Questions tagged [rational-functions]
Rational functions are ratios of two polynomials, for example $(x+5)/(x^2+3)$.
1,318 questions
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How to prove $\frac{f(x)}{g(x)}=\sum_{i=1}^{n}\frac{A_{i} }{x-r_{i} } $ using Bezout Identity
(I'm actually learning calculus.)Before I started working on this problem,I went to read this proof:
Partial Fractions Proof
I think I understand what the proof tried to do(And I can complete some ...
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Is the set of finite Blaschke products a graded ring?
I have heard tell that there are many analogies between Blaschke products and polynomials.
A finite Blaschke product is a rational function on the complex unit disk $B(z)=\mu\prod_{i=1}^{\deg B}\frac{...
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Decompose rational expressions [closed]
Partial fraction decomposition applies only when the degree of the numerator is less than the degree of the denominator.
A. True
B. False
It was in an exam. And the teacher answered A. But still, I ...
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Function fields over finite field.
Suppose $\mathbb{F}_q$ is a finite field of characteristic $p>0$ and $t$ is transcendental over $\mathbb{F}_q$. Then is this field $\mathbb{F}_q(t)$ a Hilbertian field?
The definition of ...
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Is there an easy way to know if a rational function is an n-th (compositional) iteration of a power series with indices in $\mathbb{Z}$?
I am asking this question mainly to probe the knowledge of people already familiar with this problem, otherwise I would advise caution to the unfamiliar trying to use computation, this can be really ...
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$\frac{a\left(b+c+2\right)}{bc+2a}+\frac{b\left(c+a+2\right)}{ca+2b}+\frac{c\left(a+b+2\right)}{ab+2c}\ge 3.$
I'm looking for some ideas to solve the following inequality.
Problem. Let $a,b,c\ge 0$ with $ab+bc+ca=1.$ Prove that$$\color{black}{\frac{a\left(b+c+2\right)}{bc+2a}+\frac{b\left(c+a+2\right)}{ca+2b}...
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Approximating roots of the equation $x^p - N = 0$ using approriate rational functions of rational numbers
This question is inspired by this one Choice of $q$ in Baby Rudin's Example 1.1, which is a specific case for $p=2$ and $n=2$. There, it is shown that the rational function $f(p) = \frac{2p + 2}{p+...
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Lack of closed form of Rayleigh functions for $n > 10$
The Rayleigh function is defined as follows for integers $n$: $\displaystyle \sigma_n(\nu) = \sum_{k=1}^{\infty} j_{\nu,k}^{-2n}\ $, where the $j_{\nu,k}$ are the zeros of the Bessel function of the ...
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Show that $|a_n(x)/a_n(1/x)| = 1$
Assume that $a_0(x)$ is a rational function such that $|a_0(x)/a_0(1/x)| = 1$ and define the sequence $a_n(x)$ such that $a_n(x) = x(a_{n-1}(x))'$ if $n \in \mathbb{N}$. It follows that $|a_n(x)/a_n(1/...
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Can I substitute x = ±1 to find partial fraction coefficients when the original function is undefined there? [duplicate]
Can I substitute $ x = \pm 1 $ to find partial fraction coefficients when the original function is undefined there?
I’m trying to compute the integral:
$$
\int \frac{x-3}{(x-1)^2(x+1)} dx
$$
using ...
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What makes one p-adic isometry be rational preserving, and another not?
What makes one p-adic isometry rational-preserving, and another not?
Consider the function $f(x)=\dfrac{ax+b}{cT(x)+d}$ where $a,b,c,d$ are 2-adic units.
Definition: A rational-preserving 2-adic ...
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Help with $\int_0^1 \frac{1 - x^n}{(1 - x)(1 + x)^n} \, dx$
I want to solve the integral $$
\int_0^1 \frac{1 - x^n}{(1 - x)(1 + x)^n} \, dx
$$ but I don't know how to solve it. This got shared on a math group. This is what I tried
\begin{align*}
& \...
user1683002
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Is a regular function on $U\subseteq \mathbb{A}^n$ a fraction of polynomials?
Let $k$ be an algebraically closed field and let $U$ be an open subset of $\mathbb{A}^n_k$, affine $n$-space. Is every regular function on $U$ necessarily expressible globally as a fraction of ...
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Can the graph of a rational function be a straight line except for one undefined point (a hole)? [closed]
Can the graph of a rational function be a straight line except for one undefined point (a hole)? Is there a way to recognize this directly from the equation, or could it be hidden in a more ...
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Proof that every rational function has an algebraic addition theorem (AAT)
In Article 42 (page 45) of Hancock (free PDF), the author writes:
Let $\phi(u)$ be a rational function of finite degree and let
$$x = \phi(u),\quad y = \phi(v),\quad z = \phi(u+v)$$
By means of these ...
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