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Question 1

2votes

1answer

26views

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 ProofI think I understand what the proof tried to do(And I can complete some ...

Onebytheside

121

asked1 hour ago

Question 2

0votes

0answers

35views

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{...

Kepler's Triangle

3,735

askedyesterday

Question 3

0votes

0answers

48views

Decompose rational expressions [closed]

Partial fraction decomposition applies only when the degree of the numerator is less than the degree of the denominator.A. TrueB. FalseIt was in an exam. And the teacher answered A. But still, I ...

Berhanu Baleh

1

askedyesterday

Question 4

0votes

0answers

53views

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 ...

Math123

1,273

asked2 days ago

Question 5

2votes

0answers

67views

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 ...

Evariste

2,911

askedNov 11 at 1:35

Question 6

-1votes

1answer

87views

$\frac{a\left(b+c+2\right)}{bc+2a}+\frac{b\left(c+a+2\right)}{ca+2b}+\frac{c\left(a+b+2\right)}{ab+2c}\ge 3.$

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}...

Danh Trung

504

askedOct 17 at 8:49

Question 7

3votes

1answer

96views

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+...

MushroomTea

437

askedOct 3 at 16:29

Question 8

3votes

1answer

244views

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 ...

Arurikku Burumanto

172

askedSep 23 at 16:01

Question 9

3votes

1answer

95views

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/...

John

911

askedSep 23 at 2:31

Question 10

1vote

1answer

84views

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 ...

mahler

71

askedSep 14 at 18:32

Question 11

1vote

0answers

69views

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 ...

Robert Frost

9,768

askedSep 11 at 7:44

Question 12

20votes

6answers

1kviews

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

askedSep 8 at 4:42

Question 13

3votes

1answer

103views

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 ...

Anon

1,173

askedSep 8 at 2:00

Question 14

-3votes

1answer

63views

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 ...

Anushka_Grace

188

askedSep 7 at 18:23

Question 15

2votes

2answers

88views

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 ...

Moe

355

askedAug 9 at 11:03

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Questions tagged [rational-functions]

How to prove $\frac{f(x)}{g(x)}=\sum_{i=1}^{n}\frac{A_{i} }{x-r_{i} } $ using Bezout Identity

Is the set of finite Blaschke products a graded ring?

Decompose rational expressions [closed]

Function fields over finite field.

Is there an easy way to know if a rational function is an n-th (compositional) iteration of a power series with indices in $\mathbb{Z}$?

$\frac{a\left(b+c+2\right)}{bc+2a}+\frac{b\left(c+a+2\right)}{ca+2b}+\frac{c\left(a+b+2\right)}{ab+2c}\ge 3.$

Approximating roots of the equation $x^p - N = 0$ using approriate rational functions of rational numbers

Lack of closed form of Rayleigh functions for $n > 10$

Show that $|a_n(x)/a_n(1/x)| = 1$

Can I substitute x = ±1 to find partial fraction coefficients when the original function is undefined there? [duplicate]

What makes one p-adic isometry be rational preserving, and another not?

Help with $\int_0^1 \frac{1 - x^n}{(1 - x)(1 + x)^n} \, dx$

Is a regular function on $U\subseteq \mathbb{A}^n$ a fraction of polynomials?

Can the graph of a rational function be a straight line except for one undefined point (a hole)? [closed]

Proof that every rational function has an algebraic addition theorem (AAT)

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