Dirichlet beta functionIn mathematics, the Dirichlet beta function (also known as the Catalan beta function) is a special function, closely related to the Riemann zeta function. It is a particular Dirichlet L-function, the L-function for the alternating character of period four. The Dirichlet beta function is defined as or, equivalently, In each case, it is assumed that Re(s) > 0. Alternatively, the following definition, in terms of the Hurwitz zeta function, is valid in the whole complex s-plane: Another equivalent definition, in terms of the Lerch transcendent, is: which is once again valid for all complex values of s.
Riemann zeta functionThe Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter ζ (zeta), is a mathematical function of a complex variable defined as for , and its analytic continuation elsewhere. The Riemann zeta function plays a pivotal role in analytic number theory, and has applications in physics, probability theory, and applied statistics. Leonhard Euler first introduced and studied the function over the reals in the first half of the eighteenth century.
Cross-coupling reactionIn organic chemistry, a cross-coupling reaction is a reaction where two different fragments are joined. Cross-couplings are a subset of the more general coupling reactions. Often cross-coupling reactions require metal catalysts. One important reaction type is this: (R, R' = organic fragments, usually aryle; M = main group center such as Li or MgX; X = halide) These reactions are used to form carbon–carbon bonds but also carbon-heteroatom bonds. Cross-coupling reaction are a subset of coupling reactions.
PropagatorIn quantum mechanics and quantum field theory, the propagator is a function that specifies the probability amplitude for a particle to travel from one place to another in a given period of time, or to travel with a certain energy and momentum. In Feynman diagrams, which serve to calculate the rate of collisions in quantum field theory, virtual particles contribute their propagator to the rate of the scattering event described by the respective diagram.
Polygamma functionIn mathematics, the polygamma function of order m is a meromorphic function on the complex numbers defined as the (m + 1)th derivative of the logarithm of the gamma function: Thus holds where ψ(z) is the digamma function and Γ(z) is the gamma function. They are holomorphic on . At all the nonpositive integers these polygamma functions have a pole of order m + 1. The function ψ(1)(z) is sometimes called the trigamma function. Digamma function#Integral representations When m > 0 and Re z > 0, the polygamma function equals where is the Hurwitz zeta function.
RenormalizationRenormalization is a collection of techniques in quantum field theory, statistical field theory, and the theory of self-similar geometric structures, that are used to treat infinities arising in calculated quantities by altering values of these quantities to compensate for effects of their self-interactions. But even if no infinities arose in loop diagrams in quantum field theory, it could be shown that it would be necessary to renormalize the mass and fields appearing in the original Lagrangian.
Coupling reactionIn organic chemistry, a coupling reaction is a type of reaction in which two reactant molecules are bonded together. Such reactions often require the aid of a metal catalyst. In one important reaction type, a main group organometallic compound of the type R-M (where R = organic group, M = main group centre metal atom) reacts with an organic halide of the type R'-X with formation of a new carbon-carbon bond in the product R-R'. The most common type of coupling reaction is the cross coupling reaction. Richard F.
Hiyama couplingThe Hiyama coupling is a palladium-catalyzed cross-coupling reaction of organosilanes with organic halides used in organic chemistry to form carbon–carbon bonds (C-C bonds). This reaction was discovered in 1988 by Tamejiro Hiyama and Yasuo Hatanaka as a method to form carbon-carbon bonds synthetically with chemo- and regioselectivity. The Hiyama coupling has been applied to the synthesis of various natural products.
Branch pointIn the mathematical field of complex analysis, a branch point of a multi-valued function (usually referred to as a "multifunction" in the context of complex analysis) is a point such that if the function is n-valued (has n values) at that point, all of its neighborhoods contain a point that has more than n values. Multi-valued functions are rigorously studied using Riemann surfaces, and the formal definition of branch points employs this concept.
Kumada couplingIn organic chemistry, the Kumada coupling is a type of cross coupling reaction, useful for generating carbon–carbon bonds by the reaction of a Grignard reagent and an organic halide. The procedure uses transition metal catalysts, typically nickel or palladium, to couple a combination of two alkyl, aryl or vinyl groups. The groups of Robert Corriu and Makoto Kumada reported the reaction independently in 1972. The reaction is notable for being among the first reported catalytic cross-coupling methods.
