Mathematical Writing - Vivaldi Franco 2014
Symmetries
Describing Functions
Real functions may have symmetries, expressing invariance with respect to changes of the argument. A function is even or odd, respectively, if
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 , the function is even, and so is the function for any even function . So the function is even. To construct odd functions, we first define the sign function:
(5.3)
The sign function is odd: . Then, for any real function , the function 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 is periodic with period if
(5.4)
for some non-zero real number (see Fig. 5.2).
Fig. 5.2
A periodic function
For instance, the sine function is periodic with period . If a function is periodic with period , then it is also periodic with period , , etc. For this reason one normally requires the period 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 is periodic—without reference to a specific period—then the existence of the period must be required explicitly:
(5.5)
where is the set of positive real numbers—see (2.12). Note that the period 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 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.