Analytic function

2020 Mathematics Subject Classification: Primary: 30-XX Secondary: 32-XX [MSN][ZBL]

A function that can be locally represented by power series. Such functions are usually divided into two important classes: the real analytic functions and the complex analytic functions, which are commonly called holomorphic functions. This entry concerns the latter: the reader is referred to Real analytic function for the first class.

The exceptional importance of the class of analytic functions is due to the following reasons. First, the class is sufficiently large; it includes the majority of functions which are encountered in the principal problems of mathematics and its applications to science and technology. Secondly, the class of analytic functions is closed with respect to the fundamental operations of arithmetic, algebra and analysis. Finally, an important property of an analytic function is its uniqueness: Each analytic function is an "organically connected whole" , which represents a "unique" function throughout its natural domain of existence. This property, which in the 18th century was considered as inseparable from the very notion of a function, became of fundamental significance after a function had come to be regarded, in the first half of the 19th century, as an arbitrary correspondence. The theory of analytic functions originated in the 19th century, mainly due to the work of A.L. Cauchy, B. Riemann and K. Weierstrass. The "transition to the complex domain" had a decisive effect on this theory. The theory of analytic functions was constructed as the theory of functions of a complex variable; at present (the 1970's) the theory of analytic functions forms the main subject of the general theory of functions of a complex variable.

Analytic functions of one complex variable

There are different approaches to the concept of analyticity. One definition, which was originally proposed by Cauchy, and was considerably advanced by Riemann, is based on a structural property of the function — the existence of a derivative with respect to the complex variable, i.e. its complex differentiability. This approach is closely connected with geometric ideas. Another approach, which was systematically developed by Weierstrass, is based on the possibility of representing functions by power series; it is thus connected with the analytic apparatus by means of which a function can be expressed. A basic fact of the theory of analytic functions is the identity of the corresponding classes of functions in an arbitrary domain of the complex plane.

Complex differentiability

Let $D$ be a domain (that is, an open set) in the complex plane $\mathbb C$. If to each point $z\in D$ there has been assigned some complex number $w$, then one says that on $D$ a (single-valued) function $f$ of the complex variable $z$ has been defined and one writes: $w=f(z), z\in D$ (or $f:D\to\mathbb C$). The function $w=f(z)=f(x+iy)$ may be regarded as a complex function of two real variables $x$ and $y$, defined in the domain $D\subset\mathbb R^2$ (where $\mathbb R^2$ is the Euclidean plane). To define such a function is tantamount to defining two real functions \begin{equation*} u=\phi(x,y),\quad v=\psi(x,y),\quad (x,y)\in D\quad (w = u+iv). \end{equation*}

Having fixed a point $z\in D$, one gives $z$ the increment $\Delta z = \Delta x+ i\Delta y$ (such that $z+\Delta z \in D$) and considers the corresponding increment of the function $f$: \begin{equation} \Delta f(z) = f(z+\Delta z) - f(z). \end{equation} If \begin{equation} \Delta f(z) = A\Delta z + o(\Delta z) \end{equation} as $\Delta z\to 0$, or in other words, if \begin{equation} \lim_{\Delta z\to 0}\frac{\Delta f(z)}{\Delta z} = A \end{equation} exists, the function $f$ is said to be complex-differentiable at $z$; $A = f'(z)$ is the complex derivative of $f$ at $z$, and \begin{equation} A\Delta z = f'(z)dz = df(z) \end{equation} is its complex differential at that point. A function $f$ which is complex-differentiable at every point of $D$ is called holomorphic in the domain $D$.

Cauchy-Riemann equations

One may compare the concepts of differentiability of $f$, considered as a function of two real variables variables, and its complex differentiability. In the former case the differential $df$, which is a linear map from $\mathbb R^2$ to $\mathbb C$ has the form \begin{equation} \frac{\partial f}{\partial x}dx + \frac{\partial f}{\partial y}dy, \end{equation} where \begin{equation} \frac{\partial f}{\partial x} = \frac{\partial \phi}{\partial x} + i\frac{\partial \psi}{\partial x},\quad \frac{\partial f}{\partial y} = \frac{\partial \phi}{\partial y} + i\frac{\partial \psi}{\partial y}, \end{equation}

are the partial derivatives of $f$. Passing from the independent variables $x, y$ to the variables $z, \overline{z}$, which may formally be considered as new independent variables, related to the old ones by the equations $z = x+iy$, $\overline{z}=x-iy$ (from this point of view, the function $f$ may also be written as $f(z,\overline{z})$) and expressing $dx$ and $dy$ in terms of $dz$ and $d\overline{z}$ according to the usual rules of differential calculus, one can write $df$ in its complex form: \begin{equation} df = \frac{\partial f}{\partial z}dz + \frac{\partial f}{\partial \overline{z}}d\overline{z} \end{equation}

where \begin{equation} \frac{\partial f}{\partial z} = \frac{1}{2}\left(\frac{\partial f}{\partial x} - i\frac{\partial f}{\partial y}\right), \quad \frac{\partial f}{\partial \overline{z}} = \frac{1}{2}\left(\frac{\partial f}{\partial x} + i\frac{\partial f}{\partial y}\right), \end{equation}

are the (formal) derivatives of $f$ with respect to $z$ and $\overline{z}$, respectively. It is seen, accordingly, that $f$ is complex differentiable if and only if it is differentiable in the sense of $\mathbb R^2$ and $df$ turns out to be a linear map from $\mathbb C\to \mathbb C$. This is the case if and only if the equation $\partial f /\partial\overline{z}=0$ is satisfied, which in expanded form may be written as \begin{equation}\label{e:CR} \frac{\partial \phi}{\partial x} = \frac{\partial \psi}{\partial y} \qquad \frac{\partial \psi}{\partial x} = - \frac{\partial \phi}{\partial y}\, . \end{equation} $f$ is then holomorphic in the domain $D$ if and only $f$ is differentiable as a real-variable function and the equations \eqref{e:CR} (which are called Cauchy-Riemann equations) are satisfied at all point of the domain. These equations occurred already in the 18th century in J.L. d'Alembert's and L. Euler's studies on functions of a complex variable.

Power series expansion

Remarkably, without any further assumptions than differentiability and using just the fact that the identities \eqref{e:CR} holds everywhere, it is possible to show that holomorphic functions are extremely regular. In particular the complex derivative $f'$ can be proved to be itself an holomorphic function. This fact, applied recursively, implies that $f$ is infinitely differentiable, in fact infinitely complex-differentiable, and justifies the notation $f^{(n)} (z_0)$ for the $n$-th complex derivative of $f$ at the point $z_0$. Moreover, for every point $z_0$ in its domain of definition there is a neighbourhood $U$ of this point in which $f$ may be represented by a power series: \begin{equation}\label{e:power_series} f (z) = \sum_n a_n (z-z_0)^n\, \qquad \forall z\in U \end{equation} (where we are using two conventions which will hold through the rest of this entry: $0^0$ is set to be $1$ and when we write \eqref{e:power_series} we implicitely assume that the right hand side of \eqref{e:power_series} converges at every point where the identity holds). It can indeed be shown that \eqref{e:power_series} is the Taylor series of $f$ at the point $z_0$, namely that \[ a_n = \frac{f^{(n)} (z_0)}{n!} \] and hence that the power series takes the well-known form \begin{equation}\label{e:Taylor} f (z) = \sum_n \frac{f^{(n)} (z_0)}{n!} (z-z_0)^n\, . \end{equation} Thus, the holomorphy of a function $f$ in a domain $D$ implies that $f$ is infinitely differentiable at any point in $D$ and that its Taylor series converges to it in some neighbourhood of this point.

Viceversa, if the function $f$ is complex analytic at $z_0$, i.e. it can be expanded in a (complex) power series in the neighborhood $U$ of a point $z_0$ (namely if the identity \eqref{e:power_series} holds for some sequence of complex numbers $\{a_n\}$), then $f$ is complex-differentiable everywhere in $U$ and indeed its complex derivative $f' (z)$ equals the power series obtained by differentiating the left hand side of \eqref{e:power_series} term by term, namely \[ f' (z) = \sum_n n a_n (z-z_0)^{n-1}\, . \] In particular the two notions of holomorphy and complex analyticity are equivalent.

Cauchy integral formula

One other characteristic of an analytic function is connected with the notion of path integration. The integral of a function $f= \phi + i \psi$ along an (oriented rectifiable) arc $\gamma$ parametrized by $z: [\alpha, \beta] \to \mathbb C$ may be defined by the formula: \[ \int_\gamma f(z)\, dz := \int_\alpha^\beta f (z(t))\, z'(t)\, dt \] or (equivalently) by means of a curvilinear integral of a differential form (see also Integration on manifolds): \[ \int_\gamma f(z)\, dz := \int_\gamma (\phi\, dx - \psi\, dy) + i \int_\gamma (\psi\, dx + \phi\, dy)\, . \] A key result in the theory of analytic functions is Cauchy's integral theorem: If $f$ is holomorphic in a domain $D$ then \begin{equation}\label{e:Cauchy_1} \int_\gamma f(z)\, dz =0 \end{equation} for any closed curve $\gamma$ bounding a domain inside $D$ (hence for any closed curve when $D$ is simply connected). The converse result, Morera's theorem, is also true: If $f: D\to \mathbb C$ is continuous on an open domain $D$ and if \eqref{e:Cauchy_1} holds for any curve $\gamma$ which bounds a domain in $D$, then $f$ is holomorphic in $D$. In particular, in a simply-connected domain, those and only those continuous functions $f$ are analytic, whose integral along any closed curve $\gamma$ is zero (or, which is the same thing, the integral along any curve $\gamma$ connecting two arbitrary points $p$ and $q$ does depend only on the points $p$ and $q$ themselves and not on the shape of the curve). This characterization of analytic functions forms the basis of many of their applications.

Cauchy's integral theorem yields Cauchy's integral formula, which expresses the values of an analytic function inside a domain in terms of its values on the boundary. More precisely, if $D$ is an open domain whose boundary consists of a finite number of non-intersecting rectifiable curves (oriented positively with respect to $D$) and $f: D\to \mathbb C$ is holomorphic, then \begin{equation}\label{e:Cauchy_Formula} f(z) = \frac{1}{2\pi i} \int_{\partial D} \frac{f (\zeta)}{\zeta-z}\, d\zeta \qquad \forall z\in D\, . \end{equation} This formula makes it possible, in particular, to reduce the study of many problems connected with analytic functions to the corresponding problems for a very simple function — the Cauchy kernel $\frac{1}{\zeta-z}$ (where $\zeta\in \partial D$ and $z\in D$). For more details see Integral representation of an analytic function.

Unique continuation

A very important property of analytic functions is expressed by the following uniqueness theorem: Two functions which are analytic in a domain $D$ and which coincide on some set with an accumulation point in $D$ are identical. In particular, if $D$ is a connected open set and $f:D\to \mathbb C$ an holomorphic function which is not identically zero, then each zero $z_0$ of $f$ is isolated. In addition, for some neighborhood $U$ of $z_0$, there are an holomorphic function $g: U \to \mathbb C$ which never vanishes and a natural number $n$, (called the multiplicity of the zero $z_0$, or order of vanishing of $f$ at $z_0$) such that $f (z) = (z-z_0)^n g(z)$ for every $z\in U$.

Singularities and Laurent series

An important role in the theory of analytic functions is played by the points at which the function cannot be prolonged — the so-called singular points of the analytic function. Here, only isolated singular points of (single-valued) analytic functions are considered; for more details cf. Singular point. If $f$ is an holomorphic function function in a punctured disk $D: = \{z: 0<|z-z_0|<r\}$, then $f$ may be expanded there in a Laurent series \begin{equation}\label{e:Laurent} f (z) = \sum_{n=-\infty}^\infty a_n (z-z_0)^n\, , \end{equation} which contains, as a rule, not only positive but also negative powers of $z-z_0$ (the order of the summation in \eqref{e:Laurent} does not play any role, since the series converges absolutely and uniformly on any compact region of $D$). The sum of the terms of the Laurent series for $n$ corresponding to the negative indices, \[ \sum_{n=-\infty}^{-1} a_n (z-z_0)^n \] is known as the principal part of the Laurent series (or of the function $f$) at the point $z_0$. This principal part determines the nature of the singularity of $f$ at $z_0$.

If there are no terms with negative powers, then $z_0$ is a removable singularity, namely if we define $f (z_0) = a_0$, then such extension is indeed holomoprhic in the whole disk $\{|z-z_0|<r\}$. A removable singularity can be characterized by the fact that the original function $f$ is bounded in some punctured neighborhood $\{0<|z-z_0|<\rho\}$. If the Laurent series of the function contains only a finite number of terms with negative powers of $(z-z_0)$, namely it takes the form \begin{equation}\label{e:Laurent_pole} f (z) = \sum_{n=-\mu}^\infty a_n (z-z_0)^n\, \end{equation} for some $\mu>0$ with $a_\mu\neq 0$, then the point $z_0$ is called a pole of $f$ (of multiplicity, or order, $\mu$); a pole $z_0$ is characterized by \[ \lim_{z\to z_0} |f(z)| = \infty\, . \] The function $f$ has a pole at the point $z_0$ if and only if the function $\frac{1}{f}$ can be extended to an holomorphic function in some neighborhood of $z_0$ by setting $\frac{1}{f} (z_0) = 0$. Moreover, the multiplicity of $z_0$ as pole of $f$ equals the order of $z_0$ as zero of the (extension of) $\frac{1}{f}$. If the Laurent series contains an infinite number of negative powers of $z-z_0$ (that is $a_n\neq 0$ for an infinite set of negative indices $n$), then $z_0$ is called an essential singularity; at such points there is no finite and no infinite limit for $f$.

Residues

The coefficient $a_{-1}$ in the Laurent series for $f$ with centre at the isolated singular point $z_0$ is called the residue of $f$ at $z_0$: \[ a_{-1} = {\rm res}\, [f(z);z=z_0]\, . \] The residue can also be defined by the formula \[ {\rm res}\, [f(z);z=z_0] = \frac{1}{2\pi i} \int_\gamma f(z)\, dz \] where $\gamma = \{|z-z_0| =\rho\}$ and $\rho$ is sufficiently small (so that the disc $\{|z-z_0\leq \rho\}$ does not contain singular points of $f$ other than $z_0$). The important role of residues is made clear by the following theorem: If $f$ is an analytic function in a domain $G$, except for some set of isolated singular points, if $\gamma\subset G$ is a contour bounding a domain $D\subset G$ and not passing through any singular points of $f$, and if $z_1, \ldots, z_k$ are all the singular points of $f$ which are contained in $D$, then \[ \int_\gamma f(z)\, dz = 2\pi i \sum_{i=1}^k {\rm res}\, [f(z);z_i]\, . \] This theorem is an effective tool in calculating integrals. See also Residue of an analytic function.

Meromorphic functions

Functions which are representable as a quotient of two functions that are holomorphic in a domain $D$ are called meromorphic in the domain $D$. A function $f$ which is meromorphic in a domain is holomorphic in that domain, except possibly at a discrete set of points (that are therefore finitely or countably many) which all turn out to be poles of $f$; at the poles the values of a meromorphic function are considered to be infinite. If such values are allowed, then meromorphic functions in a domain $D$ may be defined as functions that in a neighbourhood of each point $z_0$ can be represented by a Laurent series in $z-z_0$ with a finite number of terms involving negative powers of $z-z_0$ (depending on $z_0$).

Both holomorphic and meromorphic functions in a domain $D$ are often designated as analytic in the domain $D$. In this a case holomorphic functions are also said to be regular analytic or simply regular functions.

Entire functions

The simplest class of analytic functions consists of those which are holomorphic in the whole plane, which are called entire functions. Each entire function can be represented by a single series \[ \sum_n a_n z^n \] which is convergent in the whole plane. This fact is a particular case of a more general result, which is a consequence of Cauchy's integral formula. More precisely, if $f$ is holomorphic in a domain $D$ and $\sum_n a_n (z-z_0)^n$ is the power series expansion in a neighborhood of $z_0$, then the series converges in any open disc centered at $z_0$ which is contained in $D$ (and therefore $f$ coincides with the power series on each such disk).

Notable examples of entire functions are the polynomials, the exponential and the trigonometric sine and cosine functions: \begin{eqnarray*} e^z &=& \sum_n \frac{z^n}{n!}\\ \sin z &=& \frac{e^{iz} - e^{-iz}}{2i} = \sum_n (-1)^n \frac{z^{2n+1}}{(2n+1)!}\\ \cos z &=& \frac{e^{iz} + e^{-iz}}{2} = \sum_n (-1)^n \frac{z^{2n}}{(2n)!}\, . \end{eqnarray*}

Weierstrass and Mittag-Leffler Theorems

Weierstrass' theorem states that, for any sequence of complex numbers $z_n$ without limit points in $\mathbb c$, there exists an entire function $f$ such that $\{z_n:n\in \mathbb N\} = \{f=0\}$. Moreover, if there are no repetitions in the sequence and $\{\mu_n\}$ is a sequence of positive natural numbers, then $f$ can be chosen so that it vanishes at each $z_n$ with order $\mu_n$. Such function $f$ may be represented as a (generally infinite) product of entire functions each one having only one zero.

Functions that are meromorphic in the plane, i.e. that may be represented as quotients of entire functions), are called meromorphic functions. These include rational functions, $\tan z = \frac{\sin z}{\cos z}$, elliptic functions, etc.

According to Mittag-Leffler's theorem, for any sequence $\{z_n\}\subset \mathbb C$ of distinct points without limit points in $\mathbb C$ and for each sequence of (nonconstant) polynomials $\{P_n\}$, there exists a meromorphic function $f$ which has poles only at the points $\{z_n\}$ and such that for each $n$ the principal part of its Laurent series at $z_n$ coincides with $P_n ((z-z_n)^{-1})$. The function $f$ may be represented as a (usually infinite) sum of meromorphic functions, each with one single pole in $\mathbb C$.

Theorems on the existence of a holomorphic function with pre-assigned zeros and of meromorphic functions with pre-assigned poles and principal parts are also valid for an arbitrary domain $D\subset \mathbb C$.

Conformality and Riemann's mapping theorem

In the study of analytic functions the related geometric notions are also of importance. If $D$ is a connected open set and $f:D \to \mathbb C$ is an holomorphic nonconstant function, then $D$ is also open(principle of preservation of domains): namely any holomorphic nonconstant map on a connected domain is open. Moreover, the holomorphy of a (non-constant) function $F$ can be characterized by the following geometric property: the map preserves orientation and angles (the latter property is called conformality). Thus, there exists a close connection between analyticity and the important geometric notion of conformal mapping. If $f:D\to \mathbb C$ is an injective holomorphic function (i.e. $f$ is a univalent function), then $f'$ never vanishes on $D$ and $f:D \to f(D)$ defines a one-to-one conformal mapping of the domain $D$ onto the domain $f(D)$, whose inverse is also holomorphic. Riemann's mapping theorem, which is the fundamental theorem in the theory of conformal mappings, says that on any simply connected domain $D$ whose boundary contains more than one point there exist univalent analytic functions which map $D$ onto a disc or a half-plane.

Harmonic functions

The real and imaginary parts of a function $f$ which is holomorphic in a domain $D$ satisfy the Laplace equation in that domain: \[ \Delta g = \frac{\partial^2 g}{\partial x^2} + \frac{\partial^2 g}{\partial y^2} = 0 \] i.e. they are harmonic functions. Two harmonic functions which are connected by the Cauchy–Riemann equations (and hence are the real and imaginary parts of a single holomorphic function) are called conjugate. In a simply-connected domain $D$ any harmonic function $\phi$ has a conjugate function $\psi$ and is thus the real part of some holomorphic function $f$ in $D$.

The connections with conformal mappings and harmonic functions form the basis of many applications of the theory of analytic functions.

Analyticity on non-open domains

A function $f:E\to \mathbb C$, where $E$ is an arbitrary subset of $\mathbb C$, is called analytic at a point $z_0\in E$ if there exists a neighbourhood of this point such that $f$ may be represented by a convergent power series on the intersection of this neighbourhood with $E$. The function $f$ is called analytic on the set $E$ if it has an holomorphic extension to some open set which contains $E$. For open sets the notion to analyticity coincides with the notion of (complex) differentiability with respect to the set. However, this is not the case in general; in particular, on the real line there exist functions which not only have a derivative, but which are infinitely differentiable at every point and are not analytic even at a single point of this line (see also Real analytic function). The property of connectedness of the set $E$ is necessary in order that the uniqueness theorem for analytic functions holds. This is why analytic functions are usually considered in domains, i.e. on connected open sets.

Analytic continuation and Riemann surfaces

All the preceding refers to "single-valued" analytic functions $f$, considered in a given domain $D$ (or on a given set $E$) of the complex plane. In considering the possible extension of a function $f$, as an analytic function, to a larger domain, one arrives at the concept of the analytic function considered as a whole, i.e. throughout its whole natural domain of existence. If the function is thus extended, its domain of analyticity becomes larger, and may overlap itself, supplying new values of the function at points in the plane where it already was defined. Accordingly, an analytic function considered as a whole is generally multi-valued. Many problems in analysis (inversion of a function, the determination of a primitive and the construction of an analytic function with a given real part in multiply-connected domains, the solution of algebraic equations with analytic coefficients, etc.) require the study of multi-valued functions; such functions include $\sqrt[n]{z}$, $\ln z$, $\arcsin z$, $\arctan z$, algebraic functions, etc.

A regular process which yields the complete analytic function, considered throughout its natural domain of existence, was proposed by Weierstrass; it is known as Weierstrass' method of analytic continuation.

The initial concept is that of an element of an analytic function, viz. a power series with a non-zero radius of convergence. Such an element $W_0$: \begin{equation}\label{e:series_0} \sum_n a_n (z-z_0)^n \end{equation} defines a certain analytic function $f$ on its disc of convergence $K_0$. Let $z_1$ be a point of $K_0$ different from $z_0$. Expanding $f$ in a series with centre at $z_1$, one obtains a new element $W_1$: \begin{equation}\label{e:series_1} \sum_n b_n (z-z_1)^n \end{equation} whose disc of convergence will be denoted by $K_1$. In the intersection $K_0\cap K_1$ the series $W_2$ (i.e. \eqref{e:series_1}) converges to the same function as the series $W_1$ (i.e.\eqref{e:series_0}). If $K_1$ extends beyond the boundary of $K_0$, the series $W_1$ defines the function determined by $W_0$ on some set outside $K_0$ (where the series $W_1$ is divergent). In such a case the element $W_1$ is called a direct analytic continuation of the element $W_0$. Let $W_1, \ldots, W_N$ be a chain of elements in which $W_{n+1}$ is a direct analytic continuation of $W_n$ for every $n\in \{0, \ldots, N-1\}$; the element $W_N$ is then said to be an analytic continuation of the element $W_0$ (by means of the given chain of elements). When the centre of the disc $K_N$ belongs to $K_0$ it may happen that the element $W_N$ is not a direct analytic continuation of the element $W_0$. In such a case the sums of the series $W_0$ and $W_N$ will have different values in the intersection $K_0\cap K_N$; thus analytic continuation may lead to new values of the function inside $K_0$.

The totality of all elements which may be obtained by analytic continuation of an element $W_0$ forms the complete analytic function (in the sense of Weierstrass) generated by $W_0$; the union of their discs of convergence represents the (Weierstrass) domain of existence of this function. It follows from the uniqueness theorem for analytic functions that an analytic function in the sense of Weierstrass is completely determined by the given element $W_0$. The initial element may be any other element belonging to this function; the complete analytic function will not be affected.

A complete analytic function $f$, considered as a function of the points in the plane belonging to its domain of existence $D$, is generally multi-valued. In order to eliminate this feature, $f$ is considered not as a function of the points in the plane domain $D$, but rather as a function of the points on some multi-sheeted surface $R$ (lying above $D$) such that to each point of $D$ correspond as many points of the surface $R$ (projecting onto the given point of $D$) as there are different elements with centre at this point in the complete analytic function $f$ (see also Covering); on the surface $R$ the function $f$ becomes single-valued. The idea of passing to such surfaces is due to Riemann, and the surfaces themselves are known as Riemann surfaces. The abstract definition of the notion of a Riemann surface has made it possible to replace the theory of multi-valued analytic functions by the theory of single-valued analytic functions on Riemann surfaces.

Now fix a domain $\Delta$ belonging to the domain of existence $D$ of the complete analytic function $f$, and fix some element $W$ of $f$ with centre at a point in $\Delta$. The totality of all elements which may be obtained by analytic continuation of $W$ by means of chains with centres belonging to $\Delta$ is called a branch of the analytic function $f$. A branch of a multi-valued analytic function may turn out to be a single-valued analytic function in the domain $\Delta$ (this is indeed always the case when $\Delta$ is simply connected). Thus, arbitrary branches of the functions $\sqrt[n]{z}$ and $\ln z$ which correspond to an arbitrary simply-connected domain not containing the point 0, are single-valued functions. The function $\sqrt[n]{z}$ has exactly $n$ different branches in such a domain, while $\ln z$ has an infinite set of such branches. The selection of single-valued branches (using some cuts in the domain of existence) and their study by the theory of single-valued analytic functions constitute one of the principal methods of studying specific multi-valued analytic functions.

Analytic functions of several complex variables.

The complex space $\mathbb C^n$, consisting of the points $z= (z_1, \ldots, z_n)$, $z_k = x_k+ iy_k$, is a vector space over the field of complex numbers with the Euclidean metric \begin{equation}\label{e:euclidean} |z| = \sqrt{\sum_{k=1}^n |z_k|^2}\, . \end{equation} Obviously $\mathbb C^1 = \mathbb C$. The complex differs from the $2n$-dimensional Euclidean space $\mathbb R^{2n}$ by a certain asymmetry: on passing from $\mathbb R^{2n}$ to $\mathbb C^n$ (i.e. on introducing a complex structure in $\mathbb R^{2n}$), the coordinates are subdivided into pairs which appear in the complex combinations $z_k = x_k + i y_k$.

Complex differentiability

If a complex function $f = \phi + i \psi$ is defined in a domain $D$ (i.e. an open subset) of $\mathbb C^n$ and is differentiable at some point $z_0\in D$ as a real-varialbe function, its differential at $z_0$ (which is an $\mathbb R$-linear map from $\mathbb C^n$ to $\mathbb C^n$) may be represented in the form \begin{equation}\label{e:total_differential} \left.df\right|_{z_0} = \sum_{k=1}^n \frac{\partial f}{\partial z_k} (z_0)\, dz_k + \sum_{k=1}^n \frac{\partial f}{\partial \bar{z}_k} (z_0)\, d\bar{z}_k \end{equation} Here $dz_k := dx_k + i dy_k$ and $d\bar{z}_k = dx_k - i dy_k$, whereas $\frac{\partial f}{\partial z_k}$ and $\frac{\partial f}{\partial \bar{z}_k}$ are defined as in the $1$-dimensional case by the formulas \[ \frac{\partial f}{\partial z_k} (z_0) = \frac{1}{2} \left(\frac{\partial f}{\partial x_k} (z_0) - i \frac{\partial f}{\partial y_k} (z_0)\right) \qquad \frac{\partial f}{\partial \bar{z}_k} (z_0) = \frac{1}{2} \left(\frac{\partial f}{\partial x_k} (z_0) + i \frac{\partial f}{\partial y_k} (z_0)\right)\, . \] The map $f$ is said to be complex differentiable at $z_0$ if the differential in \eqref{e:total_differential} is $\mathbb C$-linear. It can be easily seen that this is equivalent to the condition \begin{equation}\label{e:del_bar} \frac{\partial f}{\partial z_k} (z_0) = 0 \qquad \forall k\, , \end{equation} and hence that in such a case the differential takes the form \[ df = \sum_{k=1}^n \frac{\partial f}{\partial z_k} dz_k\, . \] If this requirement is satisfied at every $z_0\in D$, then $f$ is said to be complex-differentiable or holomorphic or analytic in the domain $D$.

Cauchy-Riemann system

The system \eqref{e:del_bar} is equivalent to the system of $2n$ first-order partial differential equations \begin{equation}\label{e:CR_system} \frac{\partial \phi}{\partial x_k} = \frac{\partial \psi}{\partial y_k} \qquad \frac{\partial \phi}{\partial y_k} = -\frac{\partial \psi}{\partial x_k}\, , \end{equation} which is called Cauchy-Riemann system.

In the case of space ($n>1$), as distinct from that of the plane ($n=1$) this system is overdetermined, since the number of equations is larger than that of the unknown functions. It remains overdetermined on passing to the (geometrically more natural) spatial analogue of a holomorphic function of one complex variable — a holomorphic mapping $f$, which is realized by a vector $f= (f_1, \ldots, f_k)$ of $n$ functions $f_k:D\to \mathbb C$ which are holomorphic in the domain $D$. The mapping $f:D\to G$ is called biholomorphic if it is one-to-one and if it is holomorphic together with its inverse $f^{-1}$. The conditions for holomorphy of a mapping $f:D\to\mathbb C^n$ are expressed by a system of $2n^2$ real partial differential equations involving $2n$ real unknown functions. The overdeterminacy of the conditions of holomorphy for $n>1$ is the cause of a number of effects typical of the spatial case — such as the absence of a spatial analogue of the Riemann mapping theorem on the existence of conformal mappings. According to Riemann's theorem, if $n=1$, any two simply-connected domains whose boundaries do not reduce to a single point are isomorphic. However, if $n>1$, even such simple simply-connected domains as the ball $\{|z|<1$ and the product of discs (polydisc) $\{|z_k|<1 : k \in \{1, \ldots , n\}\}$ are non-isomorphic. The non-isomorphism is brought to light on comparing the groups of automorphisms of these domains (i.e. their biholomorphic mappings onto themselves, cf. Biholomorphic mapping) — the groups prove to be algebraically non-isomorphic, whereas a biholomorphic mapping of one domain onto another, if it existed, would establish an isomorphism of these groups. Owing to this, the theory of biholomorphic mappings of domains in complex space is substantially different from the theory of conformal mappings in the plane.

Hartogs' theorem

A function $f$ is called holomorphic at a point $\alpha$ if it is holomorphic in some neighbourhood of this point. According to the Cauchy–Riemann criterion, a function of several variables which is holomorphic at a point $\alpha$ is holomorphic with respect to each variable (if the values of the other variables are fixed). The converse proposition is also true: If, in a neighbourhood of some point, a function $f$ is holomorphic with respect to each variable separately, then it is holomorphic at this point (Hartogs' fundamental theorem).

Power series

In analogy with the case of the plane the holomorphy of a function $f$ at a point $\alpha\in \mathbb C^n$ is equivalent to its expandability in a multiple power series in a neighbourhood of this point \begin{equation}\label{e:power_series_nd} f(z) = \sum_{k_1, \ldots, k_n =1}^\infty c_{k_1\ldots k_n} (z_1-\alpha_1)^k_1 \ldots (z_n - \alpha_n)^{k_n}\, \end{equation} or, using multi-index notation \begin{equation}\label{e:poer_series_nd_multi} \sum_I c_I (z-\alpha)^I\, , \end{equation} (where $I = (k_1, \ldots k_n)\in \mathbb N^n$ and $z-\alpha)^I = (z_1-\alpha_1)^k_1 \ldots (z_n - \alpha_n)^{k_n}$). A holomorphic function is infinitely differentiable, and the series \eqref{e:power_series_nd} is its Taylor series, i.e. \[ c_I = c_{k_1\ldots k_n} = \frac{1}{k_1!\ldots k_n!} \frac{\partial^{k_1+\ldots + k_n} f}{\partial z_1^{k_1} \ldots \partial z_n^{k_n}} (\alpha)\, . \]

Weierstrass preparation theorem

The fundamental facts of the theory of holomorphic functions of one variable extend to holomorphic functions of several variables, sometimes in an altered form. An instance of this is the Weierstrass preparation theorem, which extends to several variables the property of holomorphic functions of one variable to be presentable as $g (z) (z-z_0)^n$ in a neighborhood of a zero $z_0$ of $f$ (where $g$ is holomorphic and $g (z_0)\neq 0$). The theorem is formulated as follows: If a function $f$ is holomorphic at a point $\alpha$, vanishes at that point, but it is not identically $0$ in any open neighborhood of $\alpha$, then (after a non-degenerate linear transformation of the independent variables) it may be represented, in a neighbourhood $U$ of $\alpha$, in the form \begin{equation}\label{e:Weiertsrass_prep} f (z) = \underbrace{\left((z_n -\alpha_n)^k + c_1 (z') (z_n -\alpha_n)^{k-1} + \ldots + c_k (z')\right)}_{=W(z)} \phi (z)\, , \end{equation} where

• $k\geq 1$ is an integer (depending on $\alpha$;
• each $c_i$ is a function of $z' = (z_1, \ldots z_{n-1})$ which is holomorphic in a neighborhood of $\alpha' = (\alpha_1, \ldots \alpha_{n-1})\in \mathbb C^{n-1}$ and vanishes at $\alpha'$;
• $\phi$ is an holomorphic function with $\phi (\alpha)\neq 0$.

A function which has the form of $W$ is often called a Weiertstrass polynomial in the variable $z_n$

Analytic varieties

This theorem is of fundamental importance in the study of analytic varieties, which are described locally, in a neighbourhood of each one of their points, as sets of common zeros of a finite number of functions which are holomorphic at this point. By Weierstrass' preparation theorem such sets may be locally described as sets of common zeros of Weierstrass polynomials, i.e. a special class of polynomials in the variable $z_n$, with coefficients from the ring of holomorphic functions in the other variables $(z_1, \ldots, z_{n-1})$. This description permits extensive use of algebraic methods in the local study of analytic sets.

Cauchy's integral theorem

Cauchy's integral theorem must also be slightly modified in the spatial case, and is then known as the Cauchy–Poincaré theorem: Let a function $f$ be holomorphic in a domain $D$; then, for any compact $n+1$-dimensional (real!) submanifold $G$ in $D$ with piecewise-smooth boundary $\partial G$, \begin{equation}\label{e:Cauchy-Poincare'} \int_{\partial G} f(z) dz = 0\, . \end{equation} Here the integral in \eqref{e:Cauchy-Poincare'} is the integral of the differential form $\omega = f(z) dz_1\wedge \ldots \wedge dz_n$, which in the real variables $x_1, y_1, \ldots, x_n, y_n$ can be simply written as $\omega_r + i \omega_i$, where $\omega_r$ and $\omega_i$ are real differential forms.

As in the planar case, this integral can be defined by a parametric representation of the given set: if $z: \mathbb R^n \supset \Omega\to \partial G$ is a parametrization of a portion $A$ of $\partial G$, then \[ \int_A f dz = \int_\Omega f (z(t)) \frac{\partial (z_1, \ldots, z_n)}{\partial (t_1, \ldots, t_n)}\, dt_1\ldots dt_n\, , \] where \[ \frac{\partial (z_1, \ldots, z_n)}{\partial (t_1, \ldots, t_n)} \] denotes the determinant of the $n\times n$ complex matrix \[ \frac{\partial z_i}{\partial t_j}\, . \] The integral in \eqref{e:Cauchy-Poincare'} can then be defined using charts.

Observe that, in contrast with the case $n=1$, when $n>1$ the dimension of the surface $G$ (which is $n+1$) is strictly smaller than the (topological) dimension of the ambient domain $D$ (which is $2n$). Observe also that \eqref{e:Cauchy-Poincare'} can be concluded from Stokes theorem: the holomorphy of $f$ implies in fact that $d (f dz_1\wedge\ldots dz_n) =0$

Integral representations

The spatial analogue of the Cauchy integral formula can be written in a particularly simple form for polycylinder domains, i.e. for products of plane domains. Let $D= D_1\times D_2\times \ldots \times D_n$ be a domain in which each $D_k$ is a domain in the complex plane with a piecewise-smooth boundary $\partial D_k$ (where $k= 1, \ldots, n$), while the function <$f$ is holomorphic in a domain which compactly contains $D$. Successive application of Cauchy's formula for one variable then yields, for any point $z\in D$, \[ f (z) = \frac{1}{(2\pi i)^n} \int_\Gamma \frac{f(\zeta) d\zeta}{\zeta-z}\, \] where

• $\Gamma = \partial D_1 \times \ldots \times \partial D_n$;
• the form $\frac{d\zeta}{\zeta-z})$ is given by

\[ \frac{d\zeta}{\zeta-z} = \frac{d\zeta_1 \wedge \ldots \wedge d\zeta_n}{(\zeta_1-z_1)\ldots (\zeta_n -z_n)} \]

• the integral is understood as the integral of an $n$-dimensional form on a (piecewise smooth) $n$-dimensional (real!) manifold (cf. Integration on manifolds).

Martinelli-Bochner formula

However, polycylinder domains are only a very special class, and in general domains such a separation of variables is not possible. The role of Cauchy's integral for arbitrary domains with a piecewise-smooth boundary is played by the Martinelli–Bochner integral formula: For any function which is holomorphic in a domain containing , and for any point ,

where , and

This is Green's formula for a pair of functions, one of which is holomorphic in , while the other is a fundamental solution of the Laplace equation in the space with singular point . If , this is the ordinary Cauchy integral. If , the formula differs from Cauchy's multiple integral for a product of plane domains in that, first, the integration is not over an -dimensional part of the boundary, but over the whole -dimensional boundary of the domain, and, secondly, its kernel (the factor multiplying under the integral sign) does not depend analytically on the parameter .

Leray formula

An analytic kernel, however, is essential in a number of problems, and it is therefore desirable to construct an integral formula with such a kernel for as large a class of domains as possible. An ample supply of integral formulas, including formulas with an analytic kernel for many domains, is contained in the general Leray formula. This formula is

where is a smooth vector function depending also on , and are defined above, and ; it is assumed that for any fixed and running over . The value of the integral in this formula does not depend on the choice of the vector function (provided that for all , ), and if , this integral coincides with the Martinelli–Bochner integral. By varying the choice of for different classes of domains, the Leray formula will yield various integral formulas. In the theory of analytic functions of several variables other integral representations, which are valid only for certain classes of domains, are also considered. An important class of this kind consists of the so-called Weil domains, which are a generalization of the product of plane domains. For such domains one has the Bergman–Weil representation with a kernel which also depends analytically on the parameter.

Hartog's extension theorem

As in the planar case, the study of the singularities of analytic functions is of fundamental interest; the main difference between the planar and the spatial cases is expressed by the Hartog's extension theorem (also known as Osgood–Brown theorem) on the removability of compact singularities, according to which any function which is holomorphic in , where is a domain in () and is compact subset of which does not subdivide , extends holomorphically to the whole domain . By this theorem, holomorphic functions of several variables cannot have isolated singular points. These are replaced in () by singular sets which are analytic if their dimension is lower than .

Multi-dimensional residues

This fact is essential in the theory of multi-dimensional residues. This theory deals with the problem of computing the integral of a function , which is holomorphic everywhere in a domain except for an analytic set , over a closed -dimensional surface not intersecting . Since the dimension of the singular set is lower than the dimension of by at least two, does not subdivide . If the surface is not linked with , i.e. bounds an -dimensional surface compactly belonging to , then by the Cauchy–Poincaré theorem . In order to calculate this integral in the general case, it is necessary to clarify how is linked with the singular set , and to calculate the integrals over special -dimensional surfaces associated with separate portions of the set (residues).

The solution of this problem involves considerable topological and analytic difficulties. These may often be overcome by the methods proposed by E. Martinelli and J. Leray. The Martinelli method is based on the use of the topological Aleksander–Pontryagin duality principle, and reduces the study of the -dimensional homologies of the set to the study of the -dimensional homologies of the singular set . The Leray method is more general: it is based on the examination of special homology classes and on the calculation of certain differential forms (residue forms). The multi-dimensional theory of residues has also found applications in theoretical physics (cf. Feynman integral).

Domains of holomorphy

The Osgood–Brown theorem reveals an important fundamental difference between the spatial and the planar theories. In the plane one can, for any domain , construct a function which is holomorphic in but which cannot be extended analytically beyond its boundary, i.e. is the natural domain of existence. In space the situation is different: thus, the spherical shell cannot be the domain of existence of any holomorphic function, since by the Osgood–Brown theorem, any function which is holomorphic in it will certainly extend analytically to the entire ball .

Thus arises the problem of the characterization of the natural domains of existence of holomorphic functions — the so-called domains of holomorphy. A simple sufficient condition may be formulated with the aid of the concept of a barrier at a boundary point of the domain, i.e. a function which is holomorphic in this domain and which increases without limit as tends to . will be a domain of holomorphy if it is possible to construct a barrier for an everywhere-dense set of points in its boundary. This condition is satisfied, in particular, by any convex domain; for any point it is sufficient to select, in the -dimensional supporting plane to at the point , a -dimensional plane of the form

the function will then be a barrier. Consequently, every convex domain in is a domain of holomorphy. However, convexity is not a necessary condition for holomorphy: A product of plane domains is always a domain of holomorphy, and such a product need not be convex. Nevertheless, if the notion of convexity is suitably generalized, it is possible to arrive at a necessary and sufficient condition. One such generalization is based on the observation that the convex hull of a set may be described as the set of points at which the value of any linear function does not exceed the supremum of the values of this function on . By analogy, the holomorphically convex hull of a set is defined by

where denotes the set of all functions which are holomorphic in . A domain is called holomorphically convex if, for every compact subset of , the hull also is a compact subset of . Holomorphic convexity is a necessary and sufficient condition for a domain of holomorphy. However, this criterion is not very effective, since holomorphic convexity is difficult to verify.

Plurisubharmonic functions

Another generalization is connected with the notion of a plurisubharmonic function, which is the complex analogue of a convex function. A convex function in a domain $D$ of $\mathbb R^n$ may be defined as a function for which the restrictions to the segments in $D$, i.e. sets of the form $x = x_0 + \omega t$ (where $x_0, \omega \in \mathbb R^n$ and $t$ is a real parameter), are convex functions of $t$. A real function $\phi$, upper semi-continuous in a domain $D\subset \mathbb C^n$, is said to be plurisubharmonic in $D$ if for each complex line $z = z_0 + \omega \zeta$ (where $z_0, \omega \in \mathbb C^n$ and $\zeta$ is a complex parameter), its restriction to the parts of this line in $D$ is a subharmonic function of $\zeta$. If $\phi$ is twice continuously differentiable, then the condition of plurisubharmonicity, in accordance with the rules of differentiation of composite functions, is that the Hermitian form \[ H (v, \bar{v}) = \sum_{j,k=1}^n \frac{\partial^2 \phi}{\partial z_j \partial\bar{z}_k} v_j \bar{v}_k \] — the so-called Levi form — be non-negative.

A domain $D\subset \mathbb C^n$ is called pseudo-convex if the function $\ln {\rm dist}\, (z, \partial D$), (where ${\rm dist}\, (z, \partial D)$ denotes the Euclidean distance of the point $z$ to the boundary $\partial D$) is plurisubharmonic in this domain). Pseudo-convexity is also a necessary and sufficient condition for a domain to be a domain of holomorphy.

In some cases it is possible to verify effectively the pseudo-convexity of a domain.

Envelope of holomorphy

As regards domains $D$ which are not domains of holomorphy, there arises the problem of describing their envelope of holomorphy, i.e. the smallest domain of holomorphy to which any function holomorphic in $D$ extends analytically. For domains of the simplest types envelopes of holomorphy can be effectively constructed, but in the general case the problem is unsolvable within the class of single-sheeted domains. Under analytic continuation of functions beyond the boundary of a given domain $D$, multi-valuedness may result, which can be avoided by introducing multi-sheeted covering domains over $D$, analogous to Riemann surfaces). In the class of covering domains, the problem of constructing envelopes of holomorphy is always solvable. This problem also has applications in theoretical physics, especially in quantum field theory.

Stein manifolds

The transition from the plane to a complex space substantially increases the variety of geometrical problems related to holomorphic functions. In particular, such functions are naturally considered not only in domains, but also on complex manifolds — smooth manifolds of even real dimension, the neighbourhood relations of which are biholomorphic. Among these, Stein manifolds (cf. Stein manifold) — natural generalizations of domains of holomorphy — play a special role

Cousin problems

Several problems in analysis may be reduced to the problem of constructing, in a given domain, a holomorphic function with given zeros or a meromorphic function with given poles and principal parts of the Laurent series. In the plane case, these problems have been solved for arbitrary domains by the theorems of Weierstrass and Mittag-Leffler and their generalizations. The spatial case is different — the solvability of the corresponding problems, the so-called Cousin problems, depends on certain topological and analytic properties of the complex manifolds considered.

The key step in the solution of the Cousin problems is to construct — starting from locally-defined functions with given properties — a global function, defined on the whole manifold under consideration and having the same local properties. Such kinds of constructions are very conveniently effected using the theory of sheaves, which arose from the algebraic-topological treatment of the concept of an analytic function, and which has found important applications in various branches of mathematics. The solution of the Cousin problems by methods of the theory of sheaves was realized by H. Cartan and J.-P. Serre.

Further developments

Complex analysis is one of the most important branches of analysis, it is closely connected with quite diverse branches of mathematics and it has numerous applications in theoretical physics, mechanics and technology.

Fundamental investigations on the theory of analytic functions have been carried out by Soviet mathematicians. Extensive interest in the theory of functions of a complex variable emerged in the Soviet Union at the beginning of the 20th century. This was in connection with noteworthy investigations by Soviet scientists on applications of the theory of analytic functions to various problems in the mechanics of continuous media. N.E. Zhukovskii and S.A. Chaplygin solved very important problems in hydrodynamics and aerodynamics by using methods of the theory of analytic functions. In the works of G.V. Kolosov and N.I. Muskhelishvili these methods were applied to fundamental problems in the theory of elasticity. In subsequent years the theory of functions of a complex variable underwent extensive development. The development of various aspects of the theory of analytic functions was determined by the fundamental research of, among others, V.V. Golubev, N.N. Luzin, I.I. Privalov and V.I. Smirnov (boundary properties), M.A. Lavrent'ev (geometric theory, quasi-conformal mappings and their applications to gas dynamics), M.V. Keldysh, M.A. Lavrent'ev and L.I. Sedov (applications to problems in the mechanics of continuous media), D.E. Men'shov (theory of monogeneity), M.V. Keldysh, M.A. Lavrent'ev and S.N. Mergelyan (approximation theory), I.N. Vekua (theory of generalized analytic functions and their applications), A.O. Gel'fond (theory of interpolation), N.N. Bogolyubov and V.S. Vladimirov (theory of analytic functions of several variables and its application to quantum field theory).

Complex analysis has been an active field of research up to the present time also in the West. Some of the history may be found in the extensive article in the Encyclopaedia Britannica entitled Analysis, complex. The modern proof, due to L. Ehrenpreis, of Hartogs' extension theorem makes use of the inhomogeneous Cauchy–Riemann or $\partial$-equations. The so-called $\partial$-method, extensively developed by J.J. Kohn and L. Hörmander, has become one of the three most powerful techniques available in complex analysis today. Among other things, it can be used to obtain solutions of the Levi problem (are domains of holomorphy the same as pseudo-convex domains?) and the Cousin problems, cf. [a15]. These fundamental problems had been settled previously by methods now belonging to sheaf theory, which go back to K. Oka (1936–1954), and have culminated in the sheaf-cohomology theory of H. Cartan, J.-P. Serre, H. Grauert and others, cf. the elegant treatment in [a12]. The third and most recent technique consists in the use of suitable integral representations as developed by G.M. Henkin [G.M. Khenkin] and E. Ramirez de Arellano, cf. [a14].

The development of the theory of analytic functions of one and several complex variables and their generalizations is continuing. See Boundary properties of analytic functions; Quasi-conformal mapping; Boundary value problems of analytic function theory; Approximation of functions of a complex variable.

References

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