词条 | Series multisection |
释义 |
In mathematics, a multisection of a power series is a new power series composed of equally spaced terms extracted unaltered from the original series. Formally, if one is given a power series then its multisection is a power series of the form where p, q are integers, with 0 ≤ p < q. Multisection of analytic functionsA multisection of the series of an analytic function has a closed-form expression in terms of the function : where is a primitive q-th root of unity. This solution was first discovered by Thomas Simpson.[1] This expression is especially useful in that it can convert an infinite sum into a finite sum. It is used, for example, in a key step of a standard proof of Gauss's digamma theorem, which gives a closed-form solution to the digamma function evaluated at rational values p/q. ExamplesBisectionIn general, the bisections of a series are the even and odd parts of the series. Geometric seriesConsider the geometric series By setting in the above series, its multisections are easily seen to be Remembering that the sum of the multisections must equal the original series, we recover the familiar identity Exponential functionThe exponential function by means of the above formula for analytic functions separates into The bisections are trivially the hyperbolic functions: Higher order multisections are found by noting that all such series must be real-valued along the real line. By taking the real part and using standard trigonometric identities, the formulas may be written in explicitly real form as These can be seen as solutions to the linear differential equation with boundary conditions , using Kronecker delta notation. In particular, the trisections are and the quadrusections are Binomial theoremMultisection of a binomial expansion at x = 1 gives the following identity for the sum of binomial coefficients with step q: References1. ^{{cite journal |last1=Simpson |first1=Thomas |date=1757 |title=CIII. The invention of a general method for determining the sum of every 2d, 3d, 4th, or 5th, &c. term of a series, taken in order; the sum of the whole series being known |url= |journal=Philosophical Transactions of the Royal Society of London |volume=51 |pages=757-759 |doi=10.1098/rstl.1757.0104}}
5 : Algebra|Combinatorics|Mathematical analysis|Complex analysis|Mathematical series |
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