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Analyticity of complex functions is a more restrictive property, as it has more restrictive necessary conditions and complex analytic functions have more structure than their real-line counterparts. [6] According to Liouville's theorem, any bounded complex analytic function defined on the whole complex plane is constant. The corresponding ...
As in complex analysis of functions of one variable, which is the case n = 1, the functions studied are holomorphic or complex analytic so that, locally, they are power series in the variables z i. Equivalently, they are locally uniform limits of polynomials; or locally square-integrable solutions to the n-dimensional Cauchy–Riemann equations.
In complex analysis, the monodromy theorem is an important result about analytic continuation of a complex-analytic function to a larger set. The idea is that one can extend a complex-analytic function (from here on called simply analytic function ) along curves starting in the original domain of the function and ending in the larger set.
More precisely, if : is a function which is analytic in each variable z i, 1 ≤ i ≤ n, while the other variables are held constant, then F is a continuous function. A corollary is that the function F is then in fact an analytic function in the n -variable sense (i.e. that locally it has a Taylor expansion ).
In mathematics, algebraic geometry and analytic geometry are two closely related subjects. While algebraic geometry studies algebraic varieties, analytic geometry deals with complex manifolds and the more general analytic spaces defined locally by the vanishing of analytic functions of several complex variables. The deep relation between these ...
Download as PDF; Printable version; ... an analytic function is a function that is locally given by a convergent power series. ... (complex analysis)
The latter property is the basis of the principle of analytic continuation which allows extending every real analytic function in a unique way for getting a complex analytic function whose domain is the whole complex plane with a finite number of curve arcs removed.
In mathematics, the Borel–Carathéodory theorem in complex analysis shows that an analytic function may be bounded by its real part. It is an application of the maximum modulus principle. It is named for Émile Borel and Constantin Carathéodory.