Often when studying the phenomena of nature, chemical andphysical properties of various substances, as well as solving complex technical problems, one has to deal with 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 is a function of a special type - a periodic function.
The most simple and clear example for everyone is an appeal.of our planet around the Sun, in which the distance between them all the time varies according to annual cycles. Similarly, returns to its place, having made a full turn, the turbine blade. All such processes can be described by such a mathematical quantity as a periodic function. By and large, our entire world is cyclical. This means that the periodic function takes an important place in the system of human coordinates.
The need of mathematical science in number theory,topology, differential equations and exact geometric calculations led to the emergence 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 used in many branches of mathematics and other sciences. For example, when studying various vibrational effects in wave physics.
In various mathematical textbooks are givendifferent definitions of a 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, the numerical indices of which are not subject to change, if we add to their argument a certain number other than 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 if X has a certain period value (T). It follows from this definition that if the numerical value of a function having a period (T) is determined at one of the points (x), then its value also becomes known at the points x + T, x - T. The important point here is that with T equal to zero the function turns into an identity. A periodic function may have an infinite number of different periods. In the majority of cases, among 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 multiples of it. This is another interesting and very important property for various fields of science.
График периодической функции обладает тоже several features. For example, if T is the main period of the expression: y = f (x), then when plotting the graph of this function, it suffices to build a branch on one of the intervals of the period length, and then transfer it along the x axis to the following values: ± T, ± 2Т , ± 3T and so on. In conclusion, it should be noted that not every periodic function has a main period. A classic example of this is the function of the following German Dirichlet mathematician: y = d (x).
