Mathematics Stack Exchange
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Questions tagged [rational-functions]

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

In mathematics, a rational function is any function which can be defined by a rational fraction, i.e. an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be rational numbers; they may be taken in any field $K$. In this case, one speaks of a rational function and a rational fraction over $K$. The values of the variables may be taken in any field $L$ containing $K$. Then the domain of the function is the set of the values of the variables for which the denominator is not zero and the codomain is $L$.

The set of rational functions over a field $K$ is a field, the field of fractions of the ring of the polynomial functions over $K$.

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Is the "determinant" that shows up accidental?

Consider the class of rational functions that are the result of dividing one linear function by another: $$\frac{a + bx}{c + dx}$$ One can easily compute that, for $\displaystyle x \neq \frac cd$ $$\frac{\mathrm d}{\mathrm dx}\left(\frac{a + bx}{c +…
stats_model
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A nasty integral of a rational function

I'm having a hard time proving the following $$\int_0^{\infty} \frac{x^8 - 4x^6 + 9x^4 - 5x^2 + 1}{x^{12} - 10 x^{10} + 37x^8 - 42x^6 + 26x^4 - 8x^2 + 1} \, dx = \frac{\pi}{2}.$$ Mathematica has no problem evaluating it while I haven't the slightest…
user54031
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Has this chaotic map been studied?

I have recently been playing around with the discrete map $$z_{n+1} = z_n - \frac{1}{z_n}$$ That is, repeatedly mapping each number to the difference between itself and its reciprocal. It shows some interesting behaviour. This map seems so…
Martin Ender
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How does partial fraction decomposition avoid division by zero?

This may be an incredibly stupid question, but why does partial fraction decomposition avoid division by zero? Let me give an example: $$\frac{3x+2}{x(x+1)}=\frac{A}{x}+\frac{B}{x+1}$$ Multiplying both sides by $x(x+1)$ we…
Argon
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Dividing an infinite power series by another infinite power series

Let's say I have two power series $\,\mathrm{F}\left(x\right) = \sum_{n = 0}^{\infty}\,a_{n}\,x^{n}$ and $\,\mathrm{G}\left(x\right) = \sum_{n = 0}^{\infty}\,b_{n}\,x^{n}$. If I define the function $\displaystyle{\,\mathrm{H}\left(x\right)…
QuantumEyedea
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Where does this approximation for $2^x-1$ come from?

I found this fairly good approximation for $2^x-1$ while reverse-engineering a fast exp2f() C++ (undocumented, uncommented) implementation: $$2^x-1\approx \frac{27.7280233}{4.84252568 - x} -0.49012907 x -5.7259425$$ This approximation is accurate…
BadHellie
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Why is every meromorphic function on $\hat{\mathbb{C}}$ a rational function?

I know that an analytic function on $\mathbb{C}$ with a nonessential singularity at $\infty$ is necessarily a polynomial. Now consider a meromorphic function $f$ on the extended complex plane $\hat{\mathbb{C}}$. I know that $f$ has only finitely…
Hana Bailey
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Lüroth's Theorem

I have just begun to read Shafarevich's Basic Algebraic Geometry. In the first section of the first chapter, he quotes Lüroth's theorem, which states that any subfield of $k(x)$ that is not just $k$ is isomorphic to $k(x)$, i.e. is generated as a…
Ben Blum-Smith
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Do rational functions eventually have monotonic derivatives?

Given a rational function $R(x)=P(x)/Q(x)$ with real coefficients, is it true that there exists an $M>0$ such that, for every $k\geq 0$, the restrictions $R^{(k)}|_{(-\infty,-M]}$ and $R^{(k)}|_{[M,\infty)}$ of the $k$-th derivatives $R^{(k)}(x)$…
Qfwfq
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Prove that: $\lfloor n^{1/2}\rfloor+\cdots+\lfloor n^{1/n}\rfloor=\lfloor \log_2n\rfloor +\cdots+\lfloor \log_nn \rfloor$, for $n > 1$

Prove that: $\lfloor n^{1/2}\rfloor+\cdots+\lfloor n^{1/n}\rfloor=\lfloor \log_2n\rfloor +\cdots+\lfloor \log_nn \rfloor$, for $n > 1,\, n\in \mathbb{N}$ For example. For $n=2$, we have $\lfloor 2^{1/2} \rfloor = \lfloor 1.414 \rfloor = 1$ whereas…
PandaMan
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How to find exact value of integral $\int_{0}^{\infty} \frac{1}{\left(x^{4}-x^{2}+1\right)^{n}}dx$?

When I first encountered the integral $\displaystyle \int_{0}^{\infty} \frac{1}{x^{4}-x^{2}+1} d x$, I am very reluctant to solve it by partial fractions and search for any easier methods. Then I learnt a very useful trick to evaluate the…
Lai
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"Taylor series" is to "Volterra series" as "Padé approximant" is to _________?

Padé approximants are often better than Taylor series at representing a function. Given a Taylor series, one can use Wynn's epsilon algorithm to easily produce the Padé approximants to it. Volterra series are a generalization of Taylor series that…
Mike Battaglia
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Does there exist a nontrivial rational function which satisfies $f(f(f(f(x))))=x$?

It's pretty well known and easy to show that for a linear fractional function $f(x)=\frac{ax+b}{cx+d}$ to be an involution, ie. $f(f(x))=x$, then $a\cdot d=-1$ is a necessary and sufficient condition. An example of a rational function which…
Geoffroi
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Proof that rational functions are an ordered field, but non-archimedean - Bartle's elements of real analysis

I read that the set of rational functions with rational coefficients forms an ordered field, yet it is non-archimedean. I tried googling this, but I don't think I understood the solution. How does one define an order on rational functions of the…
Quasar
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How did Artin discover the function $f(x)=\frac{(x^2-x+1)^3}{x^2(x-1)^2}$ with the properties $f(x)=f(1-x)=f(\frac{1}{x})$?

In Artin's "Galois Theory" P38, he said the function $$f(x) = \frac{(x^2 - x + 1)^3}{x^2(x-1)^2}$$ satisfies the properties of $f(x)=f(1-x)=f(\frac{1}{x})$. Is the function given by some rational step or just by a flash of insight？ If $f(0)$ is a…
Marine
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