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)