Rational Expressions¶
Recall (doc/postdoc). Not a first course.
Job: field of fractions \(K(x)=\operatorname{Frac}(K[x])\); domain excludes poles; partial fractions for integration / residues.
Core¶
\[
\frac{p}{q},\quad p,q\in K[x],\ q\neq0;\qquad
\frac{p}{q}=\frac{p'}{q'}\iff pq'=p'q
\]
Ops: same as fractions with polynomial arithmetic. Simplify by cancelling \(\gcd(p,q)\) — domain of original expression is \(q\neq0\), not the reduced denom alone if you care about removable discontinuities.
Proper: \(\deg p<\deg q\). Else polynomial division first: \(p/q=f+r/q\).
Partial fractions (char \(0\), complete factorization of \(q\)):
\[
\frac{r}{\prod(x-a_i)^{m_i}\prod q_j^{n_j}}=\sum_{i,\ell}\frac{A_{i\ell}}{(x-a_i)^\ell}+\sum_{j,k}\frac{\text{linear}}{q_j^k}
\]
Used later for¶
- Integrals: rational function integration
- Complex analysis: residues at poles
- Laplace transforms: rational \(F(s)\)
- Algebraic geometry: rational functions on varieties
Check¶
Does simplifying \(\frac{x^2-1}{x-1}\) to \(x+1\) give equal functions \(\mathbb{R}\to\mathbb{R}\)?
No — original undefined at \(x=1\); reduced extends continuously. Equal on \(\mathbb{R}\setminus\{1\}\).