Rational functions are ratios of two polynomials, for example $(x+5)/(x^2+3)$.
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28 views
Subfields $k\subseteq F\subseteq k(x_1,\dots,x_n)$ and their intersection with $k[x_1,\dots,x_n]$
By this thread, if I have a subfield $k\subseteq F\subseteq k(x_1,\dots,x_n)$, $F$ is of the form $F=k(\phi_1,\dots,\phi_m)$ for some rational functions $\phi_1,\dots,\phi_m\in k(x_1,\dots,x_n)$. But ...
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1answer
35 views
Express the following power series as a raional function
Consider the following power series:
$f(x) = \sum\limits_{i>=1} 2^{i-1}x^{3i}$ = $\ x^3 + 2x^6 + 4x^9 + ...$
$g(x) = \sum\limits_{i=2}^{20} f(x)^{i}$
Express both f(x) and g(x) as rational ...
2
votes
1answer
62 views
Complex Analysis - Rational Functions
I'm studying for my final exam and came across this problem:
Let f and g be entire analytic functions and |f(z)|<|g(z)| when |z|>1. Show that f/g is a rational function.
I'm not quite sure where ...
2
votes
3answers
64 views
Permutations and Cross-ratios
Pick four distinct numbers, list all 24 permutations, and compute the cross-ratio of each permutation. Show that at most six numbers have occurred, given by the cross-ratio group
$y, \frac{1}{y}, 1-y, ...
6
votes
1answer
88 views
Basis for $\mathbb{[Q(\pi):Q]}$
I'm trying to figure out whether the basis of $\mathbb{Q}(x)$ over $\mathbb{Q}$ is countable when $x$ is transcendental. I know that the elements in $\mathbb{Q}(x)$ will be rational functions in $x$ ...
0
votes
1answer
17 views
Uniqueness theorem for Rational Functions
I know that for polynomials $P,Q$, the equation $P(z) \equiv Q(z)$ implies that they are of the same degree and have the same coefficients. Is there an analogous result for rational fucntions? That ...
5
votes
3answers
81 views
Graph of $\quad\frac{x^3-8}{x^2-4}$.
I was using google graphs to find the graph of $$\frac{x^3-8}{x^2-4}$$ and it gave me:
Why is $x=2$ defined as $3$? I know that it is supposed to tend to 3. But where is the asymptote???
4
votes
3answers
64 views
When are the sections of the structure sheaf just morphisms to affine space?
Let $X$ be a scheme over a field $K$ and $f\in\mathscr O_X(U)$ for some (say, affine) open $U\subseteq X$. For a $K$-rational point $P$, I can denote by $f(P)$ the image of $f$ under the map
...
0
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0answers
24 views
$P(X)-UQ(X)$ is irreducible over $k[U]$ where $U = P/Q$
Let's $P$ and $Q$ in $k[X]$ two polynomials with no common factors, and $U = P/Q$.
How can we prove that $P(X) - U Q(X)$ is irreducible over $k[U]$ ?
I've found some things here : minimal polynomial ...
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0answers
58 views
Laurent expansion of rational functions with NOT polynomial for denominator.
I'm in trouble with the Laurent expansion (and the convergence radius) of a set of rational functions such as:
$$
\frac{1}{ \sin(1/z)}\text{ at }z=0\\
\frac{1}{\exp(1/z)-1}\text{ at }z=0
$$
or the ...
2
votes
1answer
28 views
Questions about ratios
At a school dance, each boy danced with exactly three girls and each girl danced with exactly two boys. if 100 boys attended the school dance, how many girls attended?
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1answer
29 views
Zeros of the analytic limit of complex rational function
For $n\in\mathbb{N}$ let $r_n,\ s_n$ be two polynomials of $O(n)$ degrees with real positive coefficients and set $f_n=r_n/s_n$.
Suppose there exists $c>0$ such that
$\bullet$ if $z\in\mathbb{C}$ ...
2
votes
2answers
110 views
Zeros and poles of rational functions on locally Noetherian schemes
Let $X$ be a locally Noetherian scheme and let $f$ be a rational function on $X$ (i.e. the equivalence class of a pair $(U,f)$, where $f \in \mathcal{O}_X(U)$ and $U$ contains the associated points of ...
5
votes
1answer
47 views
Roots of a polynomial and its derivative
All roots of a complex polynomial have positive imaginary part. Prove that all roots of its derivative also have positive imaginary part.
It's not a homework. This issue has been proposed in the ...
0
votes
2answers
50 views
$X^n - t$ is irreducible over $k(t)$
How can I prove (if it's true) that $X^n - t$ is irreducible over $k(t)$, the field of fractions of $k$ ?
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4answers
131 views
What goes wrong in this derivative?
$$ f(x) = \frac{2}{3} x (x^2-1)^{-2/3} $$
and f'(x) is searched.
So, by applying the product rule $ (uv)' = u'v + uv' $ with $ u=(x^2-1)^{-2/3} $ and $ v=\frac{2}{3} x $, so $ u'=-\frac{4}{3} x ...
3
votes
1answer
76 views
If a rational function is real on the unit circle, what does that say about its roots and poles? Clarification
I'm also self studying the Ahlfors Complex Analysis book.
A question asks:
Suppose $R(z)$ is some rational function which is real on the circle |z|=1 in the complex plane. The question asks, how ...
6
votes
0answers
51 views
Families of curves over number fields
Is there known a number field $K$ and a curve $F(x, y) \in K(t)[x, y]$ such that $F(x, y)$ does not have points over the field of rational functions $K(t)$ but for all but finitely many positive ...
2
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0answers
71 views
Finite/algebraic extensions of rational functions
I'm looking for results on the subject of finite/algebraic extensions of rational functions, but I only find papers who deal with algebraic geometry. I only know the basis of Galois theory. Could you ...
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2answers
56 views
Help creating a rational function
Create a rational function with vertical asymptotes $x=\pm1$ and oblique asymptote of $y=2x-3$ and a $y$-intercept of $4$.
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0answers
47 views
Limits as a representation of the Dirichlet function
I read that the Dirichlet function (1 if Rational, 0 else) can be written as:
What is the proof of that? Are those limits commutative? Is there any other closed formula for Dirichlet function? (With ...
4
votes
2answers
129 views
Finding good approximation for $x^{1/2.4}$
I would like to a good (8 bits accuracy) approximation for $x^{1/2.4}$ in the range $[0, 1]$. This transform is used for converting linear intensities to SRGB compressed values, so it's important that ...
0
votes
0answers
70 views
Proof of identity of magnitude of rational function in z-domain
In the set of rational polynomial functions $H(z)$ of a complex number $z$, there exist functions whose magnitude $|H(z)|^2$ is a constant $C$, but whose denominator and numerator are not constants. ...
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0answers
54 views
Reference request on symmetric polynomials
Let $e_k$ be the $k$th-degree elementary symmetric polynomial in $x_1,\ldots,x_n$ (and recall that $e_k=0$ if $k>n$).
I know very little about these polynomials. I've just noticed this odd ...
0
votes
3answers
151 views
What is the name of the function $f(x)=\frac{1}{x}$?
I'm facing this function:
$$f(x)=\frac{1}{x}$$
What I know is that the above equation is one of the simplest forms of "rational functions", where the numerator is $1$ and the denominator is $x$.
Is ...
3
votes
2answers
70 views
Which rational functions are derivatives of rational functions?
I thought it was interesting that $\frac{u^2+1}{(u^2-2u-1)^2}$ has the very simple integral $-\frac{u}{u^2-2u-1}$ but both of $\frac{u^2}{(u^2-2u-1)^2}$ and $\frac{1}{(u^2-2u-1)^2}$ are very ...
2
votes
3answers
112 views
Is there a general formula for the antiderivative of rational functions?
Some antiderivatives of rational functions involve inverse trigonometric functions, and some involve logarithms. But inverse trig functions can be expressed in terms of complex logarithms. So is there ...
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1answer
139 views
Finding Slant Asymptotes using synthetic division rather than long division
Is it possible to use repeated synthetic division (rather than long division) to find a slant asymptote for a rational function such as $\displaystyle \frac{2x^3 + 3x^2 + 5x + 7}{(x-1)(x-3)}$? It ...
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2answers
78 views
Solving rational inequalities
The problem is relatively simple, but I am a student teacher and the students were working on solving rational inequalities.
Such as $\frac{x+1}{x+3} \leq 1$.
I recommended that they move ...
16
votes
2answers
387 views
A nasty integral of a rational function
I'm having a hard time proving the following $$\int_0^{\infty} \frac{x^8 - 4x^6 + 9x^4 - 5x^2 + 1}{x^{12} - 10 x^{10} + 37x^8 - 42x^6 + 26x^4 - 8x^2 + 1}dx = \frac{\pi}{2}.$$
Mathematica has no ...
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vote
1answer
40 views
Asymptotics of a Product of Rational Expressions
The following is taken from page 8 of Alon and Spencer's The Probabilistic Method.
$$ \prod_{i = 0}^{n-1} \frac{v - 2i}{v-i} \sim e^{-n^2/2v} $$ as long
as $v \gg n^{3/2}$, estimating
...
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votes
1answer
86 views
Analytical function taking rationals to rationals.
Suppose $f:I \rightarrow \Bbb R$ is an analytic function defined on the interval $I\subset \Bbb R$ with the property that for every $q \in \Bbb Q:f(q)\in \Bbb Q$. Does this already imply that $f\in ...
2
votes
1answer
73 views
Evaluate a certain derivative
Let $\{x_1,\dots,x_n\}$ be pairwise distinct complex numbers and $\{l_1,\dots,l_n\}$ a vector of natural numbers such that $l_1+l_2+\dots+l_n=N$. Let
$$ h_j(x)=\prod_{i\neq j,i=1,\dots, n} ...
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2answers
71 views
Questions with respect to rational functions
I am currently studying Hardy's Pure Course of Mathematics and am on chapter 2, section 24: Rational Functions. In this chapter, Hardy defines a rational function as the quotient of two polynomials ...
0
votes
2answers
129 views
Find a rational function $f: \mathbb R \rightarrow \mathbb R$ with range $f(\mathbb R)=[-1,1]$
Find a rational function $f: \mathbb R \rightarrow \mathbb R$ with range $f(\mathbb R)=[-1,1]$
(Thus $f(x)=\frac{P(x)}{Q(x)}$ for all $x \in \mathbb R$ for suitable polynomials P and Q, where Q has ...
3
votes
2answers
66 views
Connecting the intuitive way to compute divisors of rational functions to the rigorous definition.
Let $X$ be the curve $xy-z^2 \subset \mathbb{P}^2$, and let $f$ be the rational function $x/y$ (Edit: I'm trying to simplify this as much as possible, but of course $x$ itself isn't a rational ...
3
votes
1answer
121 views
Finding out a rational equation via a graph
I need to be able to find an equation from this graph
So far I have this graph with the equation $-1/((x-3)^3)$
I can see from the desired graph that there is no horizontal asymptote, compared ...
1
vote
2answers
52 views
Factorizing rational functions of curves
Let $f:X\to \mathbf{P}^1$ be a rational function of degree $d\geq 2$ on a curve $X$.
Let $n\geq 2$ be a divisor of $d$. Does there exist a curve $Y$ with a rational function $g:Y\to \mathbf{P}^1$ of ...
2
votes
1answer
194 views
Why use radical notation instead of rational exponents?
I'm helping my younger sister for her math class. She has recently been taught integer exponents, and has starteed studying radicals (mainly square roots). The next topic will be rational exponents, ...
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1answer
106 views
minimal polynomial of $x$ over $ K\left(\frac{p(x)}{q(x)}\right) \subset K(x) $
Let $K$ be a field , let's consider the field of rationals functions over x , $k(x)$. Let $t\in k(x)$ be the rational function $\frac{p(x)}{q(x)}$ , where $P,Q$ have no common factors. I have to prove ...
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3answers
2k views
Finding the range of rational functions
I have a problem that I cannot figure out how to do. The problem is:
Suppose $s(x)=\frac{x+2}{x^2+5}$. What is the range of $s$?
I know that the range is equivalent to the domain of $s^{-1}(x)$ but ...
5
votes
1answer
91 views
Why does this equation have different number of answers?
I have a simple equation:
$$\frac{x}{x-3} - \frac{2}{x-1} = \frac{4}{x^2-4x+3}$$
By looking at it, one can easily see that $x \not= 1$ because that would cause $\frac{2}{x-1} $ to become ...
2
votes
1answer
118 views
Rational Functions are Determined By Locations and Multiplicities of Zeros and Poles (why?)
There is a theorem that says rational functions in the extended complex plane are exactly the meromorphic functions.
After this, my textbook draws the corollary:
"...as a consequence, a rational ...
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votes
0answers
49 views
Rational fraction of $f(n,m,\alpha,\beta)$
Is it possible to write $f(n,m,\alpha,\beta)$ as a fraction, i.e. $\frac{a+ib}{c}$, where a,b and c are integers?
$$f(n,m,\alpha,\beta) = H_{-1-m}(\frac{\alpha + i\beta}{\sqrt 2})$$
where $i$ is the ...
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0answers
101 views
continued fraction to rational polynomial in maple?
In maple is there a way to convert a continued fraction into a rational polynomial? I'm using the minimax function and for a particular function I want to approximate it returns a continued fraction ...
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2answers
66 views
Why is decomposing rational functions by assigning selected numerical values to x mathematically consistent? [duplicate]
Possible Duplicate:
How does partial fraction decomposition avoid division by zero?
Say you have the rational function:
$\frac{x^2 + 1}{(x-1)(x-2)(x-3)}$
This means that the function is ...
1
vote
1answer
66 views
Derive a Laurent series for the function $2z/(z+j)$
First of all, I apologize for the none mathematical notations. I've only ever hanged around Stackoverflow, and never learnt how to type Mathmatical notations. It would be great if someone could teach ...
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votes
3answers
105 views
How to simplify this rational expression?
This expression should be extremely easy to simplify, but for some reason I can't do it.
$$\frac{x^4-1}{x-1}$$
I know it simplifies down to this, but I don't know how to get there
$$x^3+x^2+x+1$$
...
0
votes
1answer
311 views
How to find the equation of a graph of a rational function from a set of points?
For example, the data points are:
(1,1)
(2,1/2)
(3,1/3)
(4,1/4)
(5,1/5)
How do I find the equation from those points? Do I look at the common ratio of the y-values or something?
3
votes
3answers
120 views
Solve an equation for x where $ y = \frac{ x^2 - x + 1}{ x^2 + x + 1 } $
Solve an equation for x where
$$
y = \frac{ x^2 - x + 1}{ x^2 + x + 1 }
$$
Also, is there an easier way to find the range of the equation, rather than reversing it and finding it's domain?
