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

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Solve for $y$: $\frac{y+1}{y-1} = 10^{x^2}$ [on hold]

Can someone please show me the steps (all of them… yeah, even the obvious ones) to go from $$\begin{align}\frac{y+1}{y-1} = 10^{x^2}\end{align}$$ to ...
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0answers
21 views

Uncommon Rational Function Expansion [on hold]

I am totally surprised by this awesome expansion: $$ \frac{a_0 + a_1x + a_2x^2 + a_3x^3}{b_0 + b_1x + b_2x^2} =\\ -\frac{a_3 b_1 - a_2 b_2}{b_2^2} +\frac{a_3x}{b_2} +\frac{a_3 b_0 b_1 ...
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2answers
67 views

How to integrate $\int\frac{3x+2}{x^2-x-2}dx$

This is the indefinite integral I have to evaluate: $$\int\frac{x^3}{x^2-x-2}dx$$ so by using the long division on polynomials technique, I got to: $$\frac{x^2}{2}+x+\int\frac{3x+2}{x^2-x-2}dx$$ How ...
3
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2answers
72 views

Complex analysis: Prove a meromorphic function to be rational.

I come across a problem about complex analysis: Show that a meromorphic function on the complex plane, which achieves any complex number no more than fixed given times, must be rational. The only ...
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3answers
33 views

How do you determine the end behavior of a rational function?

Example $$\frac{6x + 2}{x^2 - 9} = \frac{6x + 2}{(x + 3)(x - 3)}$$ I know how to find the vertical and horizontal asypmtotes and everything, I just don't know how to find end behavior for a ...
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1answer
54 views

How to convert the parametric equation into implicit form?

This problem is generated from another Green's theorem related question of mine. The original equation of the plane curve is not in rational parametric form. In order to calculate the symbolic ...
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1answer
21 views

Generating function of Language is rational

Let W be the set of all words over an alphabet $\Sigma$. Let $$L=\{w\in\Sigma^* | w\neq uvu',\text{ with }u,u'\in\Sigma^*,v\in W\}$$ I have to show that the generating function of L is rational. My ...
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0answers
17 views

Properties of rational polynomials

I have experimental data points that can be modeled by two different rational polynomials. I am wondering if there is a way (e.g. by a transform or integral), to discriminate the following two ...
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1answer
9 views

Degree theorem for Runge's approximating rational functions

Suppose that $f$ is analytic on an open set $D\subset\mathbb{C}$, and one uses Runge's theorem to obtain a sequence of rational functions $\{r_n\}$ which approach $f$ uniformly on compact subsets of ...
2
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1answer
28 views

Finding a (nonidentity) rational map of the plane with period $7$

Does there exist a nonidentity (which also is not a rotation) rational map $f:\mathbb{R}^2 \rightarrow \mathbb{R}^2$ with period $7$, i.e., for which the seventh iteration $f^7$ is the identity ...
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1answer
20 views

Why is this horizontal asymptote present and how do I immediately see that from the equation?

This may seem like a stupid question, and I do feel like I should know this. I have been given a simple curve with the following equation and was asked to state the equation of the asymptote of the ...
0
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1answer
12 views

Integral of derivative of rational map on unit disk

Let $f:D \rightarrow D$ be a surjective rational map of the unit disk of degree $n$. Prove that $$\iint_D |f'(x+iy)|\:\mathrm{d}x\:\mathrm{d}y\leq \pi \sqrt{n}.$$ Attempt: We know that rational maps ...
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1answer
35 views

Using the triangle inequality to bound $\frac{x^3 + 3x + 1}{10-x^3}$ for $|x+1|<2$

How do I use the triangle inequality to bound the function $$f(x) = \frac{x^3 + 3x + 1}{10 - x^3}$$ on the interval $|x+1|<2$? I understand how the triangle inequality works, but using fractions ...
2
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4answers
42 views

Derivative of Function with Rational Exponents $f(x)= \sqrt[3]{2x^3-5x^2+x}$

I have a question following: $$f(x)=\sqrt[3]{2x^3-5x^2+x}$$ Here's what I did, $$f(x)=\sqrt[3]{2x^3-5x^2+x} \\ = (2x^3-5x^2+x)^{3\over2} \\\\f'(x) = {3\over 2}(2x^3-5x^2+x)^{3\over2}(6x^2-10x+1)$$ ...
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1answer
34 views

Finding the partial fraction decomposition of $\frac{4s^2 - 5s + 2}{s^2(s^2 +9)}$

I am trying to find the partial fraction decomposition of $\dfrac{4s^2 - 5s + 2}{s^2(s^2 +9)}$ into something of the form $A\dfrac{1}{s} + B\dfrac{1}{s^2} + C\dfrac{1}{s^2+9} + D\dfrac{s}{s^2 + 9}$. ...
1
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1answer
31 views

How do I rewrite this rational expression?

How do I rewrite the rational expression: $$\frac{x^3+5x^2+3x-10}{x+4}$$ But in the form of: $$q(x) + \frac{r(x)}{b(x)}$$
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3answers
64 views

Solve the non-linear system of equations

For real $x,y,z>0$ solve the system of equation \begin{cases} \dfrac{1}{x}-3 y+4 z=5,\\ \dfrac{1}{y}-4 z+5 x=3,\\ \dfrac{1}{z}-5 x+3 y=4, \end{cases} It is easy to check out that $$ x ...
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2answers
42 views

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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3answers
75 views

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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0answers
24 views

$\mathcal{Z}$-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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3answers
30 views

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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4answers
61 views

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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2answers
52 views

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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2answers
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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2answers
37 views

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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1answer
131 views

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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1answer
34 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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5answers
87 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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1answer
31 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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0answers
29 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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0answers
45 views

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 ...
3
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3answers
232 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 ...
3
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1answer
89 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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25 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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1answer
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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1answer
22 views

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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1answer
29 views

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?
3
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2answers
44 views

Simplifying $\frac{1/(\frac{1}{z_1}(1-t)+\frac{1}{z_2}t) - z_1}{(z_2 - z_1)}$

This drives me mad! I am not very good in math but thought I could at least do basic things like this one, but can't figure it out and I spent a day on it. I am trying to simplify: ...
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0answers
22 views

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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2answers
225 views

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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2answers
177 views

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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2answers
96 views

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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1answer
46 views

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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2answers
89 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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1answer
26 views

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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2answers
78 views

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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1answer
53 views

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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0answers
36 views

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 ...
0
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1answer
34 views

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 ...