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

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Integral of an exponential of rational function

I have an integral of the form $\int_{a}^{b} \text{exp}\left(\frac{\lambda}{\rho^2 m + \sigma^2_u}\right) \frac{1}{m^2}\text{exp}\left(-\frac{\lambda}{m}\right) dm$. Can this integral be found ...
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Why does the graph of $x^3/x^3$ not have a horizontal asymptote?

I am a graduate student studying math, and am actually teaching College Algebra right now. But every once in a while, I come upon something new in a subject that I have supposedly mastered. Why does ...
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Truncation error in Padè approximants

Suppose only the following data are known about a rational function $R(x)=P(x)/Q(x)$ (for $P,Q$ polynomials): (a) the degree of $P$ is $\leq m$ and the degree of $Q$ is $\leq n$; (b) the first $k$ ...
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Variation of the argument of a rational function along a circle

Crossposted here on MO. Let $f:\bar{\mathbb C}\to \bar{\mathbb C}$ be a rational function, and take a circle $C$ not crossing the zero- and polar-locus of $f$. The argument principle tells us the ...
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Help solve rational expression

I need help solving this rational expression. Divide $$\frac{4x^4 + 6x^3 + 3x - 1}{2x^2 + 1}$$ How do you solve this problem? Where do I start?
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Setting up word problem for finding length and width

Word Problem: The length of a rectangular sign is $3$ feet longer than the width. If the sign has space for $54$ square feet of advertising, find its length and width. I have not idea where to start. ...
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Rationals over an interval

Suppose $I$ is an interval $[a,b]$. It is noted that $a$ and $b$ are real integers. Divide the interval into $n$ parts with step size $h=(b-a)/n$. Clearly all the points $a$, $a+h$, ...
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Solving the polynominal: $s(t) = -16t^2 + 48t + 160$

The height of a ball is thrown directly upward from an initial height of $160$ ft with an initial velocity of $48$ ft per second is given by the function: $s(t) = -16t^2 + 48t + 160$, where $s(t)$ ...
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Solving for $x$: $\;\frac8{x-2}-\frac{13}2=\frac3{2x-4}.$

I have solved this sort of problem before, but frustratingly I have forgotten to. The problem is: Solve for $x$: $$\dfrac8{x-2}-\dfrac{13}2=\dfrac3{2x-4}.$$ So hence the title, How should I ...
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The value of $a/b+b/a-ab$ under the condition $ab=a-b$ (AMC 10 problem)

I am confused about the problem below. I answered correctly but used substitution to solve instead of changing everything to the same denominator as shown in the solution. The problem: Two non-zero ...
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Derivative of rational function help.

consider $$f(x)=\frac{1}{2x-4}$$ The derivative should be $\displaystyle -\frac{1}{2(2x-4)^2}$ However I get $\displaystyle -\frac{2}{(2x-4)^2}$ my workflow: $$\begin{array}{} f'(x)&= ...
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33 views

Supremum of a rational function

Let $f(z)$ be a rational function in the complex plane such that $f$ does not have any poles in $\{z:\Im z\ge0\}$. Prove that $\sup\{|f(z)|:\Im z\ge0\}=\sup\{|f(z)|:\Im z=0\}$. Let $\Gamma_r$ be a ...
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108 views

There cannot exist a rational function $f: \mathbb{R} \to \mathbb{R}$ injective, not surjective

I was looking for a rational function $f: \mathbb{R} \to \mathbb{R}$ that looks like $\arctan$, in that it is injective not surjective well-defined on all $x\in \mathbb{R}$ (no vertical ...
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35 views

Where rational functions are undefined

I have another question/comment I'd like a fresh pair of eyes on The question is "A rational function can have infinitely many x-values at which it is not continuous" I know since Q(x) in the ...
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19 views

min max of a rational function

The variable is a vector $x \in \mathbb{R}^n \times\mathbb{R}^m_+ \times \mathcal{E}$ where $\mathcal{E}$ is an ellipsoid of dimension $e$. I would like to find the min and max of the following ...
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Rational function on a noetherian scheme

Let $X$ be a noetherian scheme and $f$ a rational function on $X$, so by definition the domain of $X$ includes all associated points of $X$. I think the following is true: $f$ is regular on $X$ if and ...
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347 views

The solutions for the equation $\frac{a}{c-b+1}+\frac{b}{a-c+1}+\frac{c}{b-a+1}=0.$

How can I find the solution for the following equation in $a,b \mbox{ and } c$. $$\frac{a}{c-b+1}+\frac{b}{a-c+1}+\frac{c}{b-a+1}=0.$$ Also $b-c \neq 1$, $c-a \neq 1$ and $a-b \neq 1$. Thanks!
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Branch of mathematics that studies groups / rings or rational functions

I'm not really a mathematician, and looking for some literature which could potentially help me in research. Im interested in algebra of rational functions (ratios of polynomials) of finite order. ...
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17 views

Termwise differentiation of sequence of rational functions when the uniform limit is analytic

Given a sequence $\{f_n(x)\}$ of rational functions which converges uniformly to the analytic function $\{g(x)\}$ on $[a, b]$ ($f_n(x)$ are defined on $[a, b]$ and hence are analytic), what can we say ...
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42 views

Inverse functions problem

There are two functions $f\colon\mathbb Q \to \mathbb Q \setminus \{-1\}$ and $g\colon\mathbb Q \to \mathbb Q \setminus \{1\}$. $$g(x) = \frac{f(x)}{f(x)+1}.$$ Prove that if there is a inverse ...
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Values of $m$ for which $y^2 + 2xy + 2x -my -3$ can be factorised

For what values of $m$, will the expression $y^2 + 2xy + 2x -my -3$ be capable of resolution into two rational linear factor? This is how I did it: $$y^2 + 2xy + 2x -my -3 = y^2+(2x-m)y+2x-3$$ ...
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rational parameterization of quartic

With the curve $x^4 - 6x^2 - y^2 + 1 = 0$ in the range of $x$ inside of $(-1,1)$, I can only identify two rational points $(0,1)$ and $(0,-1)$. Is it possible to determine if there are others?
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Inverse LaPlace Transform of the square root of Rational, Monic 1st Degree Polynomials

I tried to find this in Churchill's Operational Mathematics which has a good variety of transform pairs, but no matches for what appears a simple expression. Does anyone have a solution for the ...
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33 views

Rational parametrization of points on sphere of irrational radius

I am trying to figure out if a closed form parametrization can exist for finding all of the rational points on a sphere of radius $\sqrt{2}$. ie, x=f(u,v), y=g(u,v), and z=h(u,v) where f,g,and h are ...
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integrality of certain rational numbers

Let $P,Q\in\mathbb Q[X]$ be relatively prime polynomials ($X$ being an indeterminate). Assume that $Q(0)=1$ and that $P/Q$ is in $\mathbb Z[[X]]$. Does this imply that $P$ and $Q$ are in $\mathbb ...
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recursive relation in rational expression form

I am looking for a closed form expression for the variables $n_i$ that are stationary solutions of the recursive relation: $ n_i(t+1)=n_i(t)\sum_mf_{i,m}\frac{K_m}{\sum_j f_{j,m}n_j(t)}$ i.e. the ...
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expand a rational function in a power series

$$\frac{4-x}{(2-x)(1-x)^2}$$ Expand in ascending powers of x, stating when the expansion is valid; also write down the coefficient of $x^n $
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Find the sum of the maximum and minimum

For a real number $x$ find the sum of the maximum and minimum. $$y=\frac{x^2-2x-3}{2x^2+2x+1}$$ This is a sample question for a math competition. It seems like calculus would be used to solve this, ...
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Geometry aspect of a extreme value problem

In a plain with orthogonal coordinate $XOY$, set point $A(a,a)$, and $P$ is a point in function $y=\frac{1}{x}$,where $x>0$. If the distance between $P$ and $A$ is $2\sqrt{2}$.Find all $a$ ...
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How prove this equation roots $\frac{1}{x}=\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+2\sqrt{\frac{1}{ab}+\frac{1}{ac}+\frac{1}{bc}}$

Question: Let $a,b,c>0$ are give numbers and $x>0$, such that $$ \sqrt{\dfrac{a+b+c}{x}}=\sqrt{\dfrac{b+c+x}{a}}+\sqrt{\dfrac{c+a+x}{b}}+\sqrt{\dfrac{a+b+x}{c}} $$ show that $$ ...
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Solve this simple polynomial

$$\text{Problem: }{x^3-x^2-x-2 \over x^2 + x - 6}$$ My textbook was able to come up with $(x-2)(x^2+x+1)$ $$\text{Textbook: }{(x-2)(x^2+x+1) \over (x-2)(x+3)}$$ I've tried grouping and using the ...
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Conditions under which asymptotes can be intersected/crossed?

What are the conditions under which some horizontal asymptotes can be crossed, and why is this acceptable? I'm speaking specifically to rational functions, such as in the case of the function $$ f(x) ...
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Notational issue

Let $K = F(t)$. If $r \in K: (\nexists c \in F: r(t) = c \forall t)$ is a rational function and $L = F(r(t))$, then what form does $f \in L$ have? Is it a rational function where the coefficients are ...
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150 views

How to evaluate $\int_0^1\frac{1+x^4}{1+x^6}\,dx$

$$\int_0^1\frac{1+x^4}{1+x^6}\,dx$$ Can anyone help me solve the question? I am struggling with this.
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How do i define 'complex rational function'?

http://en.wikipedia.org/wiki/Rational_function I don't get the definition in wikipedia. It would be great to define "complex rational function" with the domain $\overline{\mathbb{C}}$, namely ...
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I am not sure if the answer for the dividing rational expressions problem (attached) should be simplified

$$ \frac{x}{x+2} \div \frac{1}{x^2 - 4} $$ Original image: http://i.stack.imgur.com/EEqd4.jpg -I am not sure if the answer for the problem (which is attached) would be $\frac{x^3-4x}{x+2}$ or if it ...
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Looking for methods/results for explicitly bounding iterations of rational functions

This is a cross-post of http://mathoverflow.net/questions/155775/looking-for-methods-results-for-explicitly-bounding-iterations-of-rational-funct But I received no answer there to the actual ...
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45 views

What is the common denominator

Help me find the common denominator please! Thank you Here it is : $$\dfrac{5}{3x+2}-\dfrac{3}{3x-2}=\dfrac{7}{6x-4}+\dfrac{x+4}{18x^2-8}. 1$$
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PSD Rational function to power series

I have a Power Spectral Density given as a Rational function that is: \begin{equation} \phi(e^{i\omega}) = 1/(1-1.7464e^{-i\omega}+1.2602e^{-2i\omega}-0.4366e^{-3i\omega}+0.625e^{-4i\omega})^2 ...
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How does $\frac{x^2}{ x^2 + 1}$ simplify to $1-\frac{1}{1+x^2}$

How does $$\frac{x^2}{ x^2 + 1}\quad\text{ simplify to }\quad 1-\frac{1}{1+x^2}\;?$$ Can someone explain the steps of how to get to that alternate form?
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Laplace transform to describe a bounded function

It is easy to show that if a real function $f:\mathbb{R}\rightarrow\mathbb{R}$ is contained in a strip $[a,b]$, that is if $\forall_{x}\, a\le f(x)\le b$, then its Laplace transform is bouned by ...
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$\lfloor n^{1/2}\rfloor+\cdots+\lfloor n^{1/n}\rfloor=\lfloor \log_2n\rfloor +\cdots+\lfloor \log_nn \rfloor$

Prove that: $\lfloor n^{1/2}\rfloor+\cdots+\lfloor n^{1/n}\rfloor=\lfloor \log_2n\rfloor +\cdots+\lfloor \log_nn \rfloor$, for $n > 1,\, n\in \mathbb{N}$ For example. For $n=2$, we have $\lfloor ...
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How can I simplify this rational function for graphing by translation?

I've actually struggled with how to properly simplify this rational function, and I'm hoping someone can point me in the right direction. It's part of my precalculus class/section on graphing rational ...
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44 views

What does the notation $O(x^n)$ mean?

I am reading a book about Padé Approximations, and I am trying to understand the following line: We denote the $[L/M]$ Padé approximant to $A(x)$ by $A(x) - P_L(x)/Q_M(x) = O(x^{L+M+1})$ where ...
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how to find the integral of a rational logarithmic function

I can't seem to figure this one out, it is: $$\int\frac{\ln(x)}xdx $$ I substituted $u$ for $\ln(x)$, so $u = \ln(x)$ and $du = \frac1x dx$ then to find $x$ in terms of $u$: $e^u = x$ so ...
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Prove that there is no rational function in $F(x)$ such that its square is $x$

I came across this homework problem and honestly have no idea where to begin. Prove that there is no rational function in $F(x)$ such that its square is $x$. First of all, I am confused about ...
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51 views

When the system of equations below had a solution?

The system of equations is $$\begin{cases} \frac{c_1}{1-x_1}+\frac{c_2}{1-x_2}+\frac{c_3}{1-x_3}=0\\ \frac{c_1}{k-x_1}+\frac{c_2}{k-x_2}+\frac{c_3}{k-x_3}=0\\ ...
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118 views

I wonder whether the system of equations and inequations below have a solution.

I wonder whether the system of equations and inequations below have a solution. If there are solutions, what are they? A numerical solution is also desired. $$\begin{cases} ...
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72 views

Connection between rational functions and matrices

$$ y = \frac{ax+b}{cx+d} \Longleftrightarrow x = \frac{dy-b}{-cy+a}, ad-bc \ne 0 $$ on the other hand $$ \left( \begin{array}{cc} a & b \\ c & d \\ \end{array} \right)^{-1} = ...
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101 views

$f$ differentiable, $f(x)$ rational if $x$ rational; $f(x)$ irrational if $x$ irrational. Is $f$ a linear function?

Let $f$ be an everywhere differentiable function whose domain consists of all real numbers. Assume that $f(x)$ is rational for rational $x$ and irrational for irrational $x$. Can we conclude that $f$ ...