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In my understanding, if $f$ is holomorphic in a neighborhood $U$ of $w$, then there is a unique analytic extension from the restriction $f\vert U$ to some maximal domain $D$, right? If so, why even bother defining a germ as a class of equivalence (besides the technicality of functions with different domains are different functions).

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Yes, if you only interested in analytic continuation you don't really need germs. It depends on what you are intend to do. Analytic continuation is a global phenomena and germs encoded only local date near every point of the domain. In fact, all local properties (such as continuity, differentiability etc) of the function can be studied by analyzing its germs.

For a holomorphic function on a Riemann surface, a germ can be think as a power series and the collection of all germs of the function can be considered to be the analytic continuation. If you collect all the different germs at a point, the resulting set is called a stalk and set of stalks forms a sheaf which capture all the locally defined data attached to all the open sets of the Riemann surface.

In general, sheaves can be used to defined Riemann surfaces, manifolds, covering spaces and many more. So, the theory does not stop after analytic continuation business, it is connected to many more beautiful phenomenons in mathematics.

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