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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.
In mathematics, an exponential sum may be a finite Fourier series (i.e. a trigonometric polynomial), or other finite sum formed using the exponential function, usually expressed by means of the function = (). Therefore, a typical exponential sum may take the form
The exponential function (in blue), and its improving approximation by the sum of the first n + 1 terms of its Maclaurin power series (in red). So So n=0 gives f ( x ) = 1 {\displaystyle f(x)=1} ,
The exponential function (in blue), and the sum of the first n + 1 terms of its power series (in red) The exponential function is the sum of a power series: [2] [3] = + +! +! + = =!, where ! is the factorial of n (the product of the n first positive integers).
Similarly, in a series, any finite rearrangements of terms of a series does not change the limit of the partial sums of the series and thus does not change the sum of the series: for any finite rearrangement, there will be some term after which the rearrangement did not affect any further terms: any effects of rearrangement can be isolated to ...
(2) If G is a finite group then the sum in the exponential is a finite sum running over all subgroups of G, and continuous homomorphisms from G to S n are simply homomorphisms from G to S n. The result in this case is due to Wohlfahrt (1977). The special case when G is a finite cyclic group is due to Chowla, Herstein, and Scott (1952), and ...
A Laurent series is a generalization of the Taylor series, allowing terms with negative exponents; it takes the form = and converges in an annulus. [6] In particular, a Laurent series can be used to examine the behavior of a complex function near a singularity by considering the series expansion on an annulus centered at the singularity.
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.