Often, when studying natural phenomena, chemical and physical properties of various substances, as well as solving complex technical problems, one has to deal with processes whose characteristic feature is periodicity, that is, a tendency to repeat after a certain period of time. To describe and graphically depict such cyclicity in science, there is a special type of function - a periodic function.
The simplest and most understandable example is the revolution of our planet around the Sun, in which the distance between them, which is constantly changing, is subject to annual cycles. In the same way, the turbine blade returns to its place, having made a full revolution. 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 also occupies an important place in the human coordinate system.
The need of mathematics for 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 became 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 oscillatory effects in wave physics.
Different mathematical textbooks give different definitions of a periodic function. However, regardless of these discrepancies in formulations, they are all equivalent, since they describe the same properties of the function. The most simple and understandable may be the following definition. Functions whose numerical indicators do not change if a certain number other than zero is added to their argument, the so-called period of the function, denoted by the letter T, are called periodic. What does it all mean in practice?
For example, a simple function of the form: y=f(x) will become periodic if X has a certain period value (T). It follows from this definition that if the numerical value of a function with 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 when T equal to zero, the function turns into an identity. A periodic function can have an infinite number of different periods. ATIn the majority of cases, among the positive values of T, there is a period with the smallest numerical indicator. 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.
The graph of a periodic function also has several features. For example, if T is the main period of the expression: y \u003d f (x), then when plotting this function, it is enough just to plot a branch on one of the intervals of the period length, and then move 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 main period. A classic example of this is the following function of the German mathematician Dirichlet: y=d(x).
