Periodic function: general concepts

Often when studying the phenomena of nature, chemical and physical properties of various substances, as well as solving complex technical problems, one must encounter processes characterized by periodicity, that is, a tendency to repeat after a certain period of time. For the description and graphic representation of such cyclicity in science there exists a function of a special kind - a periodic function.

The simplest and most understandable example is the inversion of our planet around the Sun, in which all the time the distance that varies between them obeys the annual cycles. In the same way, the turbine blade returns to its place, having made a complete revolution. All such processes can be described by such a mathematical value as a periodic function. By and large, our whole world is cyclical. This means that the periodic function also occupies an important place in the system of human coordinates.

The need for mathematical science in number theory, topology, differential equations and precise geometric calculations led to the appearance in the nineteenth century of a new category of functions with unusual properties. They are periodic functions that take identical values at certain points as a result of complex transformations. Now they are applied in many branches of mathematics and other sciences. For example, in studying various vibrational effects in wave physics.

Different mathematical textbooks give different definitions of the periodic function. However, regardless of these discrepancies in the formulations, they are all equivalent, since they describe the same properties of the function. The following definition can be the simplest and most understandable. Functions whose numerical values are not subject to change, if we add to their argument a certain number different from zero, the so-called period of the function, denoted by the letter T, is called periodic. What does all this mean in practice?

For example, a simple function of the form: y = f (x) becomes periodic in the case when X has a definite value of period (T). From this definition it follows that if the numerical value of a function having a period (T) is defined at one of the points (x), then its value also becomes known at the points x + T, x = T. An important point here is that when T the function equal to zero becomes an identity. A periodic function can have an infinite number of different periods. In the majority of cases among the positive values of T there is a period with the smallest numerical index. It is called the main period. And all other values of T are always a multiple of it. This is another interesting and very important property for various fields of science.

The graph of a periodic function also has several singularities. For example, if T is the main period of the expression: y = f (x), then when constructing the graph of a given function, it is enough just to build a branch on one of the intervals of the length of the period, and then transfer it along the x axis to the following values: ± T, ± 2T , ± 3T and so on. In conclusion, it should be noted that not every periodic function has a base period. A classic example of this is the function of the German mathematician Dirichlet of the following form: y = d (x).