Editing Analytic continuation (section)

==Formal definition of a germ==
The power series defined below is generalized by the idea of a ''[[Germ (mathematics)|germ]]''. The general theory of analytic continuation and its generalizations is known as [[sheaf (mathematics)|sheaf theory]]. Let

: <math>f(z)=\sum_{k=0}^\infty \alpha_k (z-z_0)^k</math>

be a [[power series]] converging in the [[Disk (mathematics)|disk]] ''D''<sub>''r''</sub>(''z''<sub>0</sub>), ''r'' > 0, defined by

:<math>D_r(z_0) = \{z \in \Complex : |z - z_0| < r\}</math>.

Note that [[without loss of generality]], here and below, we will always assume that a maximal such ''r'' was chosen, even if that ''r'' is ∞. Also note that it would be equivalent to begin with an analytic function defined on some small open set. We say that the vector

:<math>g = (z_0, \alpha_0, \alpha_1, \alpha_2, \ldots) </math>

is a ''[[Germ (mathematics)|germ]]'' of ''f''. The ''base'' ''g''<sub>0</sub> of ''g'' is ''z''<sub>0</sub>, the ''stem'' of ''g'' is (α<sub>0</sub>, α<sub>1</sub>, α<sub>2</sub>, ...) and the ''top'' ''g''<sub>1</sub> of ''g'' is α<sub>0</sub>. The top of ''g'' is the value of ''f'' at ''z''<sub>0</sub>.

Any vector ''g'' = (''z''<sub>0</sub>, α<sub>0</sub>, α<sub>1</sub>, ...) is a germ if it represents a power series of an analytic function around ''z''<sub>0</sub> with some radius of convergence ''r'' > 0. Therefore, we can safely speak of the set of germs <math>\mathcal G</math>.