An algorithm for the incomplete decomposition of a rational function into partial fractions

1971 ◽  
Vol 22 (4) ◽  
pp. 751-755 ◽  
Author(s):  
Peter Henrici
Keyword(s):  
Rational Function ◽  
Partial Fractions

An Algorithm for Partial-Fraction Expansion

1972 ◽  
Vol 65 (3) ◽  
pp. 237-239
Author(s):  
Joseph W. Rogers ◽  
Margaret Anne Rogers
Keyword(s):  
Rational Function ◽  
Partial Fraction ◽  
Partial Fraction Expansion ◽  
Partial Fractions

We Usually expand a rational function by first reducing it to the sum of a polynomial and a proper fraction that is the quotient of two polynomials in which the degree of the numerator is less than the degree of the denominator. We expand the fraction by writing it as a sum of partial fractions with undetermined numerator coefficients.

scholarly journals Generating Functions and Generalized Dedekind Sums

10.37236/1326 ◽  
1996 ◽  
Vol 4 (2) ◽  
Author(s):  
Ira Gessel
Keyword(s):  
Rational Function ◽  
Explicit Formula ◽  
Simple Form ◽  
Generating Functions ◽  
Dedekind Sums ◽  
Roots Of Unity ◽  
Partial Fractions

We study sums of the form $\sum_\zeta R(\zeta)$, where $R$ is a rational function and the sum is over all $n$th roots of unity $\zeta$ (often with $\zeta =1$ excluded). We call these generalized Dedekind sums, since the most well-known sums of this form are Dedekind sums. We discuss three methods for evaluating such sums: The method of factorization applies if we have an explicit formula for $\prod_\zeta (1-xR(\zeta))$. Multisection can be used to evaluate some simple, but important sums. Finally, the method of partial fractions reduces the evaluation of arbitrary generalized Dedekind sums to those of a very simple form.

Short Generating Functions for some Semigroup Algebras

10.37236/1729 ◽  
2003 ◽  
Vol 10 (1) ◽  
Author(s):  
Graham Denham
Keyword(s):  
Rational Function ◽  
Generating Functions ◽  
Hilbert Series ◽  
Simple Expression ◽  
Arbitrary Field ◽  
Graded Algebra ◽  
Semigroup Algebras ◽  
Positive Integers

Let $a_1,\ldots,a_n$ be distinct, positive integers with $(a_1,\ldots,a_n)=1$, and let k be an arbitrary field. Let $H(a_1,\ldots,a_n;z)$ denote the Hilbert series of the graded algebra k$[t^{a_1},t^{a_2},\ldots,t^{a_n}]$. We show that, when $n=3$, this rational function has a simple expression in terms of $a_1,a_2,a_3$; in particular, the numerator has at most six terms. By way of contrast, it is known that no such expression exists for any $n\geq4$.

scholarly journals The differential Galois group of the rational function field

2021 ◽  
Vol 381 ◽  
pp. 107605
Author(s):  
Annette Bachmayr ◽  
David Harbater ◽  
Julia Hartmann ◽  
Michael Wibmer
Keyword(s):  
Galois Group ◽  
Rational Function ◽  
Function Field ◽  
Differential Galois Group ◽  
Rational Function Field

scholarly journals The bifurcation set of a rational function via Newton polytopes

2021 ◽  
Author(s):  
Tat Thang Nguyen ◽  
Takahiro Saito ◽  
Kiyoshi Takeuchi
Keyword(s):  
Rational Function ◽  
Newton Polytopes ◽  
Bifurcation Set

scholarly journals RG flows of integrable σ-models and the twist function

2021 ◽  
Vol 2021 (2) ◽  
Author(s):  
François Delduc ◽  
Sylvain Lacroix ◽  
Konstantinos Sfetsos ◽  
Konstantinos Siampos
Keyword(s):  
Renormalization Group ◽  
Large Class ◽  
Rational Function ◽  
Integrable Model ◽  
Renormalization Group Flow ◽  
Flow Equations ◽  
Non Linear ◽  
Group Flow

Abstract In the study of integrable non-linear σ-models which are assemblies and/or deformations of principal chiral models and/or WZW models, a rational function called the twist function plays a central rôle. For a large class of such models, we show that they are one-loop renormalizable, and that the renormalization group flow equations can be written directly in terms of the twist function in a remarkably simple way. The resulting equation appears to have a universal character when the integrable model is characterized by a twist function.

scholarly journals Primeable entire functions

1973 ◽  
Vol 51 ◽  
pp. 123-130 ◽  
Author(s):  
Fred Gross ◽  
Chung-Chun Yang ◽  
Charles Osgood
Keyword(s):  
Entire Function ◽  
Periodic Function ◽  
Rational Function ◽  
Entire Functions ◽  
Left And Right

An entire function F(z) = f(g(z)) is said to have f(z) and g(z) as left and right factors respe2tively, provided that f(z) is meromorphic and g(z) is entire (g may be meromorphic when f is rational). F(z) is said to be prime (pseudo-prime) if every factorization of the above form implies that one of the functions f and g is bilinear (a rational function). F is said to be E-prime (E-pseudo prime) if every factorization of the above form into entire factors implies that one of the functions f and g is linear (a polynomial). We recall here that an entire non-periodic function f is prime if and only if it is E-prime [5]. This fact will be useful in the sequel.

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