Questions tagged [rational-functions]
Rational functions are ratios of two polynomials, for example $(x+5)/(x^2+3)$.
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If a tetrahedron has a midsphere, then $\rho^2$ is shown to be rational in the vertex coordinates; must $\rho$ itself be rational too?
Let $p_1,p_2,p_3,p_4\in \mathbb R^3$ be a tetrahedron. By a midsphere I mean a sphere tangent to all six edges of the tetrahedron. Suppose such a midsphere exists, and let $O$ be its center and $\rho$ ...
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Partial Fraction Decomposition in Complex Analysis
I'm currently reading through "Complex Analysis" by Bak and Newman, which I've linked over here. I have questions about Theorem 9.13 in the text, whose proof I've screenshotted below:
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Are there finitely many primes of the form $\frac{2^{p^2}\pm 1}{2^p \pm 1}$?
Consider primes of the form $\dfrac{2^{p^2}+1}{2^p+1}$ where $p$ is itself an odd prime.
I know $\dfrac{2^{49}+1}{2^7+1} = 4363953127297$ is a prime number.
MAIN QUESTION : Is this the last one ?
I ...
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Function induced by a rational $f$ determines $f$ uniquely
I'm currently learning the basics of algebraic geometry, and I'm having some confusions with the following problem in Fulton's Algebraic Curves:
2.19. Let $f$ be a rational function on a variety $V$. ...
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Proof that every explicit algebraic function can be made implicit
In Hardy's book 'A Course of Pure Mathematics', he defines an explicit algebraic function thus: These are functions which can be generated from $x$ by a definite number of operations such as those ...
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Solve $\frac{1}{x^{3}(1-x^{2})}+\frac{1}{x^{2}(1+x^{2})}=\frac{11x-18}{5}$ for $x>\sqrt{2}$
Solve for real $x>\sqrt{2}$:
$$\frac{1}{x^{3}(1-x^{2})}+\frac{1}{x^{2}(1+x^{2})}=\frac{11x-18}{5}$$
This problem appeared in a regional mathematics olympiad from India
I first combined the two ...
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How to find the ratio of control point in a rational quadratic bézier to fit a curve to known points?
I'm doing an experiment in generating color palettes, given one input color in HCL format. I have the hue part handled. Now I'm trying to fit a curve to the Chroma and Lightness of the given color. I ...
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The incenter of a quadrilateral is a rational function of the vertex coordinates
Given only the coordinates of $A,B,C,D$, the incenter can be recovered purely by reflections across diagonals and perpendicular constructions, without ever touching angle bisectors.
Take diagonal $AC$...
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Pedantic question on analytic generalisation of rational functions on real numbers
Since this is a pedantic question, we will need some definitions:
A function $f$ is polynomial when it can be expressed as $\sum_{0\leq i\leq n} c_i X^i$.
A function $f$ is rational when $f$ is a ...
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For $R(x)$ a rational function, why is $R’(x)+2xR(x)$ never identically 1?
For $R(x)$ a rational function, why is $R’(x)+2xR(x)$ never identically 1? I would like a reason that doesn’t appeal to the fact that $e^{x^2}$ doesn’t have an elementary antiderivative. I can show ...
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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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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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