Search results
Results From The WOW.Com Content Network
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.
Let (,,) be a measure space, and be a Banach space.The Bochner integral of a function : is defined in much the same way as the Lebesgue integral. First, define a simple function to be any finite sum of the form = = (), where the are disjoint members of the -algebra , the are distinct elements of , and χ E is the characteristic function of .
For a formal statement of the theorem, let : be a continuous map from a compact triangulable space to itself. Define the Lefschetz number [2] of by := ((,)), the alternating (finite) sum of the matrix traces of the linear maps induced by on (,), the singular homology groups of with rational coefficients.
In mathematics, the infinite series 1 / 2 + 1 / 4 + 1 / 8 + 1 / 16 + ··· is an elementary example of a geometric series that converges absolutely. The sum of the series is 1.
[proof 1] In particular, any finite-dimensional subspace of is complemented. [ 7 ] In arbitrary topological vector spaces, a finite-dimensional vector subspace Y {\displaystyle Y} is topologically complemented if and only if for every non-zero y ∈ Y {\displaystyle y\in Y} , there exists a continuous linear functional on X {\displaystyle X ...
A summation-by-parts (SBP) finite difference operator conventionally consists of a centered difference interior scheme and specific boundary stencils that mimics behaviors of the corresponding integration-by-parts formulation. [3] [4] The boundary conditions are usually imposed by the Simultaneous-Approximation-Term (SAT) technique. [5]
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.
Pairwise summation is the default summation algorithm in NumPy [9] and the Julia technical-computing language, [10] where in both cases it was found to have comparable speed to naive summation (thanks to the use of a large base case).