Monodrome set

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

• ${\displaystyle \alpha }$and two homotopic paths in ( set of complex numbers ), ${\displaystyle {\tilde {\alpha }}}$ ${\displaystyle \mathbb {C} }$
• ${\displaystyle h:=h(t,\tau )}$a homotopy between and , ${\displaystyle \alpha }$ ${\displaystyle {\tilde {\alpha }}}$
• ${\displaystyle K}$an open circular disk around the common starting point of and , ${\displaystyle \alpha }$ ${\displaystyle {\tilde {\alpha }}}$
• ${\displaystyle f_{0}\colon K\to \mathbb {C} }$a holomorphic function on the open disk into the set of complex numbers, ${\displaystyle K}$
• ${\displaystyle K'}$another open circular disc in , ${\displaystyle \mathbb {C} }$
• ${\displaystyle f_{1},{\tilde {f}}_{1}\colon K'\to \mathbb {C} }$two functions on after . ${\displaystyle K'}$ ${\displaystyle \mathbb {C} }$

In addition, denote the -th single path of homotopy . ${\displaystyle h_{\tau }:=h(\cdot ,\tau )}$ ${\displaystyle \tau }$ ${\displaystyle h}$

sentence

Let it be analytically continued along each , then the following applies: arise and from through analytical continuation along or , so is . ${\displaystyle f_{0}}$ ${\displaystyle h_{\tau }}$ ${\displaystyle f_{1}}$ ${\displaystyle {\tilde {f}}_{1}}$ ${\displaystyle f_{0}}$ ${\displaystyle \alpha }$ ${\displaystyle {\tilde {\alpha }}}$ ${\displaystyle f_{1}={\tilde {f}}_{1}}$

Individual evidence

1. ↑ Klaus Jänisch: Introduction to Function Theory . 2nd Edition. Springer-Verlag, Berlin / Heidelberg / New York 1980, ISBN 3-540-10032-6 .