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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 .
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 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 ...
In mathematics, Cauchy's integral formula, named after Augustin-Louis Cauchy, is a central statement in complex analysis.It expresses the fact that a holomorphic function defined on a disk is completely determined by its values on the boundary of the disk, and it provides integral formulas for all derivatives of a holomorphic function.
Since f is the derivative of the holomorphic function F, it is holomorphic. The fact that derivatives of holomorphic functions are holomorphic can be proved by using the fact that holomorphic functions are analytic, i.e. can be represented by a convergent power series, and the fact that power series may be differentiated term by term. This ...
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 ...
The functions () = = as well as the gamma function and the Riemann zeta function are meromorphic on the whole complex plane. [3] The function () = is defined in the whole complex plane except for the origin, 0.