Questions tagged [rational-functions]

Rational functions are ratios of two polynomials, for example $(x+5)/(x^2+3)$.

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Julia set of finite Blaschke product

I want to compute the Julia set of finite Blaschke product $B_3$ $$B_3 = (\frac{z+1/2}{z/2+1})^3$$ First I should analyze the map so I compute first derivative : $$ (3*(z+1/2)^2)/(z/2+1)^3-(3*(z+1/2)^...
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Rational Functions Inequality

I'm trying to solve the following inequality: $\tfrac{3x}{x^2+2}≥\tfrac{1}{x-1}$. When I solve to get a zero on one side I do the following: $\tfrac{3x}{x^2+2}≥\tfrac{1}{x-1}\Leftrightarrow \tfrac{3x(...
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Finding the rational function the power series $\sum_{n=0}^\infty \frac{(-1)^n}{3^{n+1}}x^{n+2}$ converges to

Given the power series, find the rational function equation that the power series converges to. $$\sum_{n=0}^\infty \frac{(-1)^n}{3^{n+1}}x^{n+2}$$ So far I've figured out that the rational function $\...
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Rational functions with a special symmetry

Let $f\colon \mathbb{C} \to \mathbb{C}$ be a rational function that satisfies the following property: $$ f(x) = f(-1/x),\ \ \forall x\in\mathbb{C}. $$ I wonder whether functions with this property ...
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Is the following function rational?

The rational function is defined as the quotient of two polynomials. Can functions that are equal to rational functions also be called rational, because they have the same properties? One concrete ...
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When calculating the functions intersection of a horizontal asymptote, do you use the simplified function or the main function?

let’s say we have a function $$ f(x)=\frac{2x^2-5x+2}{x^2-4} $$ finding the ratio of the leading terms gives you $2x^2/x^2=2$, so we have a horizontal asymptote at $y = 2$. If you factor out the ...
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If $Q(x)$, $R(x)$, and $P(x):=Q(x)/R(x)$ are all polynomials, then is it true that $\deg(P)=\deg(Q)-\deg(R)$?

Let $Q(x)$ and $R(x)$ be polynomials. Suppose I know that $$ P(x):=\frac{Q(x)}{R(x)} $$ is also a polynomial (and not just some rational function). Is it always true that $\deg(P(X))=\deg(Q(x))-\deg(R(...
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What is known about convergence of (derivative-matching) rational approximations?

Let $R(x)$ be a rational function and $f(x)$ some smooth function. Assume we have $$ \frac{d^n}{dx^n}R(x) = \frac{d^n}{dx^n}f(x) $$ at $x=0$ for all $0 \le n \le N$. Padé approximation is an example ...
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Rational maps and homology

Let $X$ and $Y$ be algebraic varieties defined over $\mathbb{C}$. By considering them equipped with the complex topology, we can talk about singular chains and singular homology, as usual Suppose $f:X\...
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How would I make sense of the answer choices for solutions to a rational equation?

I am trying to find all of the asymptotes of a rational function, and the answer choices I am given are confusing. Because the degree of the function in the numerator was greater than the degree of ...
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How do I prove that this complex function can be approximated uniformly by rational functions?

I have the following problem: We have $K\subset \Bbb{C}$ a compact subset and $f:K\rightarrow \Bbb{C}$ a continuous function. We take $a\in \Bbb{C}\setminus K$. We assume that there exists a sequence ...
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Should I check(search) for what numbers an equation has no or infinite number of solution? [closed]

I got myself a collection of solved questions from math, and currently, I am solving linear equations with parameters. Now, in the book authors plugin 1 and -1 after they solve equations, and check ...
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Is a polynomial divided by one of it's linear factors defined at the root of the linear factor? [duplicate]

I'm sure this is a very simple question, but I still thought it would be good to get some sound clarification. As an example, if I have the function; $f(x) := \frac{x^2 - 2x + 1}{x-1}$ My question is, ...
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Finding the vertical and horizontal asymptotes of the function $f(x) = \frac{e^x(x + 1)}{e^{2x}(x^2 - 1)}$

I am to find the vertical and horizontal asymptotes of this given function: $$f(x) = \frac{e^x(x + 1)}{e^{2x}(x^2 - 1)}$$ To find the vertical asymptote,I think I equate the bottom line to zero and ...
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How to change coordinates so that $\pi|a_3,a_4,a_6$ in the second step of Tate's algorithm.

I am trying to apply Tate's algorithm for the elliptic curve $$y^2-2(7T+3)xy-36(7T+3)y-x^3-2(16T^2-7)x^2+324x+648(16T^2-7)$$ over the rational function field $\mathbb{Q}(T)$ at the primes $\pi=T+1,T-1,...
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On a lemma used in proving a regular function is not a quotient of polynomials

Lemma: Consider algebraic set $X=V(wx-yz)\subset \mathbb{A}^{4}_{k}$, where $k$ is an algebraically closed field. Then there doesn't exist $h\in k[x,y,z,w]$ s.t. $h\mid _{X}$ is not a constant and $V(...
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Why do rational functions in $\mathbb{P}^n$ have to be a ratio of homogeneous polynomials of the same degree?

To have a well defined homogeneous polynomial $f:\mathbb{P}^n\rightarrow \mathbb{P}^n$, we require that it is homogeneous so that $f(x)=f(\lambda x)=\lambda^{\deg f} f(x)=f(x)$ (so that it is well ...
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Determining for which values of $m$ and $n$ $\frac{P_n(x)}{P_m(x)}$ is injective

By injective I mean if $f(x)=f(y)$, then $y=x$. $P_n$ and $P_m$ are degree $n$ & $m$ polynomials. my approach was this: $$f(x)=f(y)\to\frac{P_m(x)}{P_n(x)}=\frac{P_m(y)}{P_n(y)}$$ $$P_m(x)P_n(y)=...
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Rational functions with Mobius symmetries

I'm interested in describing solutions to the equation $ f = f \circ \gamma $, where $ f : \mathbb{C} \rightarrow \mathbb{C} $ is a rational function of a given degree, and $ \gamma : \mathbb{C} \...
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Why is $\frac{1}{1+(\frac{-\sin x}{1+\cos x})^2} \equiv \frac{(1+\cos x)^2}{\sin^2x+1+2\cos x+\cos^2x}$?

Currently I have a problem to understand why $$\frac{1}{1+(\frac{-\sin x}{1+\cos x})^2} \equiv \frac{(1+\cos x)^2}{\sin^2x+1+2\cos x+\cos^2x}\,?$$ My calculations get me: $$\frac{1}{1+(\frac{-\sin x}{...
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Is my rational function process for calculating tan(x) correct?

I was bored in my Algebra 2 class and wanted to try to illustrate the tan(x) function as an infinite summation of rational functions. I recalled that the solution to a rational function's numerator is ...
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Julia set of $z^2+2z$ using conjugation of $z^2$

I wanted to calculate the Julia set of $S: z \mapsto z^2+2z$. I found that for $ \varphi: z \mapsto z-1 $ and $\varphi^{-1}: z \mapsto z+1$ the map $R: z \mapsto z^2$ is conjugate: $$ S(z) = z^2 + 2z =...
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Rational Function Problem [closed]

Let $P(x)$ and $Q(x)$ be two polynomials with integer coefficients. If $\frac{P(n)}{Q(n)}$ is an integer for every integer $n$, prove that there exists a polynomial $S(x)$ with rational coefficients ...
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Behaviour of a rational function in the vicinity of a pole

I find hard to prove the following result : if $F=\frac PQ$ is a rational function, and if $a$ is a pole of $F$ of order $\alpha$, then there are reals $M>0$ and $\nu>0$ such that for all $t\in\...
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Why is $F(x)\cong F[x]\otimes F(x^n)$?

Let $F$ be field. Let $F[x]$ denote the polynomial ring over $F$, and $F(x)$ its field of fractions. Let $F(x^n)$ be the field of fractions of $F[x^n]$, the subring of polynomials in $x^n$. Consider ...
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Confusion about a rational function

There's a function $f(x)=\frac{(x+2)(x+1)}{(x-4)^2(x+1)}$ I'm a little confused: is this function considered to be defined at $x=-1$? On the one hand, if you plug in $-1$, you'll get $0/0$. But on ...
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How to find exact value of integral $\int_{0}^{\infty} \frac{1}{\left(x^{4}-x^{2}+1\right)^{n}}dx$?

When I first encountered the integral $\displaystyle \int_{0}^{\infty} \frac{1}{x^{4}-x^{2}+1} d x$, I am very reluctant to solve it by partial fractions and search for any easier methods. Then I ...
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Would you like to know the formula for the integral $\int_{0}^{\infty} \frac{x^{2}}{\left(x^{4}+1\right)^{n}} d x$ where $n\in N$?

When struggling with the integral in my answer , $$ \int \frac{x^{2}}{\left(x^{4}+1\right)^{2}} d x= \frac{\pi}{8 \sqrt{2}}, $$ I encountered the simpler integral, $$\int_{0}^{\infty} \frac{x^{2}}{x^{...
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General way of determine where a rational function $\phi\in k(X)$ is not regular

So, if you have an irreducible variety $X$ over $k=\bar{k}$ and you consider a rational function $\phi=f/g\in k(X)$, is there a general way to determine where $\phi$ is not regular? At this point I ...
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Why do I have to perform polynomial division when trying to find slant asymptotes

When trying to find the slant asymptote of $\frac{2x^2+x}{x-3}$, the way I thought was correct is to divide everything by $x$ to get $\frac{2x+1}{1-\frac{3}x}$. All that was left was to say that as $x$...
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Integral of $1/(ax^2 +bx +c)^n$

How can we obtain the recursion relation for the integral of the following rational function? $$ \begin{align} \int \frac{dx}{(ax^2 +bx +c)^n} &= \frac{ 2ax+b }{ (n-1) (4ac-b^2) (ax^2 +bx +c)^{n-...
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How to prove or disprove $ |r'(a)| \leq c_1 \sup_{x \in [a-1/2,a+1/2]} |r(x)|$?

Let matrix $ A = \begin{bmatrix} a & 1 \\ 0 & a \end{bmatrix}$ is a Jordan matrix with $ -1 < a < 1 $. Let $ r(z) = \frac{p(z)}{q(z)} $ is any an irreducible rational function, where ...
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Derivative of rational function can be bounded by the rational function

I'm studying something about singular value and I came across a problem. Matrix $A = \begin{bmatrix} a & 1 \\ 0 & a \end{bmatrix}$ is a Jordan matrix with $a>1$. For an integer $k$, let $...
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What is the relevance of order 2 points for poles & zeros of Elliptic curves?

I am reading up on poles & zeros & divisors of Elliptic Curves & I found some statements about points of order 2 which aren't well explained. https://crypto.stanford.edu/pbc/notes/...
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Minimize $\dfrac{x^2+1}{x^2(1-x)(x+3)}$ on $(0,1)$

One can differentiate to obtain $$f'(x)=\frac{2(x^2+3)(x^2+x-1)}{x^3(1-x)^2(x+3)^2}.$$ Thus, the unique stationary point is $x=\dfrac{\sqrt{5}-1}{2}$, which is also the minimum point on $(0,1)$. But ...
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Generating Function Calculation Gone Wrong

Let $T$ be a function of two complex variables, analytic at the origin. Express $T$ as a power series: $$T(x,y)=\sum_{i,j}t_{ij}x^iy^j$$ I would like to find a generating function for the sequence $(...
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How do we handle the integral $I_{4}=\int\left(\frac{\cos \theta}{1+\sin ^{2} \theta}\right)^{4} d \theta$?

After struggling with $I_2,I_3$ in the post, I dare to tackle $I_4$ now. We first rewrite the integral $$I_{4}=\int\left(\frac{\cos \theta}{1+\sin ^{2} \theta}\right)^{4} d \theta =\int \frac{\sec ^{2}...
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Rational functions: unique representation as a polynomial fraction?

Let $\mathbb{F}$ be a field. And let $\mathbb{F}(\bar x)$ be the field of rational functions over the field $\mathbb{F}$ in the indeterminate $x_1,\dots,x_n$. Namely, all the functions $f:\mathbb{F}^n\...
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Determine the fixed subfield [duplicate]

This is exercise $7$ from Sec. 4.5 of Jacobson's Basic Algebra 1 (2nd Edition) : Let $E=(\mathbb{Z}/(p))(t)$ where $t$ is transcendental over $\mathbb{Z}/(p)$. Let $G$ be the group of automorphisms ...
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Rule for going from roots/asymptotes to intervals of a rational inequality

Suppose I have a rational inequality, like $$ \frac{(x-4)(x+4)}{(x-1)^2} < 0 $$ I can manually check between the roots (4,4) and asymptote (1) by plugging in value like 3 to see that: $$ −4<x&...
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Does $t^{p-1}$ have an antiderivative in $\Bbb F_q(t)$?

The formal derivative on $\mathbb{F}_p[t]$ is not onto, since $t^{p-1}$ (for instance) has no antiderivative. Does it have one if we extend the formal derivative to $\mathbb{F}_p(t)$? By the quotient ...
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Is there a general method to operate the reduction of a rational expression to a sum : $\frac {1+2x}{1-3x} \rightarrow 1+5x+\frac{15x^3}{1-3x}$

Source : page $146$ https://archive.org/details/academicalgebraf00bowsrich/page/n5/mode/2up The question I am asking does not deal with partial fraction decomposition ( as far as I can tell). In ...
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Do holes affect the type of function (even, odd or neither)

We can know whether a function is even or odd by substituting using F(-x) but what if the function has a single hole like this f(x) = $\frac{x(x-2)}{x-2}$ is such a function considered odd or neither? ...
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Understanding the Single-scale equidistribution theorem for abelian polynomial sequences using example

I wish to understand a remark given in the book 'Higher order fourier analysis' by T. Tao. The remark is related to the following proposition: Proposition 1.1.17 (Single-scale equidistribution theorem ...
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Proving that a rational function is a section of an invertible sheaf

Let $S$ be a surface, $E$ be a curve on $S$, and $H$ be a hyperplane section of $S$. Let $a\in H^0(S,\mathcal O_S(H+(k-1)E))$, and $b\in H^0(S,\mathcal O_S(H+kE))$. Let $U$ be an open subset of $S$ on ...
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Factoring when multiplying rational functions

I'm curious why multiplying the numerators of two rational functions before factoring them results in an incorrect solution. Suppose we need to find the product of the following expression in lowest ...
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Embedding of a variety

Let $D$ be a divisor on a variety $X$, and assume that $h^0(X,\mathcal O_X(D))=n+1$. So let $s_0,\dots,s_n$ be a basis of $H^0(X,\mathcal O_X(D))$. I am trying to understand the following statement : &...
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Determining values of Coefficients

I have tried to figure this question out but I am not sure if i'm doing it correctly. If somebody could help explain it would be appreciated. Question Determine the values of m and n for $f(x)= mx^3+...
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Rational functions on an affine variety are completely determined by their values where they are defined

Say $V$ be an irreducible affine variety contained in $\mathbb{A}^n_k$, where k is algebraically closed Since, the coordinate ring $\Gamma(V)=\frac{k[X_1,X_2,...,X_n]}{I(V)}$ of $V$ is an integral ...
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Factoring by grouping: how to deal when the factors extracted are of the form $(x+k)$ and $ (x-k) $.

EDITED : the original post contained a mistake regarding the factorization of the numerator. The source of the exercice I'm trying to do is : Barton's College Practice Placement Test, Q. 17 https://...
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