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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.
A weight function is a mathematical device used when performing a sum, integral, or average to give some elements more "weight" or influence on the result than other elements in the same set. The result of this application of a weight function is a weighted sum or weighted average .
In mathematics, a Riemann sum is a certain kind of approximation of an integral by a finite sum. It is named after nineteenth century German mathematician Bernhard Riemann . One very common application is in numerical integration , i.e., approximating the area of functions or lines on a graph, where it is also known as the rectangle rule .
Note that this approximation corresponds to a finite volume method representation whereby the discrete values represent averages of the state variables over the cells. Exact relations for the averaged cell values can be obtained from the integral conservation laws. Obtain the solution for the local Riemann problem at the cell interfaces. This ...
Littlewood stated the principles in his 1944 Lectures on the Theory of Functions [1] as: . There are three principles, roughly expressible in the following terms: Every set is nearly a finite sum of intervals; every function (of class L p) is nearly continuous; every convergent sequence of functions is nearly uniformly convergent.
This algorithm can easily be adapted to compute the variance of a finite population: simply divide by n instead of n − 1 on the last line.. Because SumSq and (Sum×Sum)/n can be very similar numbers, cancellation can lead to the precision of the result to be much less than the inherent precision of the floating-point arithmetic used to perform the computation.
In mathematics, the Euler–Maclaurin formula is a formula for the difference between an integral and a closely related sum. It can be used to approximate integrals by finite sums, or conversely to evaluate finite sums and infinite series using integrals and the machinery of calculus .
The operator takes a locally integrable function f : R d → C and returns another function Mf. For any point x ∈ R d, the function Mf returns the maximum of a set of reals, namely the set of average values of f for all the balls B(x, r) of any radius r at x. Formally,