Differential equations MOC

Odd and even functions

A function between appropriate modules (in particular we could take ) 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.

The names come from the fact that for , odd is an odd function and even is even.

From an abstract perspective, this is naturally seen as the decomposition of the group representation of on 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

Integration properties

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

whereas for an even

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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