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Questions tagged [rational-functions]

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

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With certain conditions, what are the properties of function f? [closed]

The function $f(x)$ is defined on the set of real numbers in such a way that we have : $f(x)f(x+1) - f(f(x+1)) = f(2x)$ a) What general properties should the function $f(x)$ have? b) Can the function $...
زهرا میرمعصومی's user avatar
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Solving $f(x) = f(\frac{a + b x}{c + d x}) = f(\frac{a' + b' x}{c' + d' x})$?

How to solve the equation $$f(x) = f(\frac{a x + b}{c x + d}) = f(\frac{a'x + b'}{c'x + d'})$$ For given real $a,a',b,b',c,c',d,d'$ ? Maybe this system of equations is a bit overdetermined in its ...
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How to integrate division of a multivariate/multivariable polynomial with respect to its variables?

as the question suggests, I have a multivariate/multivariable polynomial and I have them in a division. In example, the first multivariate polynomial is in the dividend and the second multivariate ...
aaparker's user avatar
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Cohomology of the sheaf of invertible rational functions

Let $X$ be an irreducible algebraic variety. Recall that the group of Cartier divisors is defined to be $H^0(X,\mathcal M_X^\times/\mathcal O_X^\times)$, where $\mathcal M_X^\times$ denotes the sheaf ...
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What is the name of this group-like structure?

I noticed this group-like structure popping up all over place. Below is a realization of the structure in first order rational functions. The structure has only six elements (Edit: it's tempting to ...
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Domain of injectivity of a rational map $\frac{P(z)}{Q(z)}$ where $P(z)$ and $Q(z)$ are polynomials.

Question: Let $P(z)$ and $Q(z)$ be two polynomials with degree $m$ and $n$ respectively with $m>n$. Given that both are injective in the set $A=\{z\in\mathbb{C}: \beta \leq Arg(z) \leq \frac{\pi}{2}...
Factorial_zero's user avatar
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2 answers
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Horizontal asymptote of a Rational Function [duplicate]

I didn't quite understand how the horizontal asymptote of the function $r(x) = \frac{3x^2 - 2x - 1}{2x^2 + 3x - 2}$ can be $\frac{3}{2}$, since when you use desmos, for example, to graph it, part of ...
Guilherme Cintra's user avatar
20 votes
1 answer
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Do rational functions eventually have monotonic derivatives?

Given a rational function $R(x)=P(x)/Q(x)$ with real coefficients, is it true that there exists an $M>0$ such that, for every $k\geq 0$, the restrictions $R^{(k)}|_{(-\infty,-M]}$ and $R^{(k)}|_{[M,...
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rational complex minmax

As the title says, I wanted to understand the following part of a proof in Guttel (Corollary 2, page 6). $f$ is an analytic function and $r_m \in \mathcal{P}_{m-1}/q_{m-1}$ is a rational function with ...
jacopoburelli's user avatar
3 votes
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Is there a way to show that $5a(a+1)\over 3a+4$ $\notin \mathbb{N}$ when $a\in\mathbb N\setminus\{2\}$? [duplicate]

Is there a way to show that $5a(a+1)\over 3a+4$ $\notin \mathbb{N}$ for $a\in \mathbb{N}$ (except when $a=2$)? The expression has a few different forms, but I don't see how to show this. Any hints are ...
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Is it possible to integrate $\int \frac{x^2 - 2x + 3}{x^4 - x^3 + x^2 - x + 1} \ dx$ using methods typical of undergraduate calculus?

I was doing a calculus problem and got to this final integral: $$\int\limits_{0}^{1} \frac{x^2 - 2x + 3}{x^4 - x^3 + x^2 - x + 1} \ dx.$$ Calculators are supposed to be used for these problems, so I ...
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Galois Theory and the Rational Functions Field.

Problem. Let $F$ be a field with $\operatorname{char}(F)=3$. Define the following field $$E =\{f(t)\in F(t): f(t)=f(1-t)\}$$ which is a subfield of the rational functions field $F(t)$. Prove that $F(t)...
user108580's user avatar
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Taylor Series of $\frac{2p + 2p^2}{2+2p+p^2}$

I am trying to expand the following: $$ \frac{2p + 2p^2}{2+2p+p^2}. $$ Using the Taylor series at $a=0$, I get: $p - p^3/2 + p^4/2 - p^5/4 + \dots$. But in the book, it is also equated to $p + p^3 + ...
Naitik Mundra's user avatar
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How can I demonstrate the limit of a probability function equals 1

How can I demonstrate that the limit of $$\lim_{n \to \infty} \frac{{\sum_{i=0}^{10}[(-1)^i}\cdot\frac{10!}{i!\cdot(10-i)!}\cdot(10-i)^n]}{10^n}=1$$ This function came as a result when I was designing ...
Nina's user avatar
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3 answers
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A question about rational functions in complex analysis

In Ahlfors's complex analysis $$ R(z)=\frac{P(z)}{Q(z)} $$ given as the quotient of two polynomials. We assume, and this is essential, that $P(z)$ and $Q(z)$ have no common factors and hence no ...
Mathematics enjoyer's user avatar
8 votes
5 answers
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Evaluating a rational function integral in a quick way

In an recent test I was asked to evaluate the integral $$ \int_0^1 \frac{\sqrt[3]{x^2(1-x)}}{(1+x)^3} \text{d}x$$ in 8 minutes, but I didn't have a clue what to do with it. After the test, I tried the ...
Cyankite's user avatar
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Find constants $p$, $q$, and $r$ for $\frac{16x+1}{px+1} > x+4$ where solution set is $x < q$ or $r < x < 3$ [closed]

Find constants $p$, $q$, and $r$ for $$ \frac{16x+1}{px+1} > x+4 $$ where solution set is $x < q$ or $r < x < 3$. Attempted to rearrange for quadratic, but resulted in range of values ...
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Resources for learning about graphs of rational functions?

I'm trying to learn how to create rational functions whose graphs have particular qualities. For example, I've learned how to create functions with linear asymptotes or non-linear asymptotes, but ...
user113339's user avatar
2 votes
2 answers
245 views

How can i do the following partial decomposition?

I need to prove that: $$ \frac{1}{(x-a)(x-b)} = \frac{1}{(b-a)(x-b)}- \frac{1}{(b-a)(x-a)}, $$ and I must note that I need to go from the left expression to the right (because of the exercise). So, I ...
Miguel Simões's user avatar
3 votes
1 answer
55 views

Prove that a function doesn't have a horizontal asymptote

Suppose that $$ f'(x) = \frac{x^2+8x}{x^2+8} $$ with a horizontal asymptote $y=1$. Prove that $f(x)$ doesn't have a horizontal asymptote. One of my classmates suggested that if the function $f(x)$ ...
yahlimem's user avatar
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Integrality of a quotient

Consider two positive integers $m$ and $n$ with $m>n$. I would like to prove that the quotient $$\prod_{k=0}^{n-1}\dfrac{X^{2^m-2^k}-1}{X^{2^n-2^k}-1}$$ is a polynomial in fact. What I did is to ...
joaopa's user avatar
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Characterizing rational functions satisfying some properties

We have that $\nu(x)$ for $x\in \mathbb{R}$ is a rational function which is characterized by ($\nu(x)=\frac{p(x)}{q(x)}$): $\nu(x) \geq 0$; $0 \leq q(x)-p(x)$; $q(x)$ has non-real roots. Moreover, $\...
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Cauchy principal value for a Gaussian integral of a rational function

I wish to calculate integrals of the following form, all Gaussian integrals of a rational function, on the entire real domain: $$ I_{k,n} = \int_{-\infty}^{\infty} dx \frac{e^{-x^2/2}}{\sqrt{2\pi}} \...
Uri Cohen's user avatar
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Triple Schwinger integral

I'm working on a perturbative QFT problem which requires three Schwinger parametrizations; that is, $$ \frac{1}{A^n}=\frac{1}{\Gamma(n)}\int_0^\infty dz\ e^{-A z}z^{n-1}. $$ In the end, after taking ...
y9QQ's user avatar
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Understanding a Map from rational points to the space of k-rational maps

I am currently working through this paper by Denef, and I have run into a little bit of a snafu with Lemma 3.1. The setup is as follows: let $K$ be any field of characteristic zero, and let $E_0$ be ...
peabody's user avatar
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How did Artin discover the function $f(x)=\frac{(x^2-x+1)^3}{x^2(x-1)^2}$ with the properties $f(x)=f(1-x)=f(\frac{1}{x})$?

In Artin's "Galois Theory" P38, he said the function $$f(x) = \frac{(x^2 - x + 1)^3}{x^2(x-1)^2}$$ satisfies the properties of $f(x)=f(1-x)=f(\frac{1}{x})$. Is the function given by some ...
Marine's user avatar
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1 answer
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Some properties about the equation $P(z)\phi(z)=0$

Let $\phi(z)=\sum_{m=1}^\infty (\phi_{m}z^m-\psi_{-m}z^{-m})$ where $\phi_m, \psi_{-m}\in \mathbb{C}$. If we can find one polynomial $P(z)\in \mathbb{C}[z]$ such that $P(z)\phi(z)=0$, how can I get ...
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When is the image of $\mathbb{N}$ under a bivariate rational function the same as under a univariate rational function?

Given two polynomials $P,Q$ and consider $D=\{\tfrac{P(n)}{Q(n)}:n\in\mathbb{N}\}$, can one find polynomials $P^\prime,Q^\prime$ such that $D-D=\{\tfrac{P(n)}{Q(n)}-\tfrac{P(m)}{Q(m)}:n,m\in\mathbb{N}\...
mathemagician99's user avatar
2 votes
1 answer
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Global minimization of finite Laurent series (rational function) in $\mathbb{R}^+$

Context: I am creating a special ballistics simulation with nonstandard physics, and I have reached a step where I need to find the global minimum of a rational function very fast to be able to run ...
The Innkeeper's user avatar
1 vote
1 answer
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Vertical Stretch transformation Question for rational function with variable in numerator and denominator. [closed]

I'm trying to help a student with following rational equation question: Describe the transformations of $$g(x) = \frac{-4x - 2}{7x +1}$$ from the graph of $$f(x) = \frac{1}{x}.$$ The given answers ...
DavidGslade86's user avatar
5 votes
2 answers
356 views

Why the name linear fractional map?

Fractional linear transformation is a map from extended complex plane to itself, defined by: \begin{equation} z\to \frac{az+b}{cz+d} \end{equation} with $ad-bc\ne0$. Wikipedia says that "a linear ...
Danilo Lombardo's user avatar
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1 answer
139 views

Proving that $f$ is rational if it is known that $f$ is analytic in $\mathbb{D}$ and satisfies $|f(z)| \to 1$ as $|z| \to 1$

This question was already asked here, I wouldn’t ask it again however I do not think that the question gets the point across that it’s trying to get across, and I can’t find anything else related to ...
no lemon no melon's user avatar
2 votes
2 answers
118 views

Are rational functions a vector space?

Let $\mathscr P_n[x]$ be the space of real polynomials in $x\in\mathbb R$ of degree at most $n\in\mathbb N$, and $$\mathscr P[x] := \lim_{n\to\infty} \mathscr P_n[x]$$ the set of all real polynomials ...
confusedandbemused's user avatar
1 vote
0 answers
54 views

Integral of an exponential of rational expression of the argument form $\dfrac{a}{x} + bx$

I wish to compute the integral of $$\displaystyle \int_0^{\infty} \dfrac{1}{x^2} \exp{\left(-\dfrac{a}{x}-bx\right)} dx$$ I found something similar in Gradshteyn and Ryzhik 7th Ed. p336 (Sec 3.324) in ...
Mohammed Salama Ibrahim's user avatar
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Upper bound of the derivative of $\frac{xf(x)}{(x^2+x)f(x) + g(x)}$

Let $f$ and $g$ be real nonzero polynomials with nonnegative coefficients such that $g(0) \neq 0$. Let $$ h(x) = \frac{xf(x)}{(x^2+x)f(x) + g(x)}. $$ In particular, it follows that $h$ is well defined ...
xen's user avatar
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1 vote
1 answer
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Upper bound of the derivative of a rational function $\frac{f}{f+g}$

Let $f$ and $g$ be real nonzero polynomials with nonnegative coefficients such that the degree of $g$ equals the degree of $f$ plus one, that is $\deg(f+g) = \deg(f) + 1$, $(f+g)(0) \neq 0$. Let $$ ...
xen's user avatar
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Need to find the horizontal asymptotes of a given expression.

I need to determine the horizontal asymptotes of the following expression:$$\frac{2x^{1/3}}{(1x^2+4)^{1/6}}$$ I’m not even sure where to begin. I understand I need to divide all terms by the highest ...
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1 answer
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Simplifying rational function in Sage

Let $g\geq 1$ be an integer and define three rational functions and their sum by \begin{align*} H(q,t) &= f_1(q,t) + f_2(q,t) + f_3(q,t) \\ &:= \frac{t^{8g-4}q^{2g-1}(1+tq)^{2g-1}(1+q^2t^3)^{...
Bailey's user avatar
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Why can we discard remainders when calculating slant asymptotes of rational functions?

I understand how and when to calculate slant asymptotes of rational functions with numerators with one degree higher than the denominator, but I am confused as to why we can disregard the remainder ...
Declan H's user avatar
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At which points of $X=V(x^2+y^2-1)$ is the rational function $\frac{1-y}{x}$ regular?

Let $X \subseteq \mathbb{A}^2$ be the circle of equation $x^2+y^2=1$. I have to compute the points of $X$ such that $f=\frac{1-y}{x}$ is regular. (We can suppose $\operatorname{char}(K)\neq 2$) I ...
Mario's user avatar
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2 votes
1 answer
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Does Weierstrass substitution work for different arguments in the trig functions?

Suppose we have a two dimensional rational function $R(x,y)$. Then we can use Weierstrass substitution for following integral: $$\int R(\sin(x), \cos(x)) \, dx .$$ But can we use Weierstrass ...
haifisch123's user avatar
1 vote
0 answers
111 views

Matching Requirements to a Rational Function

On a precalculus test I got recently, the following question stumped the entire class, and even weeks later, none of us could figure it out! The problem is to Write the equation of the following ...
Niero's user avatar
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Splitting a black box rational function

Suppose I can evaluate a black box rational function $f(\vec{x})$. For the purposes of the explanation, let me take an example I can actually write down \begin{equation} f(\vec{x})=\frac{x_1+x_2+x_3+...
Tanatofobico's user avatar
2 votes
0 answers
79 views

How to reliably find the global minimum of a rational function?

I am stuck at a algorithm problem. I was assigned the following problem: Given a finite Taylor expansion in some degree $n$: $$\text{TE} = \sum_{j=0}^{n}{c_j}{x^j}$$ with all real coefficients $c_j$ ...
keshav's user avatar
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$F(u,v)$ is a rational function. If $\pi$ is a period of $F(\cos x,\sin x)$, then $F(u,v)=F(-u,-v)$. "Introduction to Analysis" by Teiji Takagi.

I am reading "Introduction to Analysis" by Teiji Takagi. The author wrote the following proposition without a proof. Let $F(u,v)$ be a rational function. If $\pi$ is a period of $F(\cos x,\...
佐武五郎's user avatar
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Partial derivative of rational function is identically zero

Let $f:D\to\mathbb R$ be the following rational function $$ f(x_1,\ldots,x_n)=\frac{P(x_1,\ldots,x_n)}{Q(x_1,\ldots,x_n)}, $$ where here $P,Q$ are polynomials and $$ D=\{(x_1,\ldots,x_n):Q(x_1,\ldots,...
boaz's user avatar
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4 votes
0 answers
113 views

Approximating $\sqrt{x}$ by a rational function in the complex plane

Newman (1963) proved the following. Theorem 1. Let $d \in \mathbb{N}$. Define $$p(x) = \prod_{k=0}^{d-1} \left(x+\exp\left(\frac{-k}{\sqrt{d}}\right)\right)$$ and $$r(x) = \frac{\sqrt{x} \cdot (p(\...
Thomas's user avatar
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1 answer
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Zorich Exercise 5.7.3: Reduction of integrals of the form $\int R(x,\sqrt{ax^2+bx+c})\,dx$

Consider integrals of the form $$\int R(x,\sqrt{ax^2+bx+c})\,dx,\tag{1}$$ where $R$ is a rational function of $x$ and $\sqrt{ax^2+bx+c}$. Show that the integral $(1)$ can always be reduced to ...
Fergns Qian's user avatar
2 votes
0 answers
39 views

Is there a special name for symbolic ratios that share no common symbol in numerator and denominator

I am wondering if there is a name for the operation $f$ that separates a multivariate rational function into fractions, wherein neither numerator nor denominator share common symbols: given $y=\frac{x^...
smichr's user avatar
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Dependence on a variable involving homogeneous functions of the 1st degree

Suppose that $f(x,z)$ and $g(x,y)$ are homogeneous rational functions of the 1st degree, that is, $f(tx,tz)=tf(x,z)$ and $g(tx,ty)=tf(x,y)$ and consider the function $$ F(x,y,z)=\big[f(x,z)+ay\big]\...
boaz's user avatar
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