# Periodic function

Illustration of a periodic function with the period ${\ displaystyle P}$
Function graph of the tangent function

In mathematics , periodic functions are a special class of functions . They have the property that their function values ​​are repeated at regular intervals. The distances between the occurrence of the same function values are period mentioned. Simple examples are sine and cosine functions . So that functions with gaps in the definition range, such as B. the tangent function, to which periodic functions can be counted, one allows domains of definition with periodic gaps. A periodic function does not only have a period, because every multiple of a period is also a period. Example: The sine function is not only -periodic, but also -periodic ... When one speaks of period, one usually means the smallest possible positive period. However, there are periodic functions that have no smallest period. Example: Every constant function defined on has any number as a period. ${\ displaystyle 2 \ pi}$${\ displaystyle 4 \ pi}$${\ displaystyle \ mathbb {R}}$

Periodic functions occur naturally in physics to describe mechanical , electrical or acoustic oscillation processes . That is why a period is often referred to as (English: T ime). ${\ displaystyle T}$

Since a periodic function is known if its course within a period is known, non-trigonometric periodic functions are usually defined in a basic interval and then continued periodically.

Just as many real functions can be developed in power series, one can, under certain conditions, develop a periodic function as a series of sine and cosine functions: see Fourier series .

Periodic sequences can be understood as special cases of the periodic functions.

Functions that are not periodic are sometimes referred to as aperiodic to emphasize this .

## Real periodic functions

Periodic function
above: . (blue), below: subset of (blue), purple: period${\ displaystyle {\ mathcal {D}} _ {f} = \ mathbb {R}}$
${\ displaystyle {\ mathcal {D}} _ {f}}$${\ displaystyle \ mathbb {R}}$

### definition

A real number is a period of a function defined in , if for every out : ${\ displaystyle T}$${\ displaystyle {\ mathcal {D}} _ {f} \ subseteq \ mathbb {R}}$ ${\ displaystyle x}$${\ displaystyle {\ mathcal {D}} _ {f}}$

• ${\ displaystyle x + T}$is in and${\ displaystyle {\ mathcal {D}} _ {f}}$
• ${\ displaystyle f (x + T) = f (x) \ ;.}$

The function is periodic if it allows at least one period . One then also says, be " -periodic". ${\ displaystyle f}$${\ displaystyle T \ neq 0}$${\ displaystyle f}$${\ displaystyle T}$

For , which is often the case, the first quality is always fulfilled. ${\ displaystyle {\ mathcal {D}} _ {f} = \ mathbb {R}}$

### Characteristics of the periods

The following properties apply to the period:

• If there is a period of , then there is also a period of ;${\ displaystyle T}$${\ displaystyle f}$${\ displaystyle -T}$${\ displaystyle f}$
• Are and two periods of , is also having a period of .${\ displaystyle T_ {1}}$${\ displaystyle T_ {2}}$${\ displaystyle f}$${\ displaystyle k_ {1} T_ {1} + k_ {2} T_ {2}}$${\ displaystyle k_ {1}, k_ {2} \ in \ mathbb {Z}}$${\ displaystyle f}$

Usually one is interested in the smallest positive period . This exists for every non- constant continuous periodic function. (A constant function is periodic with any period other than 0.) If has a smallest positive period, then the periods of are multiples of . In the other case the set of periods is dense in . ${\ displaystyle f}$${\ displaystyle f}$${\ displaystyle T}$${\ displaystyle f}$ ${\ displaystyle \ mathbb {R}}$

### Examples

#### Trigonometric functions

The standard examples of periodic functions are the trigonometric functions . For example, the fully defined sine function is periodic. Their function values ​​are repeated at a distance of ( is the circle number Pi); so she has the period . The tangent function with the domain is also a trigonometric function; it has the period and not , as though they quotient of two can be represented -periodischer functions . ${\ displaystyle \ mathbb {R}}$${\ displaystyle 2 \ pi}$${\ displaystyle \ pi}$${\ displaystyle 2 \ pi}$
${\ displaystyle \ mathbb {R} \ setminus \ {k \ pi + {\ tfrac {\ pi} {2}} \; | \; k \ in \ mathbb {Z} \}}$${\ displaystyle \ pi}$${\ displaystyle 2 \ pi}$${\ displaystyle 2 \ pi}$${\ displaystyle \ tan x = {\ tfrac {\ sin x} {\ cos x}}}$

Sum of cos and sin functions

#### Sum of cos and sin functions

Sums of cos and sin functions with a common (not necessarily the smallest) period are again periodic. (In the picture is the common period .) This property of the cos and sin functions is the basis of the Fourier series. If two functions do not have a common period, the sum is not periodic. Example: is not periodic. ${\ displaystyle T}$${\ displaystyle 2 \ pi}$${\ displaystyle f (x) = \ sin x + \ sin (\ pi x)}$

### Periodic continuation

In the example for the definition, a function given on a half-open interval was continued in the upper part of the picture by simply shifting it by integer multiples of to a periodic function of the period . This type is called a direct periodic continuation, as opposed to the even and odd periodic continuation. ${\ displaystyle (a, b]}$${\ displaystyle ba}$${\ displaystyle T = ba}$

The following formal definition also provides a way of evaluating a periodically continued function with a computer, since the rounding function used is implemented directly or indirectly in many mathematical systems.

Definition
If a function is given on the interval with , then the function with ${\ displaystyle f_ {0}}$${\ displaystyle [a, b]}$${\ displaystyle f_ {0} (a) = f_ {0} (b)}$${\ displaystyle f}$

${\ displaystyle f (x) = f_ {0} {\ big (} x- \ left \ lfloor {\ frac {xa} {T}} \ right \ rfloor \ cdot T {\ big)}, \; x \ in \ mathbb {R}}$

the (direct) periodic continuation of on whole and its period. is the rounding function . The use of the rounding function ensures that the function is only evaluated for x values ​​from its definition range (see figure). ${\ displaystyle f_ {0}}$${\ displaystyle \ mathbb {R}}$${\ displaystyle \; T = ba \;}$
${\ displaystyle \ lfloor \ cdot \ rfloor}$${\ displaystyle f_ {0}}$

Periodic continuation of a parabolic arch

Example: Periodic continuation of the parabolic arc with the period . The function value at the point (e.g.)${\ displaystyle f_ {0} (x) = (x-1) (4-x), \; a = 1, b = 4}$${\ displaystyle \; T = ba = 3 \;}$${\ displaystyle x = 8}$

${\ displaystyle f (8) = f_ {0} {\ big (} 8- \ lfloor {\ frac {7} {3}} \ rfloor \ cdot 3 {\ big)} = f_ {0} (8-2 \ cdot 3) = f_ {0} (2) = 2 \ ;.}$

Since periodic functions are often developed in Fourier series and an even / odd periodic function can only be represented with cosine / sine terms, the following continuations are of particular interest:

Periodic continuation of the function in the pink area:
top: odd, bottom: even

Odd continued:
In this case, you go from one to the interval defined function with out. In a first step the function is continued by mirroring at the zero point to the interval : ${\ displaystyle [0, b]}$${\ displaystyle f_ {0}}$${\ displaystyle f_ {0} (0) = f_ {0} (b) = 0}$${\ displaystyle [-b, 0]}$

${\ displaystyle f_ {u} (x) = {\ begin {cases} \ quad f_ {0} (x) \;, \ \ quad 0 \ leq x \ leq b \\ - f_ {0} (- x) \;, \; - b \ leq x <0 \. \ end {cases}}}$

The function defined on the interval is now continued (as described above) directly periodically. This creates a defined, odd periodic function of the period . ${\ displaystyle [-b, b]}$${\ displaystyle f_ {u}}$${\ displaystyle \ mathbb {R}}$${\ displaystyle f}$${\ displaystyle T = 2b}$

Straight continuation:
The analogous procedure with

${\ displaystyle f_ {g} (x) = {\ begin {cases} f_ {0} (x), \ quad \ \ 0 \ leq x \ leq b \\ f_ {0} (- x), \; - b \ leq x <0 \ end {cases}}}$

returns an even periodic function of the period . ${\ displaystyle T = 2b}$

### Fourier series: example

The Fourier series of a periodic odd function has the form with ${\ displaystyle 2 \ pi}$${\ displaystyle f}$${\ displaystyle \; \ sum _ {k = 1} ^ {\ infty} b_ {k} \ sin \ left (kt \ right) \;}$

${\ displaystyle b_ {k} = {\ frac {1} {\ pi}} \ int _ {- \ pi} ^ {\ pi} f (t) \ cdot \ sin \ left (kt \ right) \ mathrm { d} t \ quad {\ text {for}} k \ geq 1 \ ;.}$
Fourier series: various partial sums (blue)

The goal of a Fourier series expansion is the approximation of a periodic function on (completely!) By sums of simple periodic functions. In the ideal case, the Fourier series represents the given function on . (A power series expansion approximates a function that is not a polynomial with its partial sums on a restricted (!) Interval through polynomials .) ${\ displaystyle \ mathbb {R}}$${\ displaystyle \ mathbb {R}}$

In the picture one on the interval given function (two line segments, red) un just a -periodic function continued and then developed into a Fourier series (only with sin-terms). You can see how good / bad partial sums of the Fourier series (lengths n = 3,6,12) approximate the function . While is discontinuous (it has jumps), the partial sums as sums of sin terms are all continuous. ${\ displaystyle [0, \ pi]}$${\ displaystyle f_ {0}}$${\ displaystyle 2 \ pi}$${\ displaystyle f}$${\ displaystyle f_ {0}}$${\ displaystyle f_ {0}}$

In the example is

${\ displaystyle f_ {0} (x) = {\ begin {cases} x \;, \ \ quad 0 \ leq x \ leq {\ frac {\ pi} {2}} \\ 2.5 \;, \; { \ frac {\ pi} {2}}

and the partial sum for n = 3:

${\ displaystyle \; \ sum _ {k = 1} ^ {3} b_ {k} \ sin \ left (kt \ right) = {\ frac {1} {\ pi}} {\ big (} 7 \ ​​sin x + ({\ frac {\ pi} {2}} - 5) \ sin 2x + {\ frac {13} {9}} \ sin 3x {\ big)} \.}$

## More general definition

The concept of the periodic function is not limited to real functions. It can be defined more generally for functions whose source set an addition is declared.

So be an (additive) semigroup , a set and a function. Does a with ${\ displaystyle G}$${\ displaystyle M}$${\ displaystyle f \ colon G \ to M}$${\ displaystyle T \ in G}$

${\ displaystyle f (g + T) = f (g)}$

for all , then the function is called periodic with period . ${\ displaystyle g \ in G}$${\ displaystyle f}$${\ displaystyle T}$

## Examples

### Periodic episodes

Since a real sequence is a function from the natural numbers into the real numbers , the concept of the periodic sequence can be understood as a special case of a periodic function. A sequence is said to be periodic if there is one such that equality holds for all . It was used here that the set of natural numbers is a semigroup. ${\ displaystyle (a_ {n}) _ {n \ in \ mathbb {N}}}$ ${\ displaystyle \ mathbb {N}}$${\ displaystyle \ mathbb {R}}$${\ displaystyle T}$${\ displaystyle n \ in \ mathbb {N}}$${\ displaystyle a_ {n + T} = a_ {n}}$

### Exponential function

The (complex) exponential function with is a -periodic function. This property is only evident in the case of the exponential function with a complex domain. You can prove it with Euler's formula . ${\ displaystyle \ exp \ colon \ mathbb {C} \ to \ mathbb {C}}$${\ displaystyle x \ mapsto \ mathrm {e} ^ {x}}$${\ displaystyle 2 \ pi \ mathrm {i}}$

### Periodic functions as functions on the circular line

It is the unit circle . One can identify periodic functions on with period with functions on : A function on corresponds to the -periodic function ${\ displaystyle S ^ {1} = \ {z \ in \ mathbb {C} \ mid | z | = 1 \}}$${\ displaystyle \ mathbb {R}}$${\ displaystyle T}$${\ displaystyle S ^ {1}}$${\ displaystyle f}$${\ displaystyle S ^ {1}}$${\ displaystyle T}$

${\ displaystyle x \ mapsto f (\ mathrm {e} ^ {2 \ pi \ mathrm {i} x / T})}$.

Here is a function on the unit circle, i.e. a subset of the complex numbers . Features of the functions such as limitation, continuity or differentiability are transferred to the other perspective. ${\ displaystyle \ textstyle x \ mapsto f (\ mathrm {e} ^ {2 \ pi \ mathrm {i} x / T})}$

For example, Fourier series under this map correspond to Laurent series . ${\ displaystyle \ textstyle \ sum _ {n \ in \ mathbb {Z}} c_ {n} \ mathrm {e} ^ {\ mathrm {i} n \ omega t}}$ ${\ displaystyle \ textstyle \ sum _ {n \ in \ mathbb {Z}} c_ {n} z ^ {n}}$

## Periodic functions on real vector spaces

Let it be a -dimensional real vector space, e.g. B. . A period of a continuous, real or complex-valued function on or an ( open , connected ) part of is a vector such that ${\ displaystyle V}$${\ displaystyle n}$${\ displaystyle \ mathbb {R} ^ {n}}$${\ displaystyle f}$${\ displaystyle V}$${\ displaystyle D}$${\ displaystyle V}$${\ displaystyle \ gamma \ in V}$

• is the domain of definition of invariant under the translation with , d. H.${\ displaystyle D}$${\ displaystyle f}$${\ displaystyle \ gamma}$${\ displaystyle x \ in D \; \ Leftrightarrow \; x + \ gamma \ in D}$
• for all true: .${\ displaystyle x \ in D}$${\ displaystyle f (x + \ gamma) = f (x)}$

The set of all periods of is a closed subgroup of . Each 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. ${\ displaystyle \ Gamma}$${\ displaystyle f}$ ${\ displaystyle V}$${\ displaystyle V}$

If one applies this theory to the real two-dimensional vector space and only considers holomorphic functions , then there are the following cases: ${\ displaystyle V = \ mathbb {C}}$${\ displaystyle f}$

• ${\ displaystyle \ Gamma = \ {0 \}}$: is not periodic.${\ displaystyle f}$
• ${\ displaystyle \ Gamma = \ mathbb {Z} \ cdot \ gamma}$: is an ordinary periodic function; for example the exponential function is periodic with period .${\ displaystyle f}$${\ displaystyle \ gamma = 2 \ pi \ mathrm {i}}$
• ${\ displaystyle \ Gamma}$ contains a nontrivial real subspace: A holomorphic function that is constant along a straight line is overall constant.
• ${\ displaystyle \ Gamma = \ mathbb {Z} \ cdot \ gamma _ {1} + \ mathbb {Z} \ cdot \ gamma _ {2}}$: has two real linearly independent periods. If the whole plane is meromorphic , one speaks of an elliptic function .${\ displaystyle f}$${\ displaystyle f}$