Get Expert Help for Your Rational Functions Questions

What is rational function and how do you find domain, vertical and horizontal asymptotes. Also what is "holes" with all limits and continuity and discontinuity?

"determine the form of rational function in a plane hat has a positive value on unit circle."
hint suggested me that such a rational function must have same number of zeros and poles inside unit disk. But I cannot understand that. Using symmetry(Schwarz reflection principle) I can see that number of poles(zeros) inside unit disk is equal to number of poles(zeros) outside unit disk. BUT I CANNOT SEE WHY NUMBER OF POLES AND ZEROS MUST BE EQUAL..
Who knows?

Lagrange's rational function theorem states that if one has two rational functions in multiple variables ${\displaystyle f(x_1,x_2,\dots x_n)}$ and ${\displaystyle g(x_1,x_2,\dots x_n)}$ then one can express f as a rational function in g if and only if the set of permutations that keep g unchanged is a subset of the set of permutations that preserve f.
Is anyone familiar with the proof of this theorem? While it is fairly clear that if f can be expressed in terms of g the set of permutations that keep g unchanged has to be the subset of those that keep f unchanged, the converse is far from obvious.

A polynomial with integer coefficients maps integers to integers. A rational function (which is a quotient of two polynomials) tends to map integers to rational numbers. Other than the trivial case where the denominator is a factor of the numerator, are there any rational functions that map all integers to integers? What is known about them and how can they be found?

In Hartshorne's Algebraic Geometry the function field K(Y) of a variety Y is defined as the set of equivalence classes with f being a regular function on the open subset ${\displaystyle U\ne \mathrm{\varnothing }}$. We have = if f and g agree on ${\displaystyle U\cap V}$.
How does one evaluate a "rational function on Y'' f? Is it f(P) using the Evaluation map for polynomials? In this case, I don't see that this is well defined: Let ${\displaystyle \frac{P\in U}{V}}$ and =, then g(P) might not even be defined!

Construct all rational functions f (quotients of polynomials) in ${\displaystyle \mathbb{C}}$ such that ${\displaystyle \left(f\right)=6\left[0\right]-4\left[5\right]}$, where (f) is the divisor of the function f.
My answer/attempt: From my understanding of divisors, there seems to be exactly one function, namely, ${\displaystyle f\left(z\right)=\frac{z^6}{{(z-5)}^4}}$.
1.Are there any other functions with this divisor? If not, why?
2.If we restrict ourselves to divisors of rational functions only, then if we are given a divisor of a ${\displaystyle \mathrm{△}}$ on ${\displaystyle \mathbb{C}}$, does there exists a unique rational function f with ${\displaystyle \left(f\right)=\mathrm{△}}$?

I have to find border of rational function:
${\displaystyle \underset{x\to \mathrm{\infty }}{lim}\frac{x^3+3x^2+2x}{x^2-x-6}}$
What should I do?

Help me with creating a rational function with vertical asymptotes ${\displaystyle x=\pm 1}$ and oblique asymptote of ${\displaystyle y=2x-3}$ and a ${\displaystyle y}$-intercept of 4.

"A rational function is defined as the quotient of polynomials in which the denominator has a degree of at least 1"
If we are talking merely about x, then I get the concept. A rational function f(x) could be written as "${\displaystyle \frac{p\left(x\right)}{q\left(x\right)}}$, where ${\displaystyle q\left(x\right)\ne 0}$."
The issue that I'm having is that of talking about rational functions of n variables. For instance, what would be the meaning of 'f(x,y) is a rational function of ${\displaystyle x\mathrm{\&}{y}^{\prime }}$?

Let ${\displaystyle f:X\to P^1}$ be a rational function of degree ${\displaystyle d\ge 2}$ on a curve X.
Let ${\displaystyle n\ge 2}$ be a divisor of d. Does there exist a curve Y with a rational function ${\displaystyle g:Y\to P^1}$ of degree n such that f factors through g
Can we factor f in some sense?
Note that by de Franchis

If f is quadratic function such that f(0)=1 and ${\displaystyle \int \frac{f\left(x\right)}{x^2{(x+1)}^3}dx}$ is a rational function, find the value of f′(0).
I already tried solving this question by using general quadratic equation ${\displaystyle a{x}^2+bx+c}$ and then using partial fraction method but it became very complicated.

For each of the following functions, determine if it is a polynomial, rational function, both, or neither
${\displaystyle h\left(x\right)=3x^{-9}+6x^{-4}-3}$
a. Polynomial function only
b. Rational function only
c. Both polynomial and rational function

${\displaystyle j\left(x\right)={(8x^2-5\sqrt{x}+6)}^2}$
a. Polynomial function only
b. Rational function only
c. Both polynomial and rational function

How do you create a rational function that includes the following: crosses x-axis at 4, Touches the x-axis at -3, One vertical asymptote at x=1 and another at x=6, one horizontal asymptote at y= -3?

This is a pair of similar questions.
Here is the first one:Does there exist a rational function P(x) with a,b ${\displaystyle ϵ\mathbb{Q}}$ such that ${\displaystyle {\int }_a^{b}P\left(x\right)dx}$ is equal to ${\displaystyle k\cdot e}$ for some ${\displaystyle kϵ\mathbb{Q}}$ where e is Euler's number? Prove or disprove that such a function exists. If such a function exists find it. Also, this function should be bounded in the y dimension also so there should exist some r such that |P(x)| < r for every x on [a,b].
This is the second:Let A be a region defined by the inequality 0<P(x,y) where P is a rational function and ${\displaystyle x^2+y^2<r}$ for all ${\displaystyle (x,y)ϵA}$ for some finite r. Does such an A exist such that the area of A equals ${\displaystyle k\cdot e}$ where ${\displaystyle kϵ\mathbb{Q}}$ and e is Euler's constant? Prove or disprove that such an A exists. If such an A exist find P.
Inspiration for the questions:A nice way to approximate pi is by picking random points in the region 0<x<1 and 0<y<1 and seeing if ${\displaystyle x^2+y^2<1}$. This is easy to program because all the functions involved are rational. similar things can be done with ${\displaystyle \ln \left(2\right)}$ for example. I would like to do so with e but can't think of the correct shape to use.
The coefficients of the rational functions should be rational numbers. The bounds on a and b in the first question should be rational.

Let f be a rational function on a compact connected Riemann surface X. The rational function f induces a holomorphic map ${\displaystyle \bar{f}:X\to R^1\left(C\right)}$.
Let x be a point on the Riemann sphere ${\displaystyle P^1\left(C\right)}$. ow can I check that if b is a branch point of ${\displaystyle \bar{f}}$ by looking at the derivative of f?
How does this work when ${\displaystyle X=P^1\left(C\right)}$?

What is the flaw in the answer for proving any rational function on ${\displaystyle P{P}^1}$ is constant?
Let ${\displaystyle \varphi :P{P}^1\to {\mathrm{\forall }}^1}$ be a rational function. Since ${\displaystyle {\mathrm{\forall }}^1\subset P{P}^1}$ we can think ${\displaystyle \varphi :P{P}^1\to P{P}^1}$ as rational function. As every rational map from ${\displaystyle P{P}^1\to P{P}^n}$ is regular, so ${\displaystyle \varphi }$ is regular. We also know that any regular function on ${\displaystyle P{P}^1}$ is constant, so ${\displaystyle \varphi }$ is constant.

A rational function f in n variables is a ratio of 2 polynomials,
${\displaystyle f(x_1,\dots x_n)=\frac{p(x_1,\dots x_n)}{q(x_1,\dots x_n)}}$
where q is not identically 0. The function is called symmetric if
${\displaystyle f(x_1,\dots x_n)=f(x_{\sigma \left(1\right)},\dots x_{\sigma \left(n\right)})}$
for any permutation ${\displaystyle \sigma }$ of ${\displaystyle \{1,\dots ,n\}}$.
Let F denote the field of rational functions and S denote the subfield of symmetric rational functions. Suppose the coefficients of polynomials are all real numbers.
Show that F=S(h), where ${\displaystyle h=x_1+2x^2+\dots +n{x}_n}$. In other words, show that h generates F as a field extension of S.
Attempt at Solution:
1.Can't seem to get very far with this one. I know that F is a finite extension of S of degree n! and the Galois group of the extension is ${\displaystyle S_n}$.
2.Using h and the 1st symmetric function ${\displaystyle x_1=x_1+x_2+\dots +x_n}$, we see that ${\displaystyle h-s_1=x_2+2x^3+\dots (n-1)x_nϵS\left(h\right)}$.
3.Can't seem to find a good way to use the other symmetric functions ${\displaystyle s_2,\dots ,s_n}$.

Is the section df associated to a rational function f on a curve X a global section of the canonical sheaf ${\displaystyle {\omega }_X}$? I know its zeroes are the ramification points, but does it have poles?

a) Show sin is not a rational function.
By definition of a rational function, a rational function cannot be 0 at infinite points unless it is 0 everywhere. Obviously, sin have infinite points that are 0 and infinite points that are not zero, thus not a rational function.
b) Show that there do not exist rational functions ${\displaystyle f_0,\dots ,f^{n-1}\left(x\right){\left(\sin x\right)}^{n-1}+\dots +f_0\left(x\right)=0}$ for all x
First choose ${\displaystyle x=2k\pi }$, so ${\displaystyle f_0\left(x\right)=0}$ for ${\displaystyle x=2k\pi }$. Since ${\displaystyle f_0}$ is rational ${\displaystyle \Rightarrow f_0\left(x\right)=0}$ for all x. Thus can write ${\displaystyle \sin x[{\left(\sin x\right)}^{n-1}+f_{n-1}{\left(\sin x\right)}^{n-2}\dots +f_1\left(x\right)]=0}$
Question: The second factor is 0 for all ${\displaystyle x\ne 2k\pi }$ How does this imply that it is 0 for all x? And how does this lead to the result?

In general, how does one determine if a rational function is regular? I have the particular problem of determining in which points of the circle ${\displaystyle V(x^2+y^2-1)\subseteq A^2}$ is the rational function ${\displaystyle \alpha =\frac{y-1}{x}}$ regular?