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A function having a period (cf. [[Period of a function|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|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|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

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

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|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|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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.I. Markushevich,   "Theory of functions of a complex variable" , '''2''' , Chelsea  (1977)  (Translated from Russian)</TD></TR></table>

''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|Elliptic function]]).

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  C.L. Siegel,   "Topics in complex functions" , '''1''' , Wiley, reprint  (1988)</TD></TR></table>