often in the study of natural phenomena, chemical and physical properties of various substances, as well as in solving complex technical problems encountered with the processes characteristic feature is the frequency, then there is a tendency to repeat after a certain period of time.For a description and a graphic image such cyclicality in science there is a special kind of function - a periodic function.
most simple and clear example to all - treatment of our planet around the Sun, in which changes all the time distance between them subject to the annual cycle.Similarly, he returns to his seat, having made a full turn, the blade of the turbine.All these processes can be described by a mathematical value as a periodic function.By and large, our whole world is cyclical.And that means that a periodic function takes an important place in the system of human origin.
need for mathematics in number theory, topology, differential equations and exact geometrical calculations led to the emergence in the nineteenth century, a new category of functions with unusual properties.They were 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, in studying the effects of various vibrational wave physics.
In various mathematical textbooks are different definitions of a periodic function.However, regardless of these differences in the formulation, they are all equivalent as they describe the same property of the function.The simplest and most obvious may be the following definition.Functions that the amounts are not subject to change, if we add to their argument a number other than zero, the so-called period of the function denoted by the letter T are called periodic.What does this mean in practice?
example, a simple function of the form: y = f (x) will become a periodic if X has a certain value of the period (T).From this definition it follows that if the numerical value of the function having a period (T) is defined in one of the points (x), then it also becomes a known value at x T + x - T. The important point here is that whenT is zero function becomes an identity.A periodic function can have an infinite number of different periods.In the bulk of cases among the positive values of T exists between the lowest numerical indicator.It is called the fundamental period.And all the other values of T it is always multiples.This is another interesting and very important for the different fields property.
Schedule periodic function also has several features.For example, if T is the basic period of the expression: y = f (x), then by plotting this function, just enough to build a branch in one of the periods of the period length, and then move it along the x axis for the following values: ± T, ± 2T, ± 3T and so on.In conclusion, it should be noted that not all of a periodic function is the basic period.A classic example of this is the German mathematician Dirichlet function of the following form: y = d (x).
