Analytic Continuation - Formal Definition of A Germ

Formal Definition of A Germ

Let

be a power series converging in the disc Dr(z0) defined by

.

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

g = (z0, α0, α1, α2, ...)

is a germ of f. The base g0 of g is z0, the stem of g is (α0, α1, α2, ...) and the top g1 of g is α0. The top of g is the value of f at z0, the bottom of g.

Any vector g = (z0, α0, α1, ...) is a germ if it represents a power series of an analytic function around z0 with some radius of convergence r > 0. Therefore, we can safely speak of the set of germs .

Read more about this topic:  Analytic Continuation

Famous quotes containing the words formal, definition and/or germ:

    “This is no argument against teaching manners to the young. On the contrary, it is a fine old tradition that ought to be resurrected from its current mothballs and put to work...In fact, children are much more comfortable when they know the guide rules for handling the social amenities. It’s no more fun for a child to be introduced to a strange adult and have no idea what to say or do than it is for a grownup to go to a formal dinner and have no idea what fork to use.”
    —Leontine Young (20th century)

    “According to our social pyramid, all men who feel displaced racially, culturally, and/or because of economic hardships will turn on those whom they feel they can order and humiliate, usually women, children, and animals—just as they have been ordered and humiliated by those privileged few who are in power. However, this definition does not explain why there are privileged men who behave this way toward women.”
    —Ana Castillo (b. 1953)

    “And such the trust that still were mine,
    Though stormy winds swept o’er the brine,
    Or though the tempest’s fiery breath
    Roused me from sleep to wreck and death.
    In ocean cave, still safe with Thee
    The germ of immortality!
    And calm and peaceful shall I sleep,
    Rocked in the cradle of the deep.”
    —Emma Hart Willard (1787–1870)