Any algorithm or theorem to decide whether two functions are equivalent?

Question

Any algorithm or theorem to decide whether two functions that are polynomials,rationals and analytic over $\mathbb{N}$ or $\mathbb{Q}$ or $\mathbb{R}$ or $\mathbb{C}$ are equivalent ?

Answer 1

For polynomials or for rational functions the algorithm is:

$$\text{Reduce both to a canonical form. Compare canonical forms.}$$

A canonical form can be: Sum of powers of $x$ for the case of a polynomial, and for rational functions a polynomial in canonical form plus a proper fraction with numerator and denominator in canonical form, and numerator and denominator relatively prime.

We need to show that reducing to these canonical forms is algorithmic. The one for polynomials is clear: Open parentheses and reduce common terms. Alternatively, compute the (finitely many) derivatives at the origin and write Taylor series (polynomial).

For rational functions first use long division to write as polynomial plus proper fraction. Then use Euclids algorithm to compute greatest common divisor of the numerator and denominator of the proper fraction. Finally divide numerator and denominator by the greatest common divisor.

For analytic there is no terminating algorithms. Perhaps you can compare the coefficients of a power series at certain point, but there are infinitely many coefficients to compare. This is not the proof that there is no algorithm. The proof is more complicated.