Periodic function

In mathematics one speaks of periodicity when repeating the values ​​of a function or sequence at regular intervals. The function or result is called periodically the distances between the occurrence of the same feature value period.

Furthermore period is used as a simplification term in the representation of rational numbers in the decimal system.

  • 2.1 Definitions
  • 2.2 Features
  • 2.3 Periodic digits

Real functions

Definition, properties of the set of periods and examples

Is a real number in a period of a defined function, if the following applies:

The function is periodic, if it allows at least one period. Then we can also say is " periodically ". For the period, the following properties apply:

  • Is a period of, it is also a period of;
  • Are and two periods of, it is also a period of.

Most are more interested in the smallest positive period. This exists for each non-constant continuous periodic function. ( A constant function is periodic with any period not equal to 0 )

When is a smallest positive period, the period of the multiple of. In the other case the amount of the periods is in tight.

The best-known periodic functions are the trigonometric functions, especially the sinus, which performs an ever constant oscillation between -1 and 1, at a distance of 2π ( π is the mathematical constant pi) is repeated.

Periodic functions as functions on the circumference

It is the unit circle. You can periodic functions with period identified with functions: One function corresponds to the - periodic function

Properties of functions such as boundedness, continuity or differentiability transferred respectively to the other point of view.

Fourier series corresponding Laurent series.

Follow

Definitions

A sequence is periodically if there is a natural number, such that for all. ie, a period of the sequence. When speaking of the period of a sequence, the smallest period is meant; all other periods are then multiples of this smallest period.

Finally, a sequence is generally periodically, if there are natural numbers, and so applies to all. A finite part of the result is thus freely, and from index to repeat the sequence values ​​.

Properties

If a sequence is defined recursively, that is, by a fixed function, and it takes only finitely many values ​​, so they will eventually periodic.

Example: Let and = ( 2 mod 100) for ≥ 0

Clearly, the last two digits of the decimal representation of the number of formed. This episode begins with the values ​​:

And in the following, the values ​​repeat.

Periodic sequences of digits

It should be a fixed natural number. Are and natural numbers, then the sequence of decimal places of the -adic representation of finally according to the above principle periodically because it is iteratively determined by the residues in the long division, and these radicals can only assume a finite number of values ​​.

Periodic functions on real vector spaces

It should be a - dimensional real vector space, for example. A period of a continuous, real - or complex-valued function or a ( open, coherent ) part of is a vector such that

  • The definition range is invariant under translation with, i.e.
  • Applies in all cases.

The set of all periods of is a closed subgroup of. Any such subgroup is the direct sum of a subspace of and a discrete subgroup; the latter can be described as the set of integer linear combinations of a set of linearly independent vectors.

Applying this theory to the two-dimensional real vector space and consider only holomorphic functions, so there are the following cases:

  • : Is not periodic.
  • : Is an ordinary periodic function; For example, the exponential function is periodic with period.
  • Contains a non-trivial real subspace: a holomorphic function which is constant along a straight line, is a total constant.
  • : Has two real linearly independent periods. Is on the whole plane meromorphic, it is called an elliptic function.

Periodic breaks

→ Main article: decimal there especially the section decimal expansion

In the decimal is meant by a periodic break a number representation as 0.31 = 0.3131313131 .... = 31 /99. That is, the decimal representation does not break off and from a certain point, repeated again and again the same sequence of numbers (period). Its length is called the period length.

Closed set Meromorphic function
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