# Questions tagged [rational-functions]

Rational functions are ratios of two polynomials, for example $(x+5)/(x^2+3)$.

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### Computing ratio of two sums

I'm interested in computing $$f(s) := \frac{ \sum_{n=0}^{\infty} a_n s^n } { \sum_{n=0}^{\infty} b_n s^n }$$ for some given $s \in \mathbb{C}$, where the power series in the numerator and ...
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### Evaluation of a rational function vs formal power series

Let $K$ be a complete valued field and $X$ be an indeterminate $K[X]$ be the ring of polynomials over $K$ in the indeterminate $X$ and $K(X)$ be its quotient field. For $a\in K$, consider the ...
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### With polynomial long division, why are the un-applied divisor terms still multiplied and subtracted?

Given the following for example: $$\frac{x^2+x+8}{x+2}$$ The first point of confusion: The divisor $x + 2$ when performing the division operation, only $x$ is applied. While this works with $x$ ...
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### Computing $\int \frac{{\rm d} x}{x^3 + 12}$

$$\int \frac{{\rm d} x}{x^3 + 12}$$ This is a question I came up with and have not been able to solve. I graphed this function and it is as in the picture attached. Some of the people who I have ...
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### What kind of matroidal information is contained in the ratio of the Tutte polynomial and its dual?

Let $M$ be a matroid and $T(x,y)$ its Tutte polynomial. It is well-known that $T(y, x)$ is equal to the Tutte polynomial of the dual of $M$. What kind of matroidal information (if any) is contained in ...
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### Rational fractions as geometric predicates

I read article about restricted voronoi diagram. In chapter 3.4 Exact predicates I saw the following: "sideA, sideB, sideC are rational fractions of low-degree (resp. 2, 4/2 and 6/4)." sideA ...
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### Transformation that keeps the point $(1, 1)$ but raises the horizontal asymptote of the reciprocal function?

If we are given the function $f(x) = \frac{1}{x}$, we can change the horizontal asymptote by adding a constant to the function. For example, $g(x) = f(x) + 5$, makes the asymptote at $y = 5$. How ...
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### Solving Rational Inequalities via Reciprocals

I was reviewing rational inequalities, and noticed that the solution (identifying the domain of the inequality) was obtained just through critical points and a number line. Is there a way to get the ...