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In chemistry, the haloform reaction (also referred to as the Lieben haloform reaction) is a chemical reaction in which a haloform (CHX 3, where X is a halogen) is produced by the exhaustive halogenation of an acetyl group (R−C(=O)CH 3, where R can be either a hydrogen atom, an alkyl or an aryl group), in the presence of a base.
Trihalomethanes with all the same halogen atoms are called haloforms. Many trihalomethanes find uses in industry as solvents or refrigerants . Some THMs are also environmental pollutants , and a few are considered carcinogenic .
In complex analysis of one and several complex variables, Wirtinger derivatives (sometimes also called Wirtinger operators [1]), named after Wilhelm Wirtinger who introduced them in 1927 in the course of his studies on the theory of functions of several complex variables, are partial differential operators of the first order which behave in a very similar manner to the ordinary derivatives ...
A holomorphic function resembles an entire function ("whole") in a domain of the complex plane while a meromorphic function (defined to mean holomorphic except at certain isolated poles), resembles a rational fraction ("part") of entire functions in a domain of the complex plane. [9] Cauchy had instead used the term synectic. [10]
In complex analysis, the Phragmén–Lindelöf principle (or method), first formulated by Lars Edvard Phragmén (1863–1937) and Ernst Leonard Lindelöf (1870–1946) in 1908, is a technique which employs an auxiliary, parameterized function to prove the boundedness of a holomorphic function (i.e, | | < ()) on an unbounded domain when an additional (usually mild) condition constraining the ...
Hurwitz's theorem is used in the proof of the Riemann mapping theorem, [2] and also has the following two corollaries as an immediate consequence: . Let G be a connected, open set and {f n} a sequence of holomorphic functions which converge uniformly on compact subsets of G to a holomorphic function f.
In complex analysis, a branch of mathematics, the Hadamard three-line theorem is a result about the behaviour of holomorphic functions defined in regions bounded by parallel lines in the complex plane. The theorem is named after the French mathematician Jacques Hadamard.
The wedge product of complex differential forms is defined in the same way as with real forms. Let p and q be a pair of non-negative integers ≤ n . The space Ω p,q of ( p , q )-forms is defined by taking linear combinations of the wedge products of p elements from Ω 1,0 and q elements from Ω 0,1 .