Periodic function

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)