed into the other in D.
4. If f is analytic in the region bounded by these curves (the region with two holes in it), then
we know that .................

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