Real Rational Functions

Definition: Real Rational Functions

A real rational function is a real function $f:\mathcal{D}\subseteq \mathbb{R}\to \mathbb{R}$ for which there exist real polynomials $P$ and $Q$ such that

$$f(x)=\frac{P(x)}{Q(x)}$$

for all $x\in \mathcal{D}$.

Theorem: Limits at Infinity

Let $f$ be a real rational function given by

$$f(x)=\frac{P(x)}{Q(x)}=\frac{\sum _{k=0}^m{p}_k{x}^k}{\sum _{k=0}^n{q}_k{x}^k}.$$

If $m<n$, then $f$ has a horizontal asymptote at $y=0$:

$$m<n⟹\underset{x\to \pm \infty }{\lim }f(x)=0$$

If $m=n$, then $f$ has a horizontal asymptote at $y=\frac{p_m}{q_n}$:

$$m=n⟹\underset{x\to \pm \infty }{\lim }f(x)=\frac{p_m}{q_n}$$

If $m>n$, then the limits of $f$ at ${\pm }_{\infty }$ depend on $(n-m)$ and $\frac{p_m}{q_n}$:

$$\begin{array}{rl}& \underset{x\to +\infty }{\lim }f(x)=\{\begin{array}{l}+\infty \text{when }\frac{p_m}{q_n}>0\\ -\infty \text{when }\frac{p_m}{q_n}<0\end{array}\\ & \underset{x\to -\infty }{\lim }f(x)=\{\begin{array}{l}+\infty \text{when }(n-m)\text{ is even and }\frac{p_m}{q_n}>0\text{ or }(n-m)\text{ is odd and }\frac{p_m}{q_n}<0\\ -\infty \text{when }(n-m)\text{ is even and }\frac{p_m}{q_n}<0\text{ or }(n-m)\text{ is odd and }\frac{p_m}{q_n}>0\end{array}\end{array}$$

PROOF

TODO

Theorem: Vertical Asymptotes

Let $f$ be a real rational function given by

$$f(x)=\frac{P(x)}{Q(x)}=\frac{\sum _{k=0}^m{p}_k{x}^k}{\sum _{k=0}^n{q}_k{x}^k}.$$

If $r\in \mathbb{R}$ is a root of $Q$ but not of $P$, then $f$ has a vertical asymptote at $x=r$.

If $r\in \mathbb{R}$ is a root of both $P$ and $Q$ but its multiplicity for $P$ is lower than its multiplicity for $Q$, then $f$ again has a vertical asymptote at $x=r$.

PROOF

TODO

<<<<<<< HEAD

Theorem: Partial Fraction Decomposition

Every real rational function $f(x)=\frac{P(x)}{Q(x)}$ with $\mathrm{deg}P\le \mathrm{deg}Q$ can be represented as a sum of real rational functions

$$f(x)=\sum _{i=1}\sum _{j=1}^{m_{l_i}}\frac{a_{ji}(x)}{l_i(x{)}^j}+\sum _{i=1}\sum _{j=1}^{m_{q_i}}\frac{b_{ji}(x)}{q_i(x{)}^j},$$

where:

• $l_i(x)$ are the linear factors of $Q(x)$ and $m_{l_i}$ are their respective multiplicities;

• $q_i(x)$ are the quadratic factors of $Q(x)$ and $m_{q_i}$ are their respective multiplicities;

• $\mathrm{deg}(a_{ji})=0$ or $\mathrm{deg}(a_{ji})=0$;

• $\mathrm{deg}(b_{ji})\le 1$.

PROOF

TODO

<<<<<<< HEAD

Theorem: Antidifferentiation of Elementary Rational Functions

Let $m\in \mathbb{N}$.

For all $a\in \mathbb{R}$, the antiderivatives of $\frac{1}{(x-a{)}^m}$ are given by

$$\int \frac{1}{(x-a{)}^m}\mathrm{dx}=\{\begin{array}{l}\ln |x-a|+C\text{if }m=1\\ \\ -\frac{1}{(m-1)(x-a{)}^{m-1}}+C\text{if }m\ge 2\end{array}$$

For all $p,q\in \mathbb{R}$ with $p^2-4q<0$, the antiderivatives of $\frac{1}{(x^2+px+q{)}^m}$ are given by

$$\int \frac{1}{(x^2+px+q{)}^m}\mathrm{dx}=\{\begin{array}{l}\frac{2}{\sqrt{4q-p^2}}\arctan \left(\frac{2x+p}{\sqrt{4q-p^2}}\right)+C\text{if }m=1\\ \\ \frac{2x+p}{(m-1)(4q-p^2)(x^2+px+q{)}^{m-1}}+\frac{4m-6}{(m-1)(4q-p^2)}\int \frac{1}{(x^2+px+q{)}^{m-1}}+C\text{if }m\ge 2\end{array}$$

For all $p,q\in \mathbb{R}$ with $p^2-4q<0$, the antiderivatives of $\frac{Ax+B}{(x^2+px+q{)}^m}$ are given by

$$\int \frac{Ax+B}{(x^2+px+q{)}^m}\mathrm{dx}=\{\begin{array}{l}\frac{A}{2}\ln (x^2+px+q)+\left(B-\frac{Ap}{2}\right)\int \frac{\mathrm{dx}}{x^2+px+q}+C\text{if }m=1\\ \\ -\frac{A}{2(m-1)(x^2+px+q)}+\left(B-\frac{Ap}{2}\right)\int \frac{\mathrm{dx}}{(x^2+px+q{)}^{m-1}}+C\text{if }m\ge 2\end{array}$$

PROOF

TODO

Algorithm: Antidifferentiation of Rational Functions

We are given the following real rational function:

$$f(x)=\frac{P(x)}{Q(x)}$$

If $\mathrm{deg}P<\mathrm{deg}Q$, then find the partial fraction decomposition of $f$ and use the antiderivatives of the elementary rational functions:

$$\int f(x)\mathrm{dx}=\int \sum _{i=1}\sum _{j=1}^{m_{l_i}}\frac{a_{ji}(x)}{l_i(x{)}^j}+\sum _{i=1}\sum _{j=1}^{m_{q_i}}\frac{b_{ji}(x)}{q_i(x{)}^j}\mathrm{dx}=\sum _{i=1}\sum _{j=1}^{m_{l_i}}\int \frac{a_{ji}(x)}{l_i(x{)}^j}\mathrm{dx}+\sum _{i=1}\sum _{j=1}^{m_{q_i}}\int \frac{b_{ji}(x)}{q_i(x{)}^j}\mathrm{dx}$$

If $\mathrm{deg}P\ge \mathrm{deg}Q$, then divide $P$ by $Q$ to find $f(x)=\frac{M(x)}{Q(x)}+R(x)$. Find the find the partial fraction decomposition of $\frac{M(x)}{Q(x)}$ and use the antiderivatives of the elementary rational functions:

$$\int f(x)\mathrm{dx}=\int \frac{M(x)}{Q(x)}+R(x)\mathrm{dx}=\int \frac{M(x)}{Q(x)}\mathrm{dx}+\int R(x)\mathrm{dx}$$

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