Questions tagged [rational-functions]
Rational functions are ratios of two polynomials, for example $(x+5)/(x^2+3)$.
1,318 questions
-4 votes
1 answer
58 views
Can a rational function like this be integrated? [closed]
Can a rational function like this be integrated? $$\int \frac{-ct^2 +2ct-c}{t^5 +2t^3 +t} dt$$ where c is a constant.
2 votes
1 answer
66 views
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 ...
- 121
1 vote
0 answers
44 views
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{...
- 3,715
0 votes
0 answers
70 views
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 ...
- 1,273
2 votes
0 answers
67 views
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 ...
- 2,911
-1 votes
1 answer
91 views
$\frac{a\left(b+c+2\right)}{bc+2a}+\frac{b\left(c+a+2\right)}{ca+2b}+\frac{c\left(a+b+2\right)}{ab+2c}\ge 3.$
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$$\color{black}{\frac{a\left(b+c+2\right)}{bc+2a}+\frac{b\left(c+a+2\right)}{ca+2b}...
- 504
3 votes
1 answer
98 views
Approximating roots of the equation $x^p - N = 0$ using approriate rational functions of rational numbers
This question is inspired by this one Choice of $q$ in Baby Rudin's Example 1.1, which is a specific case for $p=2$ and $n=2$. There, it is shown that the rational function $f(p) = \frac{2p + 2}{p+...
- 437
3 votes
1 answer
250 views
Lack of closed form of Rayleigh functions for $n > 10$
The Rayleigh function is defined as follows for integers $n$: $\displaystyle \sigma_n(\nu) = \sum_{k=1}^{\infty} j_{\nu,k}^{-2n}\ $, where the $j_{\nu,k}$ are the zeros of the Bessel function of the ...
3 votes
1 answer
95 views
Show that $|a_n(x)/a_n(1/x)| = 1$
Assume that $a_0(x)$ is a rational function such that $|a_0(x)/a_0(1/x)| = 1$ and define the sequence $a_n(x)$ such that $a_n(x) = x(a_{n-1}(x))'$ if $n \in \mathbb{N}$. It follows that $|a_n(x)/a_n(1/...
- 911
1 vote
1 answer
87 views
Can I substitute x = ±1 to find partial fraction coefficients when the original function is undefined there? [duplicate]
Can I substitute $ x = \pm 1 $ to find partial fraction coefficients when the original function is undefined there? I’m trying to compute the integral: $$ \int \frac{x-3}{(x-1)^2(x+1)} dx $$ using ...
- 71
1 vote
0 answers
72 views
What makes one p-adic isometry be rational preserving, and another not?
What makes one p-adic isometry rational-preserving, and another not? Consider the function $f(x)=\dfrac{ax+b}{cT(x)+d}$ where $a,b,c,d$ are 2-adic units. Definition: A rational-preserving 2-adic ...
- 9,762
20 votes
6 answers
1k views
Help with $\int_0^1 \frac{1 - x^n}{(1 - x)(1 + x)^n} \, dx$
I want to solve the integral $$ \int_0^1 \frac{1 - x^n}{(1 - x)(1 + x)^n} \, dx $$ but I don't know how to solve it. This got shared on a math group. This is what I tried \begin{align*} & \...
user1683002
3 votes
1 answer
104 views
Is a regular function on $U\subseteq \mathbb{A}^n$ a fraction of polynomials?
Let $k$ be an algebraically closed field and let $U$ be an open subset of $\mathbb{A}^n_k$, affine $n$-space. Is every regular function on $U$ necessarily expressible globally as a fraction of ...
- 1,173
-3 votes
1 answer
65 views
Can the graph of a rational function be a straight line except for one undefined point (a hole)? [closed]
Can the graph of a rational function be a straight line except for one undefined point (a hole)? Is there a way to recognize this directly from the equation, or could it be hidden in a more ...
- 188
2 votes
2 answers
90 views
Proof that every rational function has an algebraic addition theorem (AAT)
In Article 42 (page 45) of Hancock (free PDF), the author writes: Let $\phi(u)$ be a rational function of finite degree and let $$x = \phi(u),\quad y = \phi(v),\quad z = \phi(u+v)$$ By means of these ...
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