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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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A problem about the field of rational functions over finite field

Let $p$ be a prime number, and let $F_{p}$ be the finite field with $p$ elements. Let $F=F_{p}(t)$ be the field of rational functions over $F_{p}$ . Consider all subfields of $F$ such that $F/C$ is a ...
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The integrals $\int_0^\infty \frac{\mathrm{d}x}{(a+bx^c)^d(e+fx^g)^h}$ and $\int_0^\infty \frac{x^e}{(a+bx^c)^d}\mathrm{d}x$

I am interested in improper integrals of rational functions. For example, I have found that $$\large{\int_0^\infty \frac{\mathrm{d}x}{(a+bx^c)^d}=\frac{\Gamma(\frac1c+1)\Gamma(d-\frac1c)}{\Gamma(d)a^{...
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Integral involving upper incomplete gamma function, exponential and rational funcitons

Related to these questions here and here, I found a different form of the integrals, which result in $$f(x)=\frac{(-1)^n \,2^n}{\pi\,\lambda^{n+1}} \,\mathrm{e}^{-\lambda\,x}\int_{-\infty}^{\infty}\...
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Residue theorem /Integral

I want to calculate the following integral using residue theorem: $$\int_{-\infty}^{\infty} \frac{x^2}{x^4+1} $$ When I conisder the singularities, I get: $ \text{Rez}(f, z_k)=\frac{1}{4z_k}$ with $...
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Algorithm for determining all the zeros of a complex rational function without initial guesses

Given a rational function R(x) = P(x)/Q(x), where P and Q are polynomials which can have complex coefficients, is there an algorithm which allows us to determine the zeros of R without an initial ...
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34 views

LCD of 2x+1, x^2 and x

I am given the following sum: $$\frac{x}{2x+1} + \frac{3}{x^2} + \frac{1}{x}$$ In order to add these fractions, I must find a common denominator. I have been taught to factor each denominator and ...
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Complete Elliptic Integral of the First Kind Identity

Is there an identity for $\frac{K'(k)}{K(k)}=?$ where $K(k)=\int_0^{\frac{\pi}{2}}\frac{1}{\sqrt{1-k^2\sin^2(x)}}dx=\int_0^1\frac{1}{\sqrt{(1-t^2)(1-k^2t^2)}}dt$ is the Complete Elliptic Integral of ...
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Find a range of $\frac{4x - 1}{2x + 3}$

Given a function $\frac{4x - 1}{2x + 3}$ I know its Domain is $\{x \in \mathbb{R} | x \ne \frac{-3}{2}\}$ But how can I find its range? Any hint is appreciated.
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Solve the following system of equations - (2).

(a) Solve the following system of equations: $$\begin{cases} x(x + 1) + \dfrac{1}{y}\left(\dfrac{1}{y} + 1\right) = 4\\ \dfrac1{y^2}(x+1) + x^2\left(\dfrac{1}{y} + 1\right) = 4 \end{cases}$$ (b) ...
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Solving a Symbolic system of equations SYMPY

I have the following equations I want to symbolically solve for a solution. $$\lambda = \frac{C_{v,u} + \beta \Sigma_0}{\Sigma_0 \beta^2 + \sigma^2_u}$$ $$\mu = \frac{\sigma^2_u p_o - \alpha \beta \...
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Evaluating indefinite integrals of the form $\int \frac{x^2 \,dx}{a x^5 + b}$

Evaluate the indefinite integral $$\int \frac{x^2 \,dx}{a x^5 + b},$$ for real parameters $a, b \neq 0$. No apparent substitutions simplify the expression (if the exponent of $x$ were an integral ...
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Definition of a valuation of a rational fraction

Let $F$ be a field. Then the field of rational fractions over $F$ in indeterminates $x_1,...,x_n$, denoted by $F(x_1,...,x_n)$, is the field of fractions of the polynomial ring $F[x_1,...,x_n]$. P.A....
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Use algebra to prove all proper rational function can be written in partial fraction decomposition

I'm learning the general tactics to integrate all rational functions and here's a fact that is written in the notes. It can be shown using algebra that every proper rational function $f$ can be ...
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Can $p(x)\in \mathbb{F}_{3}(x)$ with $p(x)=\frac{x²+x+1}{x+1}$ be expressed as a polynomial? Is it not possible for any of the given fields?

Can $p(x)\in \mathbb{F}_{3}(x)$ with $p(x)=\frac{x²+x+1}{x+1}$ be expressed as a polynomial? I tried it with different steps, like with polynomial long division: $ (x^2 +x +1):(x+1)=x + \frac{1}{x+1}...
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Isolating a variable under square root

Given this equation: $T=\sqrt{(ugx)^2+(T_0)^2}$ You're asked to isolate $x$. My process was: $T=ugx + T_0$ (the square root cancelled the exponents) $T-T_0=ugx$ $x=\frac{T-T_0}{ug}$ But that was ...
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Computing a Contour Integral in the Complex Plane

I am trying to compute$\int \frac{z^5 + z^3 + 19}{z^3 - 3z^2 +3z - 1} dz$ where $ \gamma$ is the circle of radius 7 centered at the origin. I have that $\gamma(t)= 7e^{it}$ so that $\gamma'(t)= 7ie^{...
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Is $ \frac{x+y}{x^3 y^3 - x^3 y - x y^3 + 2 xy + 1} $ a formal group law?

Is $ \frac{x+y}{x^3 y^3 - x^3 y - x y^3 + 2 xy + 1} $ a formal group law on the interval $[-1,1]$ ? It is a lot of work to check on associativity imo. Maybe there is a shortcut around checking ...
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Is a hole at a vertical asymptote recognized?

If I have a hole in a rational function, but the hole is located at a vertical asymptote, is the hole still recognized? For example in the equation (x+3)(x+1)/(x+1)(x+1) I will have a hole at -1, but ...
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Bound on the degree of a polynomial solution to a parametrized equation

Let $K$ be a field, $F = K(T)$ a rational function field over $K$. Let $G \in F[C, X_1, \ldots, X_n]$ be a polynomial with coefficients in $F$. We can consider the polynomial $G(c, X_1, \ldots, X_n)$ ...
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Where is a mistake equating $x$ derived from both equations of a system?

I know how to solve this, but why is the below reasoning wrong and leads to a mistake (I don't see any mistake!) Step 1: From first equation $x=\dfrac{8}{y}$ , and $y$ is not zero Step 2: From ...
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how to solve the equation $\dfrac{a}{x-a}+\dfrac{a}{y-a}$ is [closed]

Please provide the steps to solve the equation, the answer to this equation is zero I am not sure how it is derived, Kindly help if $x+y=29$ then the value of $\dfrac{a}{x-a}+\dfrac{a}{y-a}$ is
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How to construct a function with these hypotheses?

I want to construct a function $f:[0,1]×[0,1]\rightarrow [0,1]$ such that $f(0,t)=t$ $f(1,t)=2t-1$ $ \forall$ $ t\geq \frac{1}{2}$ $f(s,t)=0$ $ \forall $ $0 \leq t \leq \frac{s}{2}$ $f(s,\frac{s}{2}...
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Why does this rational function have a false slant/oblique asymptote?

Let's examine the following rational function: $f(x) = \frac{3x^3+2}{x^2-x-7}$. Considering that the degree of the polynomial in the numerator is 1 greater than that of the denominator, it can be ...
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Is the sum of series $\sum_{n=0}^{\infty} \lfloor n\pi \rfloor x^n$ a rational function?

I was reading this paper by L.J. Mordell (original paper by Morris Newman). I am trying to apply first theorem from the paper with $f(x)=x$ and $k=\pi$. So the series $\displaystyle \sum_{n=0}^{\infty}...
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Find rational representation of a power series

I need to find a rational function $\frac{p(x)}{q(x)}$, where $p(x)$ and $q(x)$ are polynomials, which value is the same as $\sum_{n = 0}^\infty (2n+1)z^{2n}$ on its convergence domain. I found $\rho=...
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When does a power series converge to a rational function?

Are there any results to determine whether the given power series of real variable converges to a rational function? I mean just analyzing the coefficients of the series. One way is to find the sum ...
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How to handle an identical zero pole when expanding to a Laurent Series?

I have a function given as $f(z)=\frac{2(z-3)}{z^2-8z+15},$ which is clearly the same as $f(z)=\frac{2(z-3)}{(z-3)(z-5)}$ when $z\neq3$. Typically I would break these into two separate fractions using ...
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What can be a function where $x \neq 2, y \neq 1$ for all $x,y$?

I am trying to find a possible function to this graph below. I am really bad with graphs so if anyone can further elaborate the ways to help identify graphs, I will deeply appreciate it.
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Easy algebra manipulation, example $x^3/(1+x^2)$

I guess this is really basic but have trouble following some algebraic manipulations on fractions. For example with two cases $$\frac{x^3}{1+x^2}$$ is supposed to be: $$x -\frac{x}{1+x^2}$$ and ...
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Create rational function

I have to create a rational function with the characteristics: 3 real zeros(1 of them of multiplicity 2). y-intercept 1. vertical asymptote at $x=-2$ and $x=3$. oblique asymptote $y=2x+1$. I've tried ...
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What is meant by “rational operation”?

So I am attending a course in complex analysis and came across the following statement pretty early in the book: Let $R(a,b,c,\dots)$ stand for any rational operation applied to the complex numbers $...
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Closed subsets of rational functions when the coefficients are endowed an Euclidean topology

Suppose $\text{rat}(n)$ denotes the collection of proper rational functions such that: if $f \in \text{rat}(n)$, then \begin{align*} f(z) = \frac{ b_{n-1} z^{n-1} + \dots + b_0 }{z^n + a_{n-1}z^{n-1}...
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rational function with absolute values

How can i write a rational function with absolute values as a piecewise function, for example $$f(x)= \frac{|x+1|}{|x+2|}$$
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Simplify $\frac{x^3-x}{x^2+xy+x+y}$

$$\frac{x^3-x}{x^2+xy+x+y}$$ What I did: $$\frac{x}{xy+x+y}$$ through simplifying the $x$'s. But it's not right. What did I do wrong?
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Simplifying $\frac{ (2x+1) (x-3) }{ 2x^3 (3-x) }$

In simplifying $$\frac{2x^2-5x-3}{6x^3-2x^4}$$ I got this far $$\frac{ (2x+1) (x-3) }{ 2x^3 (3-x) }$$ but there aren't same brackets to cancel out.
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Integral of $\sqrt{a(b(x+c)^2+1)}/((x-1)x^{3/2})$

I am trying to solve $$\int_{x_0}^\infty\frac{\sqrt{a(b(x+c)^2+1)}}{(x-1)x^{3/2}}dx$$ where, $a,b,c,x_0,x\in\Bbb R$ and $b,c,x>0$ and $x_0>1$. I tried a $\sinh$ substitution without too much ...
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Inverse of power-2 rational function

I have a function $f(a,b) = \frac{ab}{(\frac{a+b}{2})^2}$, and (to me) it has some cool properties (e.g $f(a,b) = f(b,a)$, $f(x,0) = 0$, $f(x, x) = 1$, $0 \leq f \leq 1$, etc.). Now I wanted to know ...
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$\sqrt{x} \notin F(x)$

I wanted to prove that if $F$ is a field and we consider the fraction field $F(x)$, then $\sqrt{x} \notin F(x)$. I said that if $\sqrt{x} \in F(x)$ then there are $f,g \in F[x]$ such that $\bigl(\...
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Why do the “rules” of horizontal asymptotes of rational functions work?

Note: My current understanding is only at a college algebra level From what I've seen online, in layman terms, the rules for horizontal asymptotes are as follows: Rule 1) If the degree of the ...
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Asymptotes Parallel to the axes.

I need proof of this theorem. Theorem: A straight line $y=c$ is an asymptote of a curve $f(x,y)=0$ iff $y - c$ is a factor of the co-efficient of the highest power of $x$ in $f (x,y)$.
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Primitive of $\int R(\cos x, \sin x) dx$

Suppose we are seeking the primitive $$\int R(\cos x, \sin x) dx$$ where $R(u,v)=\frac {P(u,v)}{ Q(u,v)}$ is a two-variable rational function. Show that (a) if $R(−u,v)= R(u,v)$,then $R(u,v)$ ...
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Notation of a rational function

I have a problem with the notation. rational function of n variables If I go by this notation, the square root term can be taken as y and t as x. For the numerator and denominator to be ...
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Please help: Zorich Mathematical analysis I exercise 5.7.6

The problem is: The integral $\int(a+bt)^{p}t^{q}\text{d}t$ can be expressed in terms of elementary functions if one of the numbers $p$, $q$ or $p+q$ is an integer. The first two cases (either $p$ or ...
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Field of fractions and rational functions

When I have a field $k$ and take the ring of polynomials in the variables $x1, x2, ..., xn$, and subsequently take the quotient field of these polynomials, I was asking myself, is this in 1:1 ...
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Simplification of rational expression gone wrong(high school math)

Currently doing high school math and can't get this one right. I think I'm using entirely incorrect practices and am trying to pinpoint what it is. Could someone tell me where exactly I went wrong? ...
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solutions to $\int_{-\infty}^\infty \frac{1}{x^n+1}dx$ for even $n$

I was playing around with glasser's master theorem and integrals of the form $$\int_{-\infty}^\infty \frac{1}{x^n+1}dx$$ I observed that for positive, even values of n, the solution to the integral ...
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Find the value of $Q(x)$ at $x= -1$, knowing some of its properties.

Given polynomial $Q$ with real coefficients such that $Q(1)=1$ and $$ \frac{Q(2x)}{Q(x+1)}= 8 -\frac{56}{x+7}, \quad \forall x \ne -7 \text{ and } Q(x+1)\neq 0\,. $$ Find $Q(-1)$.
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Compute the limit $\lim_{x\to1}{\frac{2x^3-2x^2+x-1}{x^3-x^2+3x-3}}$

$$\lim_{x\to 1}{\frac{2x^3-2x^2+x-1}{x^3-x^2+3x-3}}$$ What I tried: divided by $x^3$ $$\lim_{x\to1}{\frac{2-\frac{2}{x}+\frac{1}{x^2}-\frac{1}{x^3}}{1-\frac{1}{x}+\frac{3}{x^2}-\frac{3}{x^3}}}$$ ...
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51 views

Fourth point of intersection of two conics

Five points in general position define a unique conic section. Let $Q_1$ be a conic through points $A,B,C,E_1,F_1$ and likewise $Q_2$ through $A,B,C,E_2,F_2$. Two conics (over an algebraically closed ...