Improper Rational Function

Summary: A rational function whose numerator has degree at least the degree of the denominator; Euler shows (§38) it can always be split into a polynomial part plus a proper rational remainder.

Sources: chapter2

Last updated: 2026-04-23

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Definition

By analogy with improper fractions in arithmetic (source: chapter2, §38):

• A proper rational function is one in which the degree of the numerator is less than the degree of the denominator.
• An improper rational function is one in which the degree of the numerator is greater than or equal to the degree of the denominator.

Reduction by polynomial division (§38)

If a rational function $M/N$ is improper, divide $M$ by $N$ “in the usual way” until the next quotient term would have negative degree. At that point:

$\frac{M}{N}=(\text{polynomial part})+\frac{M^{\prime }}{N},$

where $M^{\prime }$ has smaller degree than $N$, so that $M^{\prime }/N$ is proper.

Euler’s example

$\frac{1+z^4}{1+z^2}=z^2-1+\frac{2}{1+z^2}(\text{source: chapter2, §38}).$

Why the split matters

partial-fraction-decomposition applies directly only to proper rational functions. For an improper rational function one first extracts the polynomial part, then decomposes the proper remainder. Euler notes (source: chapter2, §46) that the polynomial part can equivalently be added before or after the partial fractions are computed, since the same partial fraction arises from an individual linear factor of $N$ whether one uses $M$ or $M$ plus a multiple of $N$.

Related pages

• partial-fraction-decomposition
• classification-of-functions
• chapter-2-on-the-transformation-of-functions