Questions tagged [rational-functions]
Rational functions are ratios of two polynomials, for example $(x+5)/(x^2+3)$.
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Computing ratio of two sums
I'm interested in computing
$$ f(s) := \frac{ \sum_{n=0}^{\infty} a_n s^n } { \sum_{n=0}^{\infty} b_n s^n } $$
for some given $s \in \mathbb{C}$, where the power series in the numerator and ...
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Evaluation of a rational function vs formal power series
Let $K$ be a complete valued field and $X$ be an indeterminate $K[X]$ be the ring of polynomials over $K$ in the indeterminate $X$ and $K(X)$ be its quotient field.
For $a\in K$, consider the ...
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Can a rational function have infinitely many vertical asymptotes?
My first approach to this problem is to make a rational function that has infinitely many distinct real roots or zeros at distinct values of $a_1,\dots,a_i$ for $n$ approaching infinity
$$\frac{p(x)}{...
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Periodic Points in Julia Set under Rational Map
Let $A_1$ be the finite set of periodic points in the Julia set for a rational map, $R$, which has lowest possible period.
Inductively, define $A_{n+1} = R^{-1} (A_n) \cup A_n$.
Why is it true that $...
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Find $\mathcal{O}(x)$ of $\frac{x^2+x^3+x^4}{(x+x^2+x^3)^{3/2}}$
Consider $$f(x)=\frac{x^2+x^3+x^4}{(x+x^2+x^3)^{3/2}}.$$
Aim: Write $f(x) = g(x) + \mathcal{O}(x)$ as $x\downarrow0$.
I can change the fraction to $$f(x)=\frac{x^{1/2}+x^{3/2}+x^{5/2}}{(1+x+x^2)^{3/2}}...
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With polynomial long division, why are the un-applied divisor terms still multiplied and subtracted?
Given the following for example:
$$\frac{x^2+x+8}{x+2}$$
The first point of confusion:
The divisor $x + 2$ when performing the division operation, only $x$ is applied. While this works with $x$ ...
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Closed-form solution to maximization of rational function with linear and bilinear form [closed]
I have the function:
$$f(\mathbf{x}) = \frac{\mathbf{x}^\mathrm{H}\mathbf{w} + \mathbf{w}^\mathrm{H}\mathbf{x}}{c + \mathbf{x}^\mathrm{H}\mathbf{A}\mathbf{x}},$$
where $\mathbf{x},\mathbf{w}\in\mathbb{...
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Finite critical points of a polynomial are preperiodic implies the Fatou set is connected and simply connected
I'm currently going through Alan Beardon's book "Iteration of Rational Functions" and I'm a little stuck on his explanation of Corollary 9.5.3. which states that "If every finite ...
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Does fundamental theorem of algebra hold in multi-variable polynomials?
Given algebraically closed field $\mathbb{F}$ and rational function in 2 variables $f(x,y)\in \mathbb{F}(x,y)$, does it factor out in linear terms $(ax+by), a,b\in \mathbb{F}$, or in other terms, is ...
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Rational Mappings of the Annulus
Suppose $R:\widetilde{\mathbb{C}} \rightarrow \widetilde{\mathbb{C}}$ where $R(A) = B$ is a rational mapping from one annulus to another. Assume that one of the components of the complement of $A$ has ...
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Confusion about the definition of local ring of an affine variety at a point and its relation with localization
I am reading Gathmann notes on Algebraic Geometry. He defines the local ring of an affine variety $X \subset \mathbb{A}^n$ (affine variety means an irreducible algebraic set here) as follows
$$
\...
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Is there a canonical form for rational expressions?
Polynomials have a canonical form $\sum a_n x^n$ which makes it easy to understand what a polynomial is. Yet rational expressions can often wear many disguises, where it's not obvious that two ...
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Can a homographic function be approximated by an exponential function?
Can the homographic function:
$$f(x)=\frac{1+\frac{x}{a}}{1-\frac{x}{1-a}}$$
where a ∈ (0,1), be approximated by an exponential function for the interval x ∈ [0,1-a] (where the function f(x) behaves ...
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Decision procedure for whether the power series of a rational function has only nonnegative coefficients
My question is about rational functions of the form $f(x) = \frac{p(x)}{q(x)}$ where $p(x) = \sum_{i=0}^n p_i x^i$ and $q(x) = \sum_{i=0}^n q_i x^i$ with $p_i, q_i \in \mathbb{Q}$ and $q_0 \ne 0$. ...
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Characterize $\frac {ax + b} {cx + d}$ as $x \to \frac {-d} c$
Consider the function $f(x) = \frac {ax + b} {cx + d}$ with $a,b,c,d \in \mathbb R, \neq 0$. If $ad \neq cb$, then as $x \to \frac {-d} {c}$, then $f(x)$ approaches $+ \infty$ from one side and $- \...
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General method to numerically solve for the absolute maximum(s) of an irregular sinusoid
I am trying to find a strictly mathematical (i.e., not manual graphing or plugging) method to solve for the absolute (global) maximum value of an irregular sinusoid.
Given a rational function $f(x)$, ...
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Exterior square of rational function of one variable
Let us consider vector space $$V = (\Lambda^2 (\mathbb C(t))^\times)\otimes_\mathbb Z\mathbb Q.$$
Here I consider the group $A=(\mathbb C(𝑡))^\times$
of non-zero rational functions under ...
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Power series expansion of $\frac{kx}{(x+1)^k-1}$ at $x=0$
The question is simple: for integer $k\geq 2$, how to calculate the power series expansion of $f(x)=\frac{kx}{(x+1)^k-1}$ at $x=0$?
As far as I am aware, there are two effective (operatable) ways to ...
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Decomposing NURBS curve into piecewise Bezier segments
I have a question concerning a paper. In it the authors try to approximate a NURBS curve by biarcs and to do so, they first need to find a polygonal approximation of the NURBS curve. They chose a ...
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Showing that $f=X^p-X+T$ is irreducible over $\mathbb{F}_p(T)[X]$
Let $K=\mathbb{F}_p(T)$ be the field of rational functions on one variable T over $\mathbb{F}_p$, and $f=X^p-X+T \in K[X]$. I want to show that $f$ is an irreducible polynomial.
I know that $T$ is a ...
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Why are the degrees of f(t) and g(z) related in this identity?
I am reading "Computing the Continuous Discretely" by Mattias Beck and Sinai Robins and am working on the following exercise (3.13) which is to prove the following:
If $$\sum_{t\geq 0}f(t)z^...
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Weierstrass elliptic function for quaternions?
What if we generalize and modify the Weierstrass elliptic function for quaternions ?
So our function could have $2,3$ or $4$ periods.
How would that theory be like ?
Is matrix representation the key ...
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Educated guess for algebraic approximation formula.
I found a very neat ancient hindi formula for approximating square roots using rational numbers. After doing some algebra on the formula, i came across with this recursive relation:
Given any number $...
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Computing $ \int \frac{{\rm d} x}{x^3 + 12} $
$$ \int \frac{{\rm d} x}{x^3 + 12} $$
This is a question I came up with and have not been able to solve. I graphed this function and it is as in the picture attached. Some of the people who I have ...
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$k$-th derivative of a rational function
Let $f$ be the function given by
$$
f(x) = \frac{1}{x+1}.
$$
As we easily check, the $k$-th derivative of $f$ is given by
$$
f^{(k)}(x) = (-1)^k \frac{k!}{(x+1)^{k+1}}.
$$
In particular $|f^{(k)}(x)| =...
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Find $b$ If $f:\mathbb R\to\mathbb R, \;f(x)=\dfrac{x^2+bx+1}{x^2+2x+b}, (b>1)$ and $f(x), \dfrac{1}{f(x)}$ have same bounded Range
Let $f:\mathbb R\to\mathbb R, \;f(x)=\dfrac{x^2+bx+1}{x^2+2x+b}, (b>1)$ and $f(x), \dfrac{1}{f(x)}$ have the same bounded set as their range, then value of $b$ is
My Approach:
Let $f(x)=y=\dfrac{x^...
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How do I find the value of $k$ such that $(x-1)/(x-3) = (k-9x)/(x+2)$ has exactly one solution?
The answer is supposed to be $k = 14$ or $54$ with the quadratic formula. I've tried cross multiplying and doing something like this:
$1. (x-1)(x+2) = (k-9x)(x-3)$
$2. x^2 + 2x - x - 2 = kx - 3k - 9x^...
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Find all possible values of $\frac{1}{{x^2 - x - 1}}$
Question: Find all possible values of $$\frac{1}{{x^2 - x - 1}}$$
My solution-> $\frac{1}{{x^2 - x - 1}}=\frac{1}{{(x-\frac{1}{2})^2-\frac{5}{4}}}$
=$(x-\frac{1}{2})^2 \geq 0 \quad \forall x \in \...
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What kind of matroidal information is contained in the ratio of the Tutte polynomial and its dual?
Let $M$ be a matroid and $T(x,y)$ its Tutte polynomial. It is well-known that $T(y, x)$ is equal to the Tutte polynomial of the dual of $M$. What kind of matroidal information (if any) is contained in ...
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"Constrained" value of a function of two variables
Let $f:\mathbb{R}^2\to\mathbb{R}$ be a rational function, whose numerator and denominator are (second-degree) polynomials of $x$ and $y$.
The problem is to decide whether, for some given $k,k',T\in\...
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Minimum value of $\frac{a^2 + b^2 +1 }{a(b+1)}$ [closed]
Find the minimum value of $\dfrac{a^2 + b^2 +1 }{a(b+1)}$ for $a$, $b$ positive reals.
I know that the solution is $\sqrt{2}$ using perfect squares, derivatives, etc. However, I want to find the ...
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Rational solutions for $x^3+y^3=1$ where both x and y are non-negative
How can I find rational solutions for $x^3+y^3=1$ where both x and y are non-negative?
Edit: One of the answer in this post for general form of solutions
$$(a,b) \mapsto \left( \frac{a(a^3 + 2b^3)}{a^...
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Partial fraction decomposition with algebraic structure ring and modulo
I know that an exercise like this
\begin{gather*}
\frac{5x^2 - 12x + 6}{(x - 2) (x - 1) (2 x - 3)}
\end{gather*}
is resolvable with several approaches the Heaviside "Cover-up method" e.g.
\...
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Coprime factorization of rational matrix over the polynomial ring
Dear esteemed colleagues,
I have a question regarding the coprime factorization of rational matrices over the polynomial ring. Consider the following rational matrix
$$
R(s) = \begin{pmatrix} \frac{1}{...
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The field of meromorphic functions over $\hat{\mathbb{C}}$ is equivalent to the field of rational functions
I’d like to prove the following statement from my lecture notes:
The field of meromorphic functions over $\hat{\mathbb{C}}$ is equivalent
to the field of rational functions, with the convention $1/0=\...
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Minimizing the sum of rational functions over simplex
Let
$$f(x) := \frac{2x+1}{x^2+3}$$
I would like to solve the following optimization problem
$$ \begin{array}{ll} \underset {x_1, x_2, x_3} {\text{minimize}} & f(x_1) + f(x_2) + f(x_3) \\ \text{...
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How to characterize the rational points on a sphere of radius $\sqrt{3}$
Let's take a $3$ dimensional sphere of radius $r$, such as $x^2+y^2+z^2 = r^2$.
For $r^2=1$, all the rational points of the sphere (i.e. points for which all coordinates are rational numbers) are ...
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Solve for constants in rational model
I have the following rational model equation which fits distance versus time data very well:
$$y = \frac{bt}{1+ct+dt^2}$$
The derivative of this equation is :
$$\frac{dy}{dt} = \frac{(b-bdt^2)}{(1+ct+...
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Rational fractions as geometric predicates
I read article about restricted voronoi diagram. In chapter 3.4 Exact predicates I saw the following: "sideA, sideB, sideC are rational fractions of low-degree (resp. 2, 4/2 and 6/4)." sideA ...
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Transformation that keeps the point $(1, 1)$ but raises the horizontal asymptote of the reciprocal function?
If we are given the function $f(x) = \frac{1}{x}$, we can change the horizontal asymptote by adding a constant to the function. For example, $g(x) = f(x) + 5$, makes the asymptote at $y = 5$.
How ...
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Solving Rational Inequalities via Reciprocals
I was reviewing rational inequalities, and noticed that the solution (identifying the domain of the inequality) was obtained just through critical points and a number line. Is there a way to get the ...
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Division by zero in Ireland and Rosen's "A Classical Introduction to Modern Number Theory"
In Ireland and Rosen's A Classical Introduction to Modern Number Theory (Second Edition) the proof of Chapter 6 Section 4 Proposition 6.4.2. starts with dividing $x^p-1=(x-1)\prod_{j=1}^{p-1} (x-\zeta^...
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Closed form for integrals of an exponential with rational function argument?
I am trying to find the closed form for the following integral:
$$I=\int_{-\infty}^{\infty}e^{-a^2(x-b)^2+\frac{c}{1+x^2}}\ {\rm d}x$$
where $a,b,c\in\mathbb{R}$, and $a\ge0$. I know that the integral ...
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Does $\mathbb{R}(X)$ have an automorphism of order 7?
The question comes from a 2020 home exam of an undergraduate course in field theory and Galois theory and is as follows:
Let $\mathbb{R}(X)$ be the field of rational functions over $\mathbb{R}$.
...
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Does there exist a rational function $g(x,y)$ that picks out the first quadrant?
Problem statement: Let $V = \{(x,y) \in \mathbb{R}^2 : x,y>0\}$ be the first quadrant of the plane.
Does there exist a rational function $g : \mathbb{R}^2 \rightarrow \mathbb{R}$ such that $g(V)$ ...
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Convexity of rational functions between poles
Problem statement
Consider the following rational function:
$$ f(x) = -\frac{(x-a)(x-d)}{(x-b)(x-c)(x-e)} $$
where $a < b < c < d < e$.
I would like to prove that $f$ is convex on $(b,c)$.
...
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(Convex) optimization of a sum of two variables quadratic fractional (rational) function subject to circle constraint
I want to solve the below optimization problem.
\begin{align}
\min_{x, y} & \frac{a_1}{x^2} + \frac{a_2}{xy} \\
\mbox{s.t. } &x^2 + y^2 \leq \epsilon \\
&x \geq 0 \\
&y \geq 0
\end{...
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If integration sends the space of all rational functions to the same space union $\ln x$, where does integration send $\mathbb Q[x,\ln x]$?
If integration sends the space of all rational functions to the same space union $\ln x$, where does integration send $\mathbb Q[x,\ln x]$?
For example, if I have any rational function, which can be ...
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Estimate integral error from root's error
Let $R(z)$ and $Q(z)$ be polynomial functions:
$$R(z) = r_0 + r_1 \cdot z + \cdots + r_n \cdot z^{n}$$
$$Q(z) = 1 + q_1 \cdot z + \cdots + q_m \cdot z^{m}$$
From fundamental theorem of algebra, $Q(z)$ ...
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Why care about excluded values of rational expressions when in lowest terms?
I'm struggling to come up with an explanation for why we specify excluded values for rational expressions that have been simplified to lowest terms, and yet do not exhibit the same division by zero ...
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