The Caseroti-Weierstrass theorem about neighborhoods in which these functions have dense image was improved by Picard's "great" theorem.
The Caseroti-Weierstrass theorem about neighborhoods in which these functions have dense image was improved by Picard's "great" theorem. If the origin is a pole of a function g, then locally g can be written as one of these functions divided by z-to-the-n. These functions over an annulus can be represented by a Laurent series about the center. The line integral over a loop of one of these functions equals zero. These functions can be decomposed into two components which are harmonic conjugates. Functions of this type whose domain is the whole plane are called entire, and these functions satisfy the Cauchy-Riemann equations. Functions of this type are infinitely differentiable and equal their Taylor series. For 10 points, identify these functions which are differentiable everywhere, and thus pretty important in complex analysis.
The Caseroti-Weierstrass theorem about neighborhoods in which these functions have dense image was improved by Picard's "great" theorem.
holomorphic functions [accept word forms, also accept analyticity before the end, prompt afterwards, prompt on any combination of the words "infinitely differentiable complex functions"]