|
Bloch's Theorem (complex Variables)
In complex analysis, a branch of mathematics, Bloch's theorem describes the behaviour of holomorphic functions defined on the unit disk. It gives a lower bound on the size of a disk in which an inverse to a holomorphic function exists. It is named after André Bloch. Statement Let ''f'' be a holomorphic function in the unit disk , ''z'', ≤ 1 for which :, f'(0), =1 Bloch's Theorem states that there is a disk S ⊂ D on which f is biholomorphic and f(S) contains a disk with radius 1/72. Landau's theorem If ''f'' is a holomorphic function in the unit disk with the property , ''f′''(0), = 1, then let ''Lf'' be the radius of the largest disk contained in the image of ''f''. Landau's theorem states that there is a constant ''L'' defined as the infimum of ''Lf'' over all such functions ''f'', and that ''L'' ≥ ''B''. This theorem is named after Edmund Landau. Valiron's theorem Bloch's theorem was inspired by the following theorem of Georges Valiron: Theorem. If '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebraic geometry, number theory, analytic combinatorics, applied mathematics; as well as in physics, including the branches of hydrodynamics, thermodynamics, and particularly quantum mechanics. By extension, use of complex analysis also has applications in engineering fields such as nuclear, aerospace, mechanical and electrical engineering. As a differentiable function of a complex variable is equal to its Taylor series (that is, it is analytic), complex analysis is particularly concerned with analytic functions of a complex variable (that is, holomorphic functions). History Complex analysis is one of the classical branches in mathematics, with roots in the 18th century and just prior. Important mathematicians associated with comple ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |