Odd and even functions

A function $𝑓 : 𝑉 \to 𝑊$ between appropriate modules (in particular we could take $𝑉 = 𝑊 = ℝ$ ) is said to be even if it symmetric about 0, whereas it is said to be odd if it is antisymmetric about 0, i.e.

𝑓 i s e v e n ⟺ 𝑓 (𝑥) = 𝑓 (- 𝑥) 𝑓 i s o d d ⟺ 𝑓 (𝑥) = - 𝑓 (- 𝑥)

The names come from the fact that for $𝑓 (𝑥) = 𝑥 𝑛$ , odd $𝑛$ is an odd function and even $𝑛$ is even.

From an abstract perspective, this is naturally seen as the decomposition of the group representation of $S 2$ on $𝑉 𝑊$ into homogenous components of the trivial and alternating characters.

Properties

Addition

Two functions of the same parity add or divide to a function of that parity. Functions of different parity add to a function of no parity.¹

Multiplication

The properties of odd and even functions under multiplication are analogous to those of integers under addition:¹

$o d d 𝑓 \times o d d 𝑔 = e v e n 𝑓 𝑔$
$e v e n 𝑓 \times e v e n 𝑔 = e v e n 𝑓 𝑔$
$o d d 𝑓 \times e v e n 𝑔 = o d d 𝑓 𝑔$

Integration properties

The properties of odd and even functions can be used to greatly simplify integration.¹ For an odd function,

\int 𝑎 - 𝑎 o d d 𝑓 (𝑥) 𝑑 𝑥 = 0

whereas for an even

\int 𝑎 - 𝑎 e v e n 𝑓 (𝑥) 𝑑 𝑥 = 2 \int 𝑎 0 e v e n 𝑓 (𝑥) 𝑑 𝑥

Series

It follows that for any series with linearly independent terms, such as a Fourier series or Taylor series, every term will be odd for an odd function or even or an even function. In particular see Properties.

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2017. Elementary differential equations and boundary value problems, pp. 482ff. (§10.4) ↩ ↩² ↩³