Single-valued and Multi-valued Functions

Summary: A function is single-valued if each input determines one output, and $n$-valued if it determines $n$ outputs. Euler introduces multi-valued functions as those defined by degree-$n$ polynomial equations with single-valued coefficients, and states Vieta’s relations for them.

Sources: chapter1

Last updated: 2026-04-23

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Definitions

A single-valued function is one for which, no matter what value is assigned to the variable $z$, a single value of the function is determined. … A multiple-valued function is one such that, for some value substituted for the variable $z$, the function determines several values. (source: chapter1, §10)

Euler uses capital letters $P,Q,R,S,T$ for generic single-valued functions of $z$ throughout what follows.

What is single- vs. multi-valued

• All non-irrational functions (polynomial, rational) are single-valued (source: chapter1, §10).
• All irrational functions are multi-valued, because radicals are ambiguous ($\sqrt{\dots }$ carries a $\pm$ sign).
• Transcendental functions can be single-valued, multi-valued, or even infinite-valued. Euler’s example of an infinite-valued function is the arcsine, since “there are infinitely many circular arcs with the same sine” (source: chapter1, §10).

n-valued functions by polynomial equations

The central construction (§§11-14): if $P,Q,R,S,\dots$ are single-valued functions of $z$, then the equation

$$Z^n-P{Z}^{n-1}+Q{Z}^{n-2}-R{Z}^{n-3}+S{Z}^{n-4}-\cdots =0$$

defines $Z$ as an $n$-valued function of $z$. Each choice of $z$ gives $n$ values of $Z$ — the roots of the equation.

Low-degree cases

Two-valued (§11): $Z^2-PZ+Q=0$ gives $Z=\frac{1}{2}P\pm \sqrt{\frac{1}{4}{P}^2-Q}$. Both values are real or both are complex. Their sum is $P$, their product is $Q$.

Three-valued (§12): $Z^3-P{Z}^2+QZ-R=0$. The three values are either all real, or one real and two complex. Sum = $P$, sum of pairwise products = $Q$, product = $R$.

Four-valued (§13): $Z^4-P{Z}^3+Q{Z}^2-RZ+S=0$. The four values are all real, two real and two complex, or all complex. Sum = $P$, sum of pairwise products = $Q$, sum of triple products = $R$, product = $S$.

These are Vieta’s formulas stated for coefficients that are themselves functions of $z$.

Rules for reducing to rationality and counting values

To determine how many values $Z$ has as a function of $z$, the defining equation must first be “reduced to rationality”; then the largest power of $Z$ is the count (source: chapter1, §14). If any of $P,Q,R,S,\dots$ is itself multi-valued, the total number of values of $Z$ is larger than the apparent degree.

Parity of complex roots: complex roots come in conjugate pairs. Consequences (source: chapter1, §14):

• If $n$ is odd, at least one value of $Z$ is real.
• If $n$ is even, it is possible that no value of $Z$ is real.

Multi-valued functions that “imitate” single-valued ones (§15)

If a multi-valued function always has exactly one real value among its $n$ values, it can often be treated as single-valued. Fractional powers $P^{m/n}$:

• $n$ odd, any $m$: one real value, others complex $\to$ may be treated as single-valued.
• $n$ even, $m/n$ in lowest terms: either no real value or two $\to$ two-valued.

Reciprocity and valuedness

If $y$ is a function of $z$, then $z$ is a function of $y$, but the two counts of values may differ (source: chapter1, §16). Example: $y^3=ayz-b{z}^2$ makes $y$ three-valued in $z$ and $z$ two-valued in $y$.

Related pages

• function
• classification-of-functions
• even-and-odd-functions
• vietas-formulas
• chapter-1-on-functions-in-general