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

learn more… | top users | synonyms

1
vote
1answer
30 views

Polynomial inequalities vs rational inequalities

A question from one of the comprehension questions I have is: How would the intervals of the solution set differ between a polynomial inequality and a rational inequality? I have tried to research ...
0
votes
3answers
45 views

Factoring Trick - Adding Up Coefficients

My professor told me this for factoring polynomials: Add up the coefficients and if they equal 0 then the polynomial has root of 1. Add up, but switch the signs of the coefficients with odd ...
1
vote
2answers
25 views

A question about the domain of definition of rational functions

Is the function $f(x) = \frac{x^2}{x}$ defined for every $x \in \mathbb R$, or only defined on $\mathbb R \setminus \{0 \}$? Background: Say we are given $P(x) = x^2 - 4x + 3$ and $Q(x) = (x - ...
-1
votes
2answers
46 views

How can one calculate the limit of $\frac{1}{x^2-9}$ as x approaches -3 and 3 by hand? [closed]

Reviewing math for college after a gap year and so I know this is probably a pretty elementary question, but let me know if it has any interesting implications or alternative solutions or if it ...
-1
votes
3answers
50 views

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

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 ...
0
votes
2answers
69 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
votes
2answers
74 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 ...
0
votes
3answers
34 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 ...
1
vote
1answer
58 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 ...
0
votes
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 ...
1
vote
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 ...
0
votes
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
votes
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 ...
0
votes
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
votes
1answer
13 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 ...
1
vote
1answer
36 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
votes
4answers
44 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, $$f(x)=\sqrt[3]{2x^3-5x^2+x} \\ = (2x^3-5x^2+x)^{3\over2} \\\\f'(x) = {3\over 2}(2x^3-5x^2+x)^{3\over2}(6x^2-10x+1)$$ ...
1
vote
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
vote
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: $$q(x) + \frac{r(x)}{b(x)}$$
2
votes
3answers
68 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 ...
0
votes
2answers
48 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 ...
0
votes
3answers
76 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. ...
0
votes
0answers
27 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): $$\mathcal{Z}\{y(n+2)\}=z^2Y(z)-0z^2-2z=z^2Y(z)-2z$$ ...
1
vote
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.
2
votes
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 ...
0
votes
2answers
53 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.
0
votes
2answers
31 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, ...
1
vote
2answers
39 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 ...
3
votes
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 ...
0
votes
0answers
14 views

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: ...
3
votes
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. ...
2
votes
5answers
90 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 ...
1
vote
1answer
37 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 ...
0
votes
0answers
30 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 ...
1
vote
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
votes
3answers
244 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
votes
1answer
97 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 ...
0
votes
0answers
27 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?
0
votes
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 ...
0
votes
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 ...
0
votes
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
votes
2answers
45 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: ...
0
votes
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 ...
2
votes
2answers
231 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 ...
7
votes
2answers
178 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 ...
0
votes
2answers
113 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 ...
0
votes
1answer
47 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: ...
0
votes
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.
0
votes
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.
1
vote
2answers
79 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! $$\int\frac{36}{4x^2-12x+9}dx$$