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A function having a period (cf. [[Period of a function|Period of a function]]).
 
A function having a period (cf. [[Period of a function|Period of a function]]).
  
Let a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072170/p0721701.png" /> be defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072170/p0721702.png" /> and have period <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072170/p0721703.png" />. To obtain the graph of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072170/p0721704.png" /> it is sufficient to have the graph of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072170/p0721705.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072170/p0721706.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072170/p0721707.png" /> is a certain number, and shift it along <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072170/p0721708.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072170/p0721709.png" />. If a periodic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072170/p07217010.png" /> with period <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072170/p07217011.png" /> has a finite derivative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072170/p07217012.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072170/p07217013.png" /> is a periodic function with the same period. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072170/p07217014.png" /> be integrable over any segment and have period <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072170/p07217015.png" />. The indefinite integral <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072170/p07217016.png" /> has period <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072170/p07217017.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072170/p07217018.png" />, otherwise it is non-periodic, such as for example for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072170/p07217019.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072170/p07217020.png" />.
+
Let a function $  f $
 +
be defined on $  X \subset  \mathbf R $
 +
and have period $  T $.  
 +
To obtain the graph of $  f $
 +
it is sufficient to have the graph of $  f $
 +
on $  [ a, a+ T] \cap X $,  
 +
where $  a $
 +
is a certain number, and shift it along $  \mathbf R $
 +
over $  \pm  T, \pm  2T ,\dots $.  
 +
If a periodic function $  f $
 +
with period $  T $
 +
has a finite derivative $  f ^ { \prime } $,  
 +
then $  f ^ { \prime } $
 +
is a periodic function with the same period. Let $  f $
 +
be integrable over any segment and have period $  T $.  
 +
The indefinite integral $  F( x)= \int _ {0}  ^ {x} f( t)  dt $
 +
has period $  T $
 +
if  $  \int _ {0}  ^ {T} f( t)  dt = 0 $,  
 +
otherwise it is non-periodic, such as for example for $  f( x) = \cos  x+ 1 $,  
 +
where $  F( x) = \sin  x + x $.
  
 
''A.A. Konyushkov''
 
''A.A. Konyushkov''
  
A periodic function of a complex variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072170/p07217021.png" /> is a single-valued analytic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072170/p07217022.png" /> having only isolated singular points (cf. [[Singular point|Singular point]]) in the complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072170/p07217023.png" />-plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072170/p07217024.png" /> and for which there exists a complex number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072170/p07217025.png" />, called a period of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072170/p07217026.png" />, such that
+
A periodic function of a complex variable $  z $
 +
is a single-valued analytic function $  f( z) $
 +
having only isolated singular points (cf. [[Singular point|Singular point]]) in the complex $  z $-
 +
plane $  \mathbf C $
 +
and for which there exists a complex number $  p \neq 0 $,  
 +
called a period of the function $  f( z) $,  
 +
such that
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072170/p07217027.png" /></td> </tr></table>
+
$$
 +
f( z+ p)  = f( z),\ \
 +
z \in \mathbf C .
 +
$$
  
Any linear combination of the periods of a given periodic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072170/p07217028.png" /> with integer coefficients is also a period of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072170/p07217029.png" />. The set of all periods of a given periodic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072170/p07217030.png" /> constitutes a discrete Abelian group under addition, called the period group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072170/p07217031.png" />. If the basis of this group consists of one unique basic, or primitive, period <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072170/p07217032.png" />, i.e. if any period <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072170/p07217033.png" /> is an integer multiple of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072170/p07217034.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072170/p07217035.png" /> is called a [[Simply-periodic function|simply-periodic function]]. In the case of a basis consisting of two basic periods <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072170/p07217036.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072170/p07217037.png" />, one has a [[Double-periodic function|double-periodic function]]. If the periodic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072170/p07217038.png" /> is not a constant, then a basis of its period group cannot consist of more than two basic independent periods (Jacobi's theorem).
+
Any linear combination of the periods of a given periodic function $  f( z) $
 +
with integer coefficients is also a period of $  f( z) $.  
 +
The set of all periods of a given periodic function $  f( z) \neq \textrm{ const } $
 +
constitutes a discrete Abelian group under addition, called the period group of $  f( z) $.  
 +
If the basis of this group consists of one unique basic, or primitive, period $  2 \omega = 2 \omega _ {1} \neq 0 $,  
 +
i.e. if any period p $
 +
is an integer multiple of $  2 \omega $,  
 +
then $  f( z) $
 +
is called a [[Simply-periodic function|simply-periodic function]]. In the case of a basis consisting of two basic periods $  2 \omega _ {1} , 2 \omega _ {3} $,
 +
$  \mathop{\rm Im} ( \omega _ {1} / \omega _ {3} ) \neq 0 $,  
 +
one has a [[Double-periodic function|double-periodic function]]. If the periodic function $  f( z) $
 +
is not a constant, then a basis of its period group cannot consist of more than two basic independent periods (Jacobi's theorem).
  
 
Any strip of the form
 
Any strip of the form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072170/p07217039.png" /></td> </tr></table>
+
$$
 +
\{ {z = ( \tau e ^ {i \alpha } + t) 2 \omega } : {- \infty < \tau < \infty , 0 \leq  t < 1 ,\
 +
0 < \alpha \leq  \pi /2 } \}
 +
,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072170/p07217040.png" /> is one of the basic periods of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072170/p07217041.png" /> or is congruent to it, is called a period strip of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072170/p07217042.png" />; one usually takes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072170/p07217043.png" />, i.e. one considers a period strip with sides perpendicular to the basic period <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072170/p07217044.png" />. In each period strip, a periodic function takes any of its values and moreover equally often.
+
where $  2 \omega $
 +
is one of the basic periods of $  f( z) $
 +
or is congruent to it, is called a period strip of $  f( z) $;  
 +
one usually takes $  \alpha = \pi /2 $,  
 +
i.e. one considers a period strip with sides perpendicular to the basic period $  2 \omega $.  
 +
In each period strip, a periodic function takes any of its values and moreover equally often.
  
Any entire periodic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072170/p07217045.png" /> can be expanded into a Fourier series throughout <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072170/p07217046.png" />:
+
Any entire periodic function $  f( z) $
 +
can be expanded into a Fourier series throughout $  \mathbf C $:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072170/p07217047.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
+
$$ \tag{* }
 +
f( z)  = \sum _ {k=- \infty } ^  \infty  a _ {k} e ^ {\pi ikz/ \omega } ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072170/p07217048.png" /></td> </tr></table>
+
$$
 +
a _ {k}  =
 +
\frac{1}{2 \omega }
 +
\int\limits _ {- \infty } ^  \infty  f( t) e ^ {- \pi ikt/ \omega }  dt,
 +
$$
  
which converges uniformly and absolutely on the straight line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072170/p07217049.png" /> and, in general, on any arbitrarily wide strip of finite width parallel to that line. The case when an entire periodic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072170/p07217050.png" /> tends to a certain finite or infinite limit at each of the two ends of the period strip is characterized by the fact that the series (*) contains only a finite number of terms, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072170/p07217051.png" /> should be a [[Trigonometric polynomial|trigonometric polynomial]].
+
which converges uniformly and absolutely on the straight line $  \{ {z = t \omega } : {- \infty < t < \infty } \} $
 +
and, in general, on any arbitrarily wide strip of finite width parallel to that line. The case when an entire periodic function $  f( z) $
 +
tends to a certain finite or infinite limit at each of the two ends of the period strip is characterized by the fact that the series (*) contains only a finite number of terms, i.e. $  f( z) $
 +
should be a [[Trigonometric polynomial|trigonometric polynomial]].
  
Any meromorphic periodic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072170/p07217052.png" /> throughout <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072170/p07217053.png" /> with basic period <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072170/p07217054.png" /> can be represented as the quotient of two entire periodic functions with the same period, i.e. as the quotient of two series of the form (*). In particular, the class of all [[Trigonometric functions|trigonometric functions]] can be described as the class of meromorphic periodic functions with period <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072170/p07217055.png" /> that in each period strip have only a finite number of poles and tend to a definite limit at each end of the period strip.
+
Any meromorphic periodic function $  f( z) $
 +
throughout $  \mathbf C $
 +
with basic period $  2 \omega $
 +
can be represented as the quotient of two entire periodic functions with the same period, i.e. as the quotient of two series of the form (*). In particular, the class of all [[Trigonometric functions|trigonometric functions]] can be described as the class of meromorphic periodic functions with period $  2 \pi $
 +
that in each period strip have only a finite number of poles and tend to a definite limit at each end of the period strip.
  
 
====References====
 
====References====
Line 33: Line 107:
  
 
====Comments====
 
====Comments====
In 1), the assertion that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072170/p07217056.png" /> has period <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072170/p07217057.png" /> means that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072170/p07217058.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072170/p07217059.png" /> implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072170/p07217060.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072170/p07217061.png" />.
+
In 1), the assertion that $  f $
 +
has period $  T $
 +
means that $  T \neq 0 $,  
 +
and $  x \in X $
 +
implies $  x \pm  T \in X $
 +
and $  f( x \pm  T) = f( x) $.
  
 
Double-periodic functions are also known as elliptic functions (cf. [[Elliptic function|Elliptic function]]).
 
Double-periodic functions are also known as elliptic functions (cf. [[Elliptic function|Elliptic function]]).

Latest revision as of 08:05, 6 June 2020


A function having a period (cf. Period of a function).

Let a function $ f $ be defined on $ X \subset \mathbf R $ and have period $ T $. To obtain the graph of $ f $ it is sufficient to have the graph of $ f $ on $ [ a, a+ T] \cap X $, where $ a $ is a certain number, and shift it along $ \mathbf R $ over $ \pm T, \pm 2T ,\dots $. If a periodic function $ f $ with period $ T $ has a finite derivative $ f ^ { \prime } $, then $ f ^ { \prime } $ is a periodic function with the same period. Let $ f $ be integrable over any segment and have period $ T $. The indefinite integral $ F( x)= \int _ {0} ^ {x} f( t) dt $ has period $ T $ if $ \int _ {0} ^ {T} f( t) dt = 0 $, otherwise it is non-periodic, such as for example for $ f( x) = \cos x+ 1 $, where $ F( x) = \sin x + x $.

A.A. Konyushkov

A periodic function of a complex variable $ z $ is a single-valued analytic function $ f( z) $ having only isolated singular points (cf. Singular point) in the complex $ z $- plane $ \mathbf C $ and for which there exists a complex number $ p \neq 0 $, called a period of the function $ f( z) $, such that

$$ f( z+ p) = f( z),\ \ z \in \mathbf C . $$

Any linear combination of the periods of a given periodic function $ f( z) $ with integer coefficients is also a period of $ f( z) $. The set of all periods of a given periodic function $ f( z) \neq \textrm{ const } $ constitutes a discrete Abelian group under addition, called the period group of $ f( z) $. If the basis of this group consists of one unique basic, or primitive, period $ 2 \omega = 2 \omega _ {1} \neq 0 $, i.e. if any period $ p $ is an integer multiple of $ 2 \omega $, then $ f( z) $ is called a simply-periodic function. In the case of a basis consisting of two basic periods $ 2 \omega _ {1} , 2 \omega _ {3} $, $ \mathop{\rm Im} ( \omega _ {1} / \omega _ {3} ) \neq 0 $, one has a double-periodic function. If the periodic function $ f( z) $ is not a constant, then a basis of its period group cannot consist of more than two basic independent periods (Jacobi's theorem).

Any strip of the form

$$ \{ {z = ( \tau e ^ {i \alpha } + t) 2 \omega } : {- \infty < \tau < \infty , 0 \leq t < 1 ,\ 0 < \alpha \leq \pi /2 } \} , $$

where $ 2 \omega $ is one of the basic periods of $ f( z) $ or is congruent to it, is called a period strip of $ f( z) $; one usually takes $ \alpha = \pi /2 $, i.e. one considers a period strip with sides perpendicular to the basic period $ 2 \omega $. In each period strip, a periodic function takes any of its values and moreover equally often.

Any entire periodic function $ f( z) $ can be expanded into a Fourier series throughout $ \mathbf C $:

$$ \tag{* } f( z) = \sum _ {k=- \infty } ^ \infty a _ {k} e ^ {\pi ikz/ \omega } , $$

$$ a _ {k} = \frac{1}{2 \omega } \int\limits _ {- \infty } ^ \infty f( t) e ^ {- \pi ikt/ \omega } dt, $$

which converges uniformly and absolutely on the straight line $ \{ {z = t \omega } : {- \infty < t < \infty } \} $ and, in general, on any arbitrarily wide strip of finite width parallel to that line. The case when an entire periodic function $ f( z) $ tends to a certain finite or infinite limit at each of the two ends of the period strip is characterized by the fact that the series (*) contains only a finite number of terms, i.e. $ f( z) $ should be a trigonometric polynomial.

Any meromorphic periodic function $ f( z) $ throughout $ \mathbf C $ with basic period $ 2 \omega $ can be represented as the quotient of two entire periodic functions with the same period, i.e. as the quotient of two series of the form (*). In particular, the class of all trigonometric functions can be described as the class of meromorphic periodic functions with period $ 2 \pi $ that in each period strip have only a finite number of poles and tend to a definite limit at each end of the period strip.

References

[1] A.I. Markushevich, "Theory of functions of a complex variable" , 2 , Chelsea (1977) (Translated from Russian)

E.D. Solomentsev

Comments

In 1), the assertion that $ f $ has period $ T $ means that $ T \neq 0 $, and $ x \in X $ implies $ x \pm T \in X $ and $ f( x \pm T) = f( x) $.

Double-periodic functions are also known as elliptic functions (cf. Elliptic function).

References

[a1] C.L. Siegel, "Topics in complex functions" , 1 , Wiley, reprint (1988)
How to Cite This Entry:
Periodic function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Periodic_function&oldid=48156
This article was adapted from an original article by A.A. Konyushkov, E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article