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In mathematics, especially in set theory, two ordered sets X and Y are said to have the same order type if they are order isomorphic, that is, if there exists a bijection (each element pairs with exactly one in the other set) : such that both f and its inverse are monotonic (preserving orders of elements).
In mathematics, a generating function is a representation of an infinite sequence of numbers as the coefficients of a formal power series.Generating functions are often expressed in closed form (rather than as a series), by some expression involving operations on the formal series.
Faà di Bruno's formula gives coefficients of the composition of two formal power series in terms of the coefficients of those two series. Equivalently, it is a formula for the nth derivative of a composite function. Lagrange reversion theorem for another theorem sometimes called the inversion theorem; Formal power series#The Lagrange inversion ...
Every n in M x can be written as n = m 2 r with positive integers m and r, where r is square-free. Since only the k primes p 1, ..., p k can show up (with exponent 1) in the prime factorization of r, there are at most 2 k different possibilities for r. Furthermore, there are at most √ x possible values for m.
The Dirichlet inverse of a sequence () is related to the inverse Mellin transform of its generating function: = + = (), where is a real number greater than the abscissa of convergence of the Dirichlet series = /.
The complexity of an elementary function is equivalent to that of its inverse, since all elementary functions are analytic and hence invertible by means of Newton's method. In particular, if either exp {\displaystyle \exp } or log {\displaystyle \log } in the complex domain can be computed with some complexity, then that complexity is ...
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.
If n > 1, then there are just as many even permutations in S n as there are odd ones; [3] consequently, A n contains n!/2 permutations. (The reason is that if σ is even then (1 2)σ is odd, and if σ is odd then (1 2)σ is even, and these two maps are inverse to each other.) [3] A cycle is even if and only if its length is odd. This follows ...