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An analytic function approaches any given value arbitrarily closely in any epsilon-neighborhood of an essential singularity.

If a complex function is analytic at all finite points of the complex plane C, then it is said to be entire, sometimes also called "integral" (Knopp 1996, p. 112). Any ...

There are at least three theorems known as Jensen's theorem. The first states that, for a fixed vector v=(v_1,...,v_m), the function |v|_p=(sum_(i=1)^m|v_i|^p)^(1/p) is a ...

There are at least two results known as "the area principle." The geometric area principle states that (|A_1P|)/(|A_2P|)=(|A_1BC|)/(|A_2BC|). (1) This can also be written in ...

In complex analysis, a branch (also called a sheet) is a portion of the range of a multivalued function over which the function is single-valued. Combining all the sheets ...

A set function mu possesses countable additivity if, given any countable disjoint collection of sets {E_k}_(k=1)^n on which mu is defined, mu( union ...

Let X be a set and S a collection of subsets of X. A set function mu:S->[0,infty] is said to possess countable monotonicity provided that, whenever a set E in S is covered by ...

Let F be the Maclaurin series of a meromorphic function f with a finite or infinite number of poles at points z_k, indexed so that 0<|z_1|<=|z_2|<=|z_3|<=..., then a pole ...

Let f be a finite real-valued function defined on an interval [a,b]. Then at every point in [a,b] except on a set of Lebesgue measure zero, either: 1. There is a finite ...

In real and functional analysis, equicontinuity is a concept which extends the notion of uniform continuity from a single function to collection of functions. Given ...