Search results
Results from the WOW.Com Content Network
While not derived as a Riemann sum, taking the average of the left and right Riemann sums is the trapezoidal rule and gives a trapezoidal sum. It is one of the simplest of a very general way of approximating integrals using weighted averages. This is followed in complexity by Simpson's rule and Newton–Cotes formulas.
The Weyl tensor has the same basic symmetries as the Riemann tensor, but its 'analogue' of the Ricci tensor is zero: = = = = The Ricci tensor, the Einstein tensor, and the traceless Ricci tensor are symmetric 2-tensors:
Generally speaking, Riemann solvers are specific methods for computing the numerical flux across a discontinuity in the Riemann problem. [1] They form an important part of high-resolution schemes; typically the right and left states for the Riemann problem are calculated using some form of nonlinear reconstruction, such as a flux limiter or a WENO method, and then used as the input for the ...
The Riemann zeta function can be replaced by a Dirichlet L-function of a Dirichlet character χ. The sum over prime powers then gets extra factors of χ(p m), and the terms Φ(1) and Φ(0) disappear because the L-series has no poles.
The Poisson summation formula is also useful to bound the errors obtained when an integral is approximated by a (Riemann) sum. Consider an approximation of S ( 0 ) = ∫ − ∞ ∞ d x s ( x ) {\textstyle S(0)=\int _{-\infty }^{\infty }dx\,s(x)} as δ ∑ n = − ∞ ∞ s ( n δ ) {\textstyle \delta \sum _{n=-\infty }^{\infty }s(n\delta ...
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: = = = + +.
In real analysis, the Darboux integral is constructed using Darboux sums and is one possible definition of the integral of a function.Darboux integrals are equivalent to Riemann integrals, meaning that a function is Darboux-integrable if and only if it is Riemann-integrable, and the values of the two integrals, if they exist, are equal. [1]
In mathematics, the Bernoulli numbers B n are a sequence of rational numbers which occur frequently in analysis.The Bernoulli numbers appear in (and can be defined by) the Taylor series expansions of the tangent and hyperbolic tangent functions, in Faulhaber's formula for the sum of m-th powers of the first n positive integers, in the Euler–Maclaurin formula, and in expressions for certain ...