Symmetries - Describing Functions

Mathematical Writing - Vivaldi Franco 2014

Symmetries
Describing Functions

Real functions may have symmetries, expressing invariance with respect to changes of the argument. A function $$f$$ is even or odd, respectively, if

$$ \forall x\in \mathbb {R},\,\,f(-x)=f(x) \qquad \text{ or }\qquad \forall x\in \mathbb {R},\,\,f(-x)=-f(x). $$

So the cosine is even, the sine is odd, the exponential is neither even nor odd, and the zero function is both even and odd. The property of being even or odd has a geometrical meaning: graphs of even functions have a mirror symmetry with respect to the ordinate axis, while those of odd functions are symmetrical with respect to the origin (see Fig. 5.1). There is an easy way of constructing even/odd functions. For any real function $$g$$, the function $$x\mapsto g(x)+g(-x)$$ is even, and so is the function $$g(f(x))$$ for any even function $$f$$. So the function $$x\mapsto g(|x|)$$ is even. To construct odd functions, we first define the sign function:

$$\begin{aligned} \text{ sign }(x)={\left\{ \begin{array}{ll} +1&{} \mathrm{if}\,x>0\\ 0&{} \mathrm{if}\,x=0\\ -1&{} \mathrm{if}\,x<0\\ \end{array}\right. } \qquad x\in \mathbb {R}. \end{aligned}$$

(5.3)

The sign function is odd: $$\text{ sign }(-x)=-\text{ sign }(x)$$. Then, for any real function $$g$$, the function $$x\mapsto \text{ sign }(x)\,g(|x|)$$ is odd, as easily verified. This construct ensures that our function vanishes at the origin, which is a property of all odd functions.

Fig. 5.1

An odd function

Next we turn to translational symmetry, which is called periodicity. A function $$f$$ is periodic with period $$T$$ if

$$\begin{aligned} \forall x\in \mathbb {R},\,\,f(x+T)=f(x) \end{aligned}$$

(5.4)

for some non-zero real number $$T$$ (see Fig. 5.2).

Fig. 5.2

A periodic function

For instance, the sine function is periodic with period $$T= 2{\uppi }$$. If a function is periodic with period $$T$$, then it is also periodic with period $$2T$$, $$3T$$, etc. For this reason one normally requires the period $$T$$ to be the smallest positive real number for which (5.4) is satisfied. To emphasise this point we use the terms minimal or fundamental period.

If we say that a function $$f$$ is periodic—without reference to a specific period—then the existence of the period must be required explicitly:

$$\begin{aligned} \exists T\in \mathbb {R}^+,\,\,\forall x\in \mathbb {R},\,\,f(x+T)=f(x) \end{aligned}$$

(5.5)

where $$\mathbb {R}^+$$ is the set of positive real numbers—see (2.12). Note that the period $$T$$ must be non-zero (lest this definition say nothing) and without loss of generality 1 we shall require the period to be positive. [The case of negative period is dealt with the substitution $$x\mapsto x-T$$ in (5.4).]

The presence of symmetries reduces the amount of information needed to specify a function. If a function is even or odd, knowledge of the function for non-negative values of the argument suffices to characterise it completely. Likewise, if a periodic function is known over any interval of length equal to the period, then the function is specified completely.

One should be aware that symmetries are special; a function chosen ’at random’ will have no symmetry.