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Color representation of the Dirichlet eta function. It is generated as a Matplotlib plot using a version of the Domain coloring method. [1]In mathematics, in the area of analytic number theory, the Dirichlet eta function is defined by the following Dirichlet series, which converges for any complex number having real part > 0: = = = + +.
For every odd positive integer +, the following equation holds: [3] (+) = ()!() +where is the n-th Euler Number.This yields: =,() =,() =,() =For the values of the Dirichlet beta function at even positive integers no elementary closed form is known, and no method has yet been found for determining the arithmetic nature of even beta values (similarly to the Riemann zeta function at odd integers ...
Dedekind η-function in the upper half-plane. In mathematics, the Dedekind eta function, named after Richard Dedekind, is a modular form of weight 1/2 and is a function defined on the upper half-plane of complex numbers, where the imaginary part is positive. It also occurs in bosonic string theory.
The Dirichlet function is not Riemann-integrable on any segment of despite being bounded because the set of its discontinuity points is not negligible (for the Lebesgue measure). The Dirichlet function provides a counterexample showing that the monotone convergence theorem is not true in the context of the Riemann integral.
In mathematics, eta function may refer to: The Dirichlet eta function η(s), a Dirichlet series; The Dedekind eta function η(τ), a modular form; The Weierstrass eta function η(w) of a lattice vector; The eta function η(s) used to define the eta invariant
Clinical obesity is defined as "a chronic, systemic illness characterized by alterations in the function of tissues, organs, the entire individual or a combination thereof, due to excess adiposity."
The polylogarithm is related to Dirichlet eta function and the Dirichlet beta function: = (), where η(s) is the Dirichlet eta function. For pure imaginary arguments, we have: Li s ( ± i ) = − 2 − s η ( s ) ± i β ( s ) , {\displaystyle \operatorname {Li} _{s}(\pm i)=-2^{-s}\eta (s)\pm i\beta (s),} where β ( s ) is the Dirichlet ...
There’s a number of reasons why we need a good balance of electrolytes in the body; it helps regulate fluid levels, promotes a healthy pH and supports nerve, muscle and brain function, says ...