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Abel's uniform convergence test is a criterion for the uniform convergence of a series of functions or an improper integration of functions dependent on parameters. It is related to Abel's test for the convergence of an ordinary series of real numbers, and the proof relies on the same technique of summation by parts. The test is as follows.
The notions of completely and absolutely monotone function/sequence play an important role in several areas of mathematics. For example, in classical analysis they occur in the proof of the positivity of integrals involving Bessel functions or the positivity of Cesàro means of certain Jacobi series. [ 6 ]
p3 peptide generates from the 17-40 or 17-42 sequence of the amyloid precursor protein , which is a type I integral membrane protein concerned in neurons’ synapses in many human tissues. Under normal physiological conditions, APP is processed with three different proteolytic enzymes: α-, β-and γ-secretases. At first, APP molecule is cut by ...
The technique of the previous example may also be applied to other Dirichlet series. If a n = μ ( n ) {\displaystyle a_{n}=\mu (n)} is the Möbius function and ϕ ( x ) = x − s {\displaystyle \phi (x)=x^{-s}} , then A ( x ) = M ( x ) = ∑ n ≤ x μ ( n ) {\displaystyle A(x)=M(x)=\sum _{n\leq x}\mu (n)} is Mertens function and
An infinite series of any rational function of can be reduced to a finite series of polygamma functions, by use of partial fraction decomposition, [8] as explained here. This fact can also be applied to finite series of rational functions, allowing the result to be computed in constant time even when the series contains a large number of terms.
In mathematics, Helly's selection theorem (also called the Helly selection principle) states that a uniformly bounded sequence of monotone real functions admits a convergent subsequence. In other words, it is a sequential compactness theorem for the space of uniformly bounded monotone functions. It is named for the Austrian mathematician Eduard ...
Let {f n} be a sequence of functions analytic on a simply-connected domain S. Suppose there exists a compact set Ω ⊂ S such that for each n , f n ( S ) ⊂ Ω. Forward (inner or right) Compositions Theorem — { F n } converges uniformly on compact subsets of S to a constant function F ( z ) = λ .
The most basic type of convergence for a sequence of functions (in particular, it does not assume any topological structure on the domain of the functions) is pointwise convergence. It is defined as convergence of the sequence of values of the functions at every point.