Complex Analysis (9781108526463) by Stewart Ian; Tall David

Author:Stewart, Ian; Tall, David [Stewart, Ian; Tall, David]
Language: eng
Format: epub
ISBN: 9781108436793
Publisher: CambridgeUP
Published: 2018-07-13T00:00:00+00:00

9.5

Fixed End Point Homotopy

Consider the rectangle

DEFINITION 9.8. Two paths and are fixed end point homotopic in D if there is a continuous map

and points such that

as in Figure 9.14.

Figure 9.14 A fixed end point homotopy.

If we let then is a path in D from to for all , and as s increases continuously from 0 to 1, the path ‘deforms continuously’ from to .

Example 9.9. Let . Then and are fixed end point homotopic in D, where

As a corollary of Theorem 9.6 we deduce:

THEOREM 9.10. If f is differentiable in a domain D and is fixed end point homotopic in D to , then .

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