Even and Odd Functions

Summary: Euler defines even and odd functions by their behavior under $z\to -z$, and derives their multiplicative rules. The modern definitions appear here essentially unchanged.

Sources: chapter1

Last updated: 2026-04-23

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Definitions

Even function (§18): $Z$ is even in $z$ if $Z(+k)=Z(-k)$ for every $k$. Equivalently, $z$ appears everywhere with even exponent.

Odd function (§21): $Z$ is odd in $z$ if $Z(-z)=-Z(z)$. Equivalently, $z$ appears everywhere with odd exponent.

Examples

Even (source: chapter1, §18):

• Powers $z^m$ with $m$ an even integer (positive or negative).
• $z^{m/n}$ with $m$ even, $n$ odd.
• Polynomials in even powers only: $a+b{z}^2+c{z}^4+d{z}^6+\dots$
• Rational functions whose numerator and denominator are even: $\frac{a+b{z}^2+c{z}^4+\dots }{\alpha +\beta z^2+\gamma z^4+\dots }$
• Fractional-exponent variants like $a+b{z}^{2/3}+c{z}^{4/7}+\dots$, provided every exponent is an even integer divided by an odd integer.

Odd (source: chapter1, §21):

• $z,z^3,z^5,z^7,\dots ;z^{-1},z^{-3},z^{-5},\dots$
• $z^{m/n}$ with $m$ and $n$ both odd integers.
• $az+b{z}^3,az+a{z}^{-1},z^{1/3}+a{z}^{3/5}+b{z}^{-5/3}$.

Multiplicative rules

From §§22-23:

Operation	Result
even $\times$ $z$	odd
even $\times$ odd	odd
even $÷$ odd	odd
odd $÷$ even	odd
odd $\times$ odd	even
odd $÷$ odd	even
(odd)${}^2$	even
(odd)${}^3$	odd

These are just the familiar parity rules. Euler’s proofs are direct substitutions of $-z$ for $z$.

Even functions as functions of $z^2$ (§20)

An even function of $z$ can be obtained by starting with any function $Z=f(y)$ and substituting $y=z^2$. The caveat: if $f$ contains $\sqrt{y}$ or any form that disappears under the substitution, the result is not actually even. Euler’s counterexample is $y+\sqrt{ay}$, which becomes $z^2+z\sqrt{a}$ under $y=z^2$ — odd and even terms mixed.

Multi-valued even and odd functions

A multi-valued function $Z$ of $z$ is even (§19) if the defining equation has $z$ appearing only with even exponents. Examples:

• $Z^2=az{Z}^4+b{z}^2$ — but this is not purely even, since it has $az$. Euler’s actual examples are along the lines of $Z^3-a{z}^2{Z}^2+b{z}^{4}Z-c{z}^8=0$ for three-valued even.
• General template: $Z^2-PZ+Q=0$ with $P,Q$ single-valued even functions gives a two-valued even $Z$; $Z^3-P{Z}^2+QZ-R=0$ analogously gives three-valued even.

Reciprocity for odd functions (§24)

If $y$ is an odd function of $z$, then $z$ is an odd function of $y$. Example: $y=z^3$ gives $z=y^{1/3}$.

Parity from the defining equation (§25)

If $y$ is defined implicitly by an equation in $y$ and $z$ such that in every term the sum of the exponents of $y$ and $z$ has the same parity (all even, or all odd), then $y$ is an odd function of $z$. Example: $y^3+a{y}^{2}z=by{z}^2+cy+dz$ — each term has exponent-sum 3, 3, 3, 1, 1 (all odd), so $y$ is odd in $z$.

Related pages

• function
• single-valued-and-multi-valued-functions
• chapter-1-on-functions-in-general