The monodrome theorem is an important mathematical theorem from the field of function theory and describes the homotopy - invariance of the analytical continuation of a holomorphic function .
Monodrome set
Be there
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and two homotopic paths in ( set of complex numbers ),
![{\ displaystyle \ mathbb {C}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7)
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a homotopy between and ,![\alpha](https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3)
![{\ displaystyle {\ tilde {\ alpha}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6dd79883e364e0e3e1e375b1195a0cc7ac6e4beb)
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an open circular disk around the common starting point of and ,![\alpha](https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3)
![{\ displaystyle {\ tilde {\ alpha}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6dd79883e364e0e3e1e375b1195a0cc7ac6e4beb)
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a holomorphic function on the open disk into the set of complex numbers,![K](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0)
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another open circular disc in ,![{\ displaystyle \ mathbb {C}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7)
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two functions on after .![K '](https://wikimedia.org/api/rest_v1/media/math/render/svg/63399d2b53fc472cfd349fd6167c21585d0a37f9)
![{\ displaystyle \ mathbb {C}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7)
In addition, denote the -th single path of homotopy .
![{\ displaystyle h _ {\ tau}: = h (\ cdot, \ tau)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9d9e9e6d04e8f136d80421696c4821564dab71a2)
![\dew](https://wikimedia.org/api/rest_v1/media/math/render/svg/38a7dcde9730ef0853809fefc18d88771f95206c)
![H](https://wikimedia.org/api/rest_v1/media/math/render/svg/b26be3e694314bc90c3215047e4a2010c6ee184a)
sentence
Let it be analytically continued along each , then the following applies: arise and from through analytical continuation along or , so is .
![f_ {0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6423b30a4c5770c59b5ab92dcb4ce378755440ab)
![{\ displaystyle h _ {\ tau}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c3267794495b8c3a1f85f26ecf4be30ca6681227)
![f_ {1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/50dfd257a51e037112c917f8a9e47c9c053466df)
![{\ displaystyle {\ tilde {f}} _ {1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6625e19e2c541087b9495a2743d9119fb497a34f)
![f_ {0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6423b30a4c5770c59b5ab92dcb4ce378755440ab)
![\alpha](https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3)
![{\ displaystyle {\ tilde {\ alpha}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6dd79883e364e0e3e1e375b1195a0cc7ac6e4beb)
![{\ displaystyle f_ {1} = {\ tilde {f}} _ {1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a23828c8706f2564771432cf2685ad67b45a9ce4)
Individual evidence
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↑ Klaus Jänisch: Introduction to Function Theory . 2nd Edition. Springer-Verlag, Berlin / Heidelberg / New York 1980, ISBN 3-540-10032-6 .