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

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Brilliant formulaes [on hold]

Hey Brilliant mathematician, i am very honored for having your time. I need general Formulas on breaking down a number to a different and being able to derive that number back, my requirements is to ...
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Rational function regression without poles in a interval, or polynomial regression with positivity constraint

I have some sets of experimental data for some functions $f_i$ from $I=[0,1]$ onto itself, which should satisfy the following physical constraints: $f_i(0)=1$ $f_i(x) \in I= [0,1] \; \forall x \in I ...
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Evaluate the integral $\int \frac{dx}{x^3 + 2x^2 + 2x}$ of a rational function

Evaluate $$\int \frac{dx}{x^3 + 2x^2 + 2x}.$$ I have no idea how to approach this. I know how to solve rational functions with numerator as highest degree polynomial using division and remainder. ...
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$\mathcal{Z}$-transform of differential equations $y(n+2)-3y(n+1)-10y(n)=(-2)^n$

Is defined function: $$y(n+2)-3y(n+1)-10y(n)=(-2)^n$$ with conditions: $$y(0)=0, y(1)=0 $$ And my solution is (Z-transform): $$\mathcal{Z}\{y(n+2)\}=z^2Y(z)-0z^2-2z=z^2Y(z)-2z$$ ...
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What is that function? Polynomial?

Is it a polynomial or rational polynomial or else? $y = \dfrac{a}{x^4} + \dfrac {b}{x^2} + c$ I need to fit a curve to a discrete data of that form, so I need to know what fitting to use.
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Solve the following integral: $ \int \frac{x^2}{x^2+x-2} dx $

Solve the integral: $ \int \frac{x^2}{x^2+x-2} dx $ I was hoping that writing it in the form $ \int 1 - \frac{x-2}{x^2+x-2} dx $ would help but I'm still not getting anywhere. In the example it was ...
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Partial fractions expansion problem $\frac{x^3-1}{4x^3-x}$

I want to calculate integral of the fraction, but first how to find the partial fraction expansion of $\frac{x^3-1}{4x^3-x}$. How to expand denominator? I am a bit lost here.
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29 views

What is the limit of a rational function as it approaches its vertical asymptote?

For example, take the function $f(x)=\frac{1}{(x-3)^2}$. What is the the limit as x approaches 3? (sorry, I don't know how to format this question) My teacher says that there is no limit at x=3, ...
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Is long division must in integrating improper rational functions

I came across an integration question, which I tried to solved through substitution, but my answer is wrong. I entered the same question in Wolfram Alpha engine and saw the "Step-by-Step" solution ...
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Where to find an algorithm for decomposing rational functions into elementary fractions?

Specifically I need to decompose $\frac1{(1-x)(1-x^n)^2}$ into $\frac{f(x)}{(1-x)^3}+\frac{g(x)}{1-x^n\vphantom{()^2}}+\frac{h(x)}{(1-x^n)^2}$ where $f(x)$, $g(x)$, $h(x)$ are polynomials. Surely ...
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Defining the rational function field in n variables.

Reading over an editing my dissertation "Elementary functions" and i am having trouble with my definition of a rational functions in n variables, this is what i have written but its missing one part: ...
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33 views

Rational Exponents

I'm just checking to see if have the correct answer because my teacher didn't give us an answer key and i like to know that I have done one question properly before doing the rest. Evaluate. ...
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80 views

Limit as $x$ tend to zero of: $x/[\ln (x^2+2x+4) - \ln(x+4)]$

Without making use of LHôpital's Rule solve: $$\lim_{x\to 0} {x\over \ln (x^2+2x+4) - \ln(x+4)}$$ $ x^2+2x+4=0$ has no real roots which seems to be the gist of the issue. I have attempted several ...
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30 views

How do I know if a fractional linear transformation exists?

I have a feeling I'm missing another obvious point about FLTs. How do I know if a specific fractional linear transformation exists? I think I can find specific transformations by using the ...
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26 views

Proving a property of Fractional Linear Transformations

I'm having some trouble showing that FLTs send circles and lines to circles or lines. I know that they are compositions of linear maps and inversions. Showing that the linear maps send circles to ...
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Is $\sqrt{-x^2-\frac{1}{x}}$ a rational function?

I have to construct a rational function with the range being $[-1,0)$, which is pretty much just $-1$. I came up with the solution $\sqrt{-x^2-\frac{1}{x}}$. It works for the range, but I'm not sure ...
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204 views

Integral of rational function with trigonometric functions

$$ \int \frac{dx}{(\sqrt{\cos x}+ \sqrt{\sin x})^4} $$ I saw this problem online and it looked like an interesting/difficult problem to try and tackle. My attempt so far is to use tangent half-angle ...
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67 views

Integral with logarithm - residue

Let $R(x)$ be rational function. It is any general method to calculate $\int_{0}^{\infty}R(x) \log(x)dx$ ? I can do it in special cases, but I am looking for a general method. What should be a minimal ...
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24 views

Is algebraic closure of rational functions field Puiseaux series?

Consider a field of rational functions over algebraicly closed field. Is its algebraic closure isomorphic to Puiseaux series over the field?
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37 views

Simplify a rational expression

Suppose I want to simplify this expression: $$\frac{bx-bc-dx+ad}{a-c}$$ More specifically, I want to minimize the number of operations. Counting each addition, subtraction, and multiplication, the ...
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Rational Functions

Part 1: You are planning a school field trip that costs $120$ dollars to rent the bus. A. How much will it cost per student if $10$ students go? $12$? $15$? B. Write an equation to represent the ...
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finding the free energy of a van der waals gas (integration)

I have the following integral, $\int{ \frac{-nrtV}{(v-nb)^{2}} dV}$ could anyone tell me how to do this?
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How do you calculate certain variables of two or more events that occur simultaneously compared to the same events happening subsequently.

Say you have two hoses, A and B, that fill up a pool of equal size at different rates. Hose A fills up a pool in 10 mins, hose B in 20 mins. Thus A = 1p/10m, B = 1p/20m. Lets say that Hose A filling ...
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Partial-Fraction Decomposition

So I was doing some integrals and ran across this one: $$\int{\frac{3x+1}{x^2+4x+4}}dx=\int{\frac{3x+1}{(x+2)(x+2)}}dx$$ Of course, I started decomposing the fraction and immediately realized it ...
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Simple Finite Continued Fraction

I am working on my senior thesis and have encountered, unexpectedly, a finite continued fraction that I would be interested in resolving. I already know the answer (by an informed guess based on where ...
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Effect of a simple pole vs complex conjugate poles

If $H(s)$ is a transfer function and it has just one pole in $s = p$, $p \in \mathbf{R}$, $$H(s) = \displaystyle \frac{H_0}{(s - p)}$$ the frequency response is $20 \log_{10} |H(j\omega)|$. With ...
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Can every rational function be represented in barycentric form?

This article about polynomial interpolation claims that (it is known that) every rational function may be represented in barycentric form: ...
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87 views

What is the integral of $\frac{x^7}{x^3+1}$?

How to find this integral? $$\int \frac{x^7}{x^3+1}\,dx$$ I think it may need a partial fraction but I'm not sure. Just need a start. Any help would be appreciated.
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Expand rational fractional expression in power series $\frac{k^2}{(k^2+\frac{1}{2})^{n+2}}$

Expression as showed in the title and n can be take arbitrary natural numbers, 0,1,2,.... How to expand it in powers of k and what coefficient of $k^m$ is? Any suggestion is much appreciated.
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How to integrate $36/(4x^2-12x+9)$?

I've just learned integration today and my teacher wasn't too helpful in explaining this. If anyone could help me here I would be most appreciative! Thank you! $$\int\frac{36}{4x^2-12x+9}dx$$
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Evaluating $\lim_{x\to -\infty} \frac{(x-1)}{(x^{2/3}-1)}$

The limit at negative infinity should not exist, right? $$\lim_{x\to -\infty} \frac{(x-1)}{(x^{2/3}-1)}$$ for positive infinity, the limit is infinity, but the function is undefined for values less ...
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Putting together a system of equations.

Someone helped me out with a problem a while back. Bout a year ago. But I still don't know how he derived the equation from or how he got it. I have two equations. $f/(f-n) = c$ $-nf/(f-n) = d$ We ...
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Is $Y=aX^b\cdot\exp(X)$ a rational or exponential function?

Is $Y=aX^b\cdot\exp(X)$ a rational or exponential function? $Y$ and $X$ are real variables, $a$ and $b$ are parameters. Someone said this is a product of polynomial and exponential function. Do we ...
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Rational functions over curves define regular maps?

I've a rather simple question that is strucking me, perhaps because I'm a newbie in algebraic geometry. If we have a projective irreducible non-singular curve $C\subseteq \mathbf{P}^n_k$ where $k$ is ...
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Evaluating$\int\frac{1}{(x^2-1)^2}$

This is the integral: $\int\frac{1}{(x^2-1)^2}$ I have tried several ways to solve this but I always end up that last parameter equals 1 and all others equals 0 so I end up where I started. Examples ...
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Solving a curve sketching task

I have some troubles with the general idea of curve sketching. I have following function that I have to sketch. $$ y = \frac{1}{2x^{2} + 12x + 6}. $$ I have the intercepts $y = 1/6$ and no ...
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How to solve this rational equation?

I'm stuck on this rational expression. I factored and simplified, by what do I do next? Should I divide x/2x and 8/4? I posted my work below. Thank you!
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Finding the equation of a rational function or a conic section given three points

I have a rational equation derived from 2 points, $(2, 2)$ and $(10, 10)$. Solving for the rational equation gives the equation $$y = \frac{20}{12-x}.$$ What I want to happen right now is that given ...
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Integration of rational functions of the form $\frac{p(x)}{\left[q(x)\right]^\alpha}$ when $\deg p < \deg q$ and $q(x)$ cannot be factored

What is the general method to evaluate integrals of the form $$\int\frac{p(x)}{\left[q(x)\right]^\alpha}\mathrm dx$$ when $\deg p < \deg q$ and $q(x)$ is irreducible? I don't see how partial ...
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Is $f(x) = \dfrac{x^2 -4x + 4}{x - 2}$ the same as $g(x) = x - 2 $?

Is $$f(x) = \dfrac{x^2 -4x + 4}{x - 2}$$ the same as $$g(x) = x - 2?$$ Why yes? Why not?
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Negative degree valuation: valuation ring and its maximal ideal

I know that the $v: f \mapsto -\deg(f)$ is a discrete valuation on the field of complex rational functions $\mathbb{C}(X)$ (the quotient field of $\mathbb{C}[X]$). The valuation ring $\mathcal{O}_v$ ...
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asymptote vs extraneous values

I am having trouble understanding the difference between a rational function with an asymptote versus having extraneous solutions. What is the difference between the two, if there is. Aren't ...
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Power series for the rational function $(1+x)^3/(1-x)^3$

Show that $$\dfrac{(1+x)^3}{(1-x)^3} =1 + \displaystyle\sum_{n=1}^{\infty} (4n^2+2)x^n$$ I tried with the partial frationaising the expression that gives me $\dfrac{-6}{(x-1)} - ...
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20th derivative of a rational function

I could not find the 20th derivative of the function below : $$f(x) = \frac{2x}{x^2 - 4}$$ I have taken 1st and 2nd derivatives but I could not succeed at generalizing the derivative function.
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How to prove that $f$ can be expressed as a ratio of polynomials, given that $|f(z)|=1$ when $|z|=1$? [duplicate]

Given: $f$ is analytic in $| z|\leq1$ and $|f(z)|=1$ when $|z|=1$. Prove that $f(z)=P(z)/Q(z)$ where $P$, $Q$ are polynomials.
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Is any rational function $R(x)$ a real analytic function in its domain?

To begin with, the definition of a rational function $R(x)$ can be found in Wiki. Suppose that $R(x)$ is defined in a subset $D \subseteq \mathbb{R}^n$. Then my question is: Is any rational ...
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Field of rational functions

Let $K$ be a field with characteristic $p>0$ and $M=K(X,Y)$ the field of rational functions in 2 variables over $K$. We consider the subfield $L=K(X^p,Y^p)\subset M$. Show that $[M:L]=p^2$. I ...
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what are the solutions of this biquadratic equation

I tried to choose this equation but I don't know how to find the LCM
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Graph shifting, compression, and stretch

Given $f(x)$, sketch $p(x) = (1/2)f(2x-6)-3$. I can't put the graph here. You can just tell me the order of transformation of the graph. What i did by myself is horizontal compressing (using $2x$ in ...
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Algebraic division with two variables? $\frac{a^3 + b^3}{a+b}$

I know there's a formula for this, but I would like to know how to do algebraic division the long way - would appreciate if you can guide me along. How can I use long division for $$\frac{a^3 + ...