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Analytic function

In complex analysis, an analytic function is a function f: D \to \mathbb{C}, where D is an open subset of the complex plane, that is complex differentiable at every point in D.^[1] These functions, equivalently known as holomorphic functions, can be locally expressed as convergent power series around any point in their domain.^[2] A key characterization is that if f(z) = u(x,y) + iv(x,y) with z = x + iy, then u and v satisfy the Cauchy-Riemann equations \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} and \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}.^[3] Analytic functions exhibit strong regularity properties: they are infinitely differentiable in their domain, and differentiation can be performed term by term within the radius of convergence of their power series expansions.^[4] This contrasts sharply with real-variable functions, where differentiability does not imply higher-order differentiability or power series representability.^[1] Fundamental theorems underpin their behavior, such as Cauchy's integral theorem, which states that if f is analytic in a simply connected domain D and \gamma is a closed contour in D, then \int_\gamma f(z) \, dz = 0.^[5] Cauchy's integral formula further allows expressing f at interior points via contour integrals: f(a) = \frac{1}{2\pi i} \int_\gamma \frac{f(z)}{z - a} \, dz for a inside \gamma.^[5] Notable consequences include the maximum modulus principle, which asserts that a non-constant analytic function on a bounded domain attains its maximum modulus on the boundary, and Liouville's theorem, implying that bounded entire functions (analytic on the whole plane) are constant.^[6] Analytic functions also have harmonic real and imaginary parts, satisfying Laplace's equation.^[5] The concept extends to several complex variables, where analyticity requires differentiability in each variable separately, leading to Hartogs's theorem on extending holomorphic functions across compact sets.^[7] Applications span physics (e.g., potential theory, fluid dynamics) and engineering, leveraging tools like residue calculus for evaluating real integrals.^[8]

Historical Development

Origins in Complex Analysis

The concept of analytic functions in complex analysis traces its origins to the 18th century, when mathematicians began to explore functions of complex variables in a formal manner without a fully rigorous framework for differentiability. Jean le Rond d'Alembert, in his 1746 work on the fundamental theorem of algebra, demonstrated that algebraic operations, including roots and powers, could be consistently applied to complex numbers, treating them as entities amenable to analysis similar to real numbers.^[9] Leonhard Euler extended this approach significantly, investigating series expansions and logarithmic functions for complex arguments; for instance, in his 1751 publication on complex logarithms, he manipulated analytic expressions involving imaginary quantities to derive trigonometric identities, laying informal groundwork for power series representations in the complex plane.^[10] Augustin-Louis Cauchy advanced the field in the 1820s through his pioneering work on complex integration, which provided a pathway to understanding derivatives via integrals without relying on explicit pointwise differentiability. In his 1825 memoir "Mémoire sur les intégrales définies prises entre des limites imaginaires," Cauchy established the integral theorem for closed contours and derived a formula expressing the value of a function (and its derivatives) inside a contour solely in terms of its boundary values, a result that implicitly characterized the smoothness of functions analytic in a domain.^[11] This integral-based perspective shifted focus from algebraic manipulation to geometric and analytic properties, forming the core of modern complex analysis. Bernhard Riemann revolutionized the study of complex functions in the 1850s by introducing Riemann surfaces to handle multi-valued functions systematically. In his 1851 doctoral thesis at the University of Göttingen, Riemann conceptualized these surfaces as multi-sheeted coverings of the complex plane, allowing multi-valued functions like the square root or logarithm to be represented as single-valued analytic mappings on a branched surface, thus resolving singularities and branch points through topological means.^[12] This geometric innovation emphasized the global structure of analytic functions, influencing subsequent developments in function theory. Karl Weierstrass, in the latter half of the 19th century, provided rigorous constructions of entire functions independent of integration, using infinite products to demonstrate their existence and properties. During his lectures in Berlin around 1860–1870 and in his 1876 publication on the theory of analytic functions, Weierstrass showed that any entire function could be expressed as a Weierstrass product over its zeros, combined with an exponential factor, thereby proving the density of such functions without invoking Cauchy's integral methods and establishing uniform convergence as a key analytic tool.^[13]^[14]

Key Contributions

Augustin-Louis Cauchy laid the foundations for the modern theory of analytic functions in his 1825 memoir "Mémoire sur les intégrales définies prises entre des limites imaginaires," where he introduced the concept of contour integrals and demonstrated how they could be used to establish key properties of analytic functions, such as the integral representation that underpins later developments like the residue theorem.^[15] Although the full residue theorem, which computes contour integrals via sums of residues at singularities within a closed curve, was formalized in his subsequent works around 1825–1831, Cauchy's 1825 contributions emphasized the role of these integrals in proving differentiability and other intrinsic properties of functions analytic in a domain.^[16] In 1851, Bernhard Riemann advanced the geometric understanding of analytic functions through his doctoral dissertation "Grundlagen für eine allgemeine Theorie der Functionen einer veränderlichen complexen Grösse," in which he stated the Riemann mapping theorem, proving the existence of a conformal map from any simply connected domain in the complex plane (not the entire plane) onto the unit disk. This theorem, relying on the Dirichlet principle for minimizing energy integrals, highlighted the conformal invariance of analytic functions and provided a powerful tool for classifying Riemann surfaces and studying domain mappings.^[17] Karl Weierstrass contributed significantly to the theory in the 1870s through his development of elliptic function theory, constructing doubly periodic meromorphic functions via infinite products and series, which offered an arithmetic foundation for understanding analytic functions on compact Riemann surfaces.^[13] His work on elliptic functions, detailed in lectures and publications from the 1860s onward, paved the way for the concept of uniformization by showing how such functions could parameterize algebraic curves, influencing later proofs of the uniformization theorem that every simply connected Riemann surface is conformally equivalent to the disk, plane, or sphere.^[18] Weierstrass also pioneered rigorous power series methods, establishing uniform convergence criteria essential for representing analytic functions locally as power series.^[13] Émile Picard extended the study of singularities in the 1880s with theorems characterizing the behavior of analytic functions near essential singularities, building on Weierstrass's results to show that such functions assume every complex value, except possibly one, infinitely often in any punctured neighborhood of the singularity.^[19] His great Picard theorem, proved in 1913 but rooted in these earlier investigations, strengthened this by affirming that near an essential singularity, the function takes all finite values except at most one infinitely many times, with the exceptional value possibly omitted entirely in some cases.^[20] In the 20th century, Kiyoshi Oka bridged single-variable analytic function theory to multivariable extensions through his work on several complex variables in the 1930s and 1940s, resolving the Levi problem by proving that pseudoconvex domains are domains of holomorphy, thus generalizing analyticity to higher dimensions.^[21] Oka's sheaf-theoretic approach and solutions to Cousin problems established foundational results for coherent sheaves of holomorphic functions, influencing the modern theory of complex manifolds.^[22]

Core Concepts

Definition

In complex analysis, an analytic function is a function f: D \to \mathbb{C} that is complex differentiable at every point in an open domain D \subset \mathbb{C}.^[1] This means that for each z_0 \in D, there exists a neighborhood around z_0 where the derivative is defined, emphasizing the local nature of differentiability.^[23] The complex derivative at a point z is given by

f'(z) = \lim_{h \to 0} \frac{f(z + h) - f(z)}{h},

where h \in \mathbb{C} approaches 0, and the limit must exist and be independent of the direction (or path) in which h approaches 0.^[1] In modern usage, the terms "analytic" and "holomorphic" are equivalent, both describing functions satisfying this condition on an open set.^[24] The domain D must be an open subset of the complex plane \mathbb{C} to ensure that every point has a surrounding disk within D, allowing the limit to be evaluated locally.^[25] This open set requirement distinguishes analytic functions from those merely differentiable at isolated points, as analyticity demands differentiability throughout a region. The existence of the complex derivative in a neighborhood implies that the real and imaginary parts of f satisfy the Cauchy-Riemann equations; conversely, satisfaction of the Cauchy-Riemann equations with continuous partial derivatives ensures analyticity. Functions that are analytic on the entire complex plane \mathbb{C} are termed entire functions; examples include polynomials and the exponential function, which exhibit this global analyticity.^[1]

Examples

Analytic functions abound in complex analysis, with many familiar forms from real analysis extending naturally to the complex plane. Polynomials provide the simplest examples; any polynomial p(z) = a_n z^n + \cdots + a_1 z + a_0, where the coefficients a_k are complex constants, is entire, meaning it is analytic everywhere in the complex plane.^[26] This follows from their finite power series representation, which converges uniformly on the entire plane. The exponential function, defined by its power series \exp(z) = \sum_{n=0}^\infty \frac{z^n}{n!}, converges everywhere and thus is entire. Similarly, the trigonometric functions sine and cosine extend to the complex domain via power series or exponential definitions, such as \sin(z) = \frac{\exp(iz) - \exp(-iz)}{2i} and \cos(z) = \frac{\exp(iz) + \exp(-iz)}{2}; both are entire functions.^[27] These satisfy the Cauchy-Riemann equations, confirming their analyticity.^[28] Rational functions, quotients of polynomials like f(z) = \frac{p(z)}{q(z)} where q is not identically zero, are analytic on the complex plane except at the poles, which are the zeros of q(z).^[29] For instance, f(z) = \frac{1}{z} is analytic everywhere except at z = 0, where it has a simple pole.^[29] To distinguish analyticity from mere differentiability, consider non-analytic examples. The complex conjugate f(z) = \bar{z} fails the Cauchy-Riemann equations everywhere and is nowhere differentiable in the complex sense.^[30] In contrast, g(z) = |z|^2 = z \bar{z} = x^2 + y^2 (with z = x + iy) is complex differentiable at the origin, where g'(0) = 0, but it is not analytic there because the Cauchy-Riemann equations hold only at that isolated point, not in any neighborhood.^[31] These cases highlight that complex differentiability at a point does not imply analyticity without satisfaction in a disk around it.

Characterizations

Cauchy-Riemann Equations

The Cauchy-Riemann equations express the condition for a complex-valued function to be complex differentiable in terms of its real and imaginary parts as functions of real variables. Consider a function f(z) = u(x, y) + i v(x, y), where z = x + i y with x, y \in \mathbb{R} and u, v: \mathbb{R}^2 \to \mathbb{R}. The function f is complex differentiable at z_0 = x_0 + i y_0 if the partial derivatives \frac{\partial u}{\partial x}(x_0, y_0), \frac{\partial u}{\partial y}(x_0, y_0), \frac{\partial v}{\partial x}(x_0, y_0), and \frac{\partial v}{\partial y}(x_0, y_0) exist and satisfy the equations

\frac{\partial u}{\partial x}(x_0, y_0) = \frac{\partial v}{\partial y}(x_0, y_0), \quad \frac{\partial u}{\partial y}(x_0, y_0) = -\frac{\partial v}{\partial x}(x_0, y_0).

^[32] These equations are necessary for the existence of the complex derivative f'(z_0) = \lim_{z \to z_0} \frac{f(z) - f(z_0)}{z - z_0}.^[32] To derive the equations, the limit defining f'(z_0) must be independent of the path of approach to z_0. Approaching along the real axis with increment h = \Delta x (real), the difference quotient becomes \frac{\Delta u}{\Delta x} + i \frac{\Delta v}{\Delta x}, which approaches \frac{\partial u}{\partial x}(x_0, y_0) + i \frac{\partial v}{\partial x}(x_0, y_0) as \Delta x \to 0. Approaching along the imaginary axis with h = i \Delta y, the quotient is \frac{\Delta u}{i \Delta y} + i \frac{\Delta v}{i \Delta y} = -i \frac{\Delta u}{\Delta y} + \frac{\Delta v}{\Delta y}, approaching -i \frac{\partial u}{\partial y}(x_0, y_0) + \frac{\partial v}{\partial y}(x_0, y_0) as \Delta y \to 0. Equating the real parts gives \frac{\partial u}{\partial x}(x_0, y_0) = \frac{\partial v}{\partial y}(x_0, y_0), and equating the imaginary parts yields \frac{\partial v}{\partial x}(x_0, y_0) = -\frac{\partial u}{\partial y}(x_0, y_0).^[32] For sufficiency, if the four partial derivatives exist and are continuous in a neighborhood of z_0 and satisfy the Cauchy-Riemann equations at z_0, then f is complex differentiable at z_0.^[33] More generally, if the partials are continuous throughout an open domain D \subset \mathbb{C} and the equations hold everywhere in D, then f is differentiable (hence analytic) at every point in D.^[34] An equivalent form of the Cauchy-Riemann equations arises in polar coordinates, where z = r e^{i \theta} with r > 0 and \theta \in \mathbb{R}, and f(z) = u(r, \theta) + i v(r, \theta). The equations become

\frac{\partial u}{\partial r} = \frac{1}{r} \frac{\partial v}{\partial \theta}, \quad \frac{\partial v}{\partial r} = -\frac{1}{r} \frac{\partial u}{\partial \theta}.

^[35] This polar version is obtained via the chain rule, relating the polar partials to the Cartesian ones through x = r \cos \theta and y = r \sin \theta.^[33] For example, the exponential function f(z) = e^z satisfies both the Cartesian and polar forms of the Cauchy-Riemann equations wherever it is defined.^[35]

Power Series Representation

A fundamental characterization of analytic functions is their local representation by power series. Specifically, if f is analytic at a point z_0 in the complex plane, then there exists a radius R > 0 such that f can be expressed as a Taylor series

f(z) = \sum_{n=0}^{\infty} \frac{f^{(n)}(z_0)}{n!} (z - z_0)^n

which converges to f(z) for all z in the open disk |z - z_0| < R.^[36] The radius R is determined by the distance from z_0 to the nearest singularity of f, ensuring uniform convergence on compact subsets within the disk.^[37] This representation arises from Cauchy's integral formula, which provides an explicit expression for the Taylor coefficients. For n \geq 0, the coefficient is given by

\frac{f^{(n)}(z_0)}{n!} = \frac{1}{2\pi i} \oint_C \frac{f(\zeta)}{(\zeta - z_0)^{n+1}} \, d\zeta,

where C is a positively oriented simple closed contour enclosing z_0 and lying within the domain of analyticity of f.^[36] Substituting these coefficients into the series yields the power series expansion, and the convergence follows from estimates on the growth of the derivatives via Cauchy's estimates.^[37] This derivation confirms that the series equals f(z) inside the disk of convergence. The power series representation is both necessary and sufficient for analyticity: a function is analytic in a domain if and only if it is representable by a convergent power series in every sufficiently small disk within that domain.^[36] Conversely, any power series \sum_{n=0}^{\infty} a_n (z - z_0)^n with positive radius of convergence defines an analytic function inside its disk of convergence, as term-by-term differentiation yields the derivatives, establishing holomorphicity.^[37] For functions analytic in an annulus or punctured disk around an isolated singularity, the Taylor series generalizes to a Laurent series \sum_{n=-\infty}^{\infty} a_n (z - z_0)^n, which converges in the region r < |z - z_0| < R and captures the behavior near the singularity through negative powers.^[38] The principal part (negative indices) distinguishes types of isolated singularities, such as removable, poles, or essential singularities.^[36]

Properties

Infinite Differentiability and Uniqueness

One fundamental property of analytic functions is their infinite differentiability. If a function f is analytic in a domain D \subseteq \mathbb{C}, then f is infinitely differentiable at every point in D, meaning all higher-order derivatives f^{(n)} exist and are continuous in D. Moreover, each derivative f^{(n)} is itself analytic in D.^[39] This smoothness arises from the local power series representation of f, where term-by-term differentiation yields convergent series for the derivatives within the disk of convergence, or alternatively from repeated application of the Cauchy-Riemann equations, which ensure the existence of higher derivatives through inductive satisfaction of the equations.^[40] Unlike real-variable functions, where infinite differentiability does not imply analyticity, complex analyticity enforces this global smoothness property.^[41] The identity theorem underscores the uniqueness of analytic functions in connected domains. Specifically, if two functions f and g are analytic in a connected domain D and agree on a subset S \subseteq D that has a limit point in D, then f \equiv g throughout D.^[42] A proof sketch relies on the uniqueness of power series expansions: at any point z_0 \in D, the coefficients of the local power series for f and g are determined by integrals or derivatives matching on S, hence identical series; by connectedness of D, this equality propagates across the domain via overlapping disks.^[39] This theorem highlights the rigid structure of analytic functions, where local agreement implies global identity, contrasting with smoother real functions that may coincide locally without being identical globally.^[43] This uniqueness extends to analytic continuation, allowing a function defined and analytic in a subdomain to be uniquely extended along paths in the larger domain while avoiding singularities. If an analytic function f is defined in a simply connected region and can be continued along a path \gamma to a larger domain, any such continuation is unique, as differing continuations would violate the identity theorem on overlapping regions.^[44] The process preserves analyticity, with the extended function remaining infinitely differentiable and satisfying the same local properties.^[45]

Liouville's Theorem

Liouville's theorem asserts that if f is an entire function—that is, analytic everywhere on the complex plane \mathbb{C}—and bounded, meaning there exists some M > 0 such that |f(z)| \leq M for all z \in \mathbb{C}, then f must be constant.^[46]^[37] To prove this, fix any point z_0 \in \mathbb{C} and consider Cauchy's estimate for the derivatives: for any r > 0,

|f^{(n)}(z_0)| \leq \frac{n! \, M}{r^n},

where the estimate arises from the Cauchy integral formula applied to a disk of radius r centered at z_0.^[37] For n \geq 1, letting r \to \infty yields |f^{(n)}(z_0)| \leq 0, so f^{(n)}(z_0) = 0. Thus, the Taylor series of f around z_0 has only the constant term, implying f is constant everywhere.^[47] A direct corollary is that no non-constant entire function can be bounded on \mathbb{C}; for example, any non-constant polynomial, being entire, must be unbounded as |z| \to \infty.^[46] This underscores the rigid global behavior imposed by analyticity on the entire plane. The theorem extends briefly to periodic entire functions: if f is entire and periodic with period \tau \neq 0 (so f(z + \tau) = f(z) for all z) and bounded, then f must be constant, as boundedness on one fundamental strip implies boundedness everywhere by periodicity.^[48]

Maximum Modulus Principle

The maximum modulus principle asserts that if f is analytic in a bounded domain D \subset \mathbb{C} and continuous on the closed set \overline{D}, then the supremum of |f(z)| for z \in \overline{D} is attained on the boundary \partial D.^[49] Moreover, if |f(z_0)| = \max_{z \in \overline{D}} |f(z)| for some z_0 in the interior of D, then f is constant throughout D.^[50] To prove the interior maximum implication, suppose |f(z_0)| = M for some interior point z_0 \in D and that f is not constant. Without loss of generality, assume f(z_0) = M (by rotating via multiplication by e^{-i\arg f(z_0)}). Since f is analytic, Cauchy's integral formula yields the mean value property: for a small disk B(z_0, r) \subset D,

f(z_0) = \frac{1}{2\pi} \int_0^{2\pi} f(z_0 + r e^{i\theta}) \, d\theta.

Taking moduli gives

M = |f(z_0)| \leq \frac{1}{2\pi} \int_0^{2\pi} |f(z_0 + r e^{i\theta})| \, d\theta \leq M,

with equality only if |f(z_0 + r e^{i\theta})| = M for almost all \theta, implying f is constant on the circle \partial B(z_0, r) by the strict convexity of the modulus. By analytic continuation and connectedness of D, f must be constant in D, a contradiction.^[51] For the boundary version, compactness of \overline{D} ensures a maximum exists; if not on \partial D, it occurs interiorly, forcing constancy.^[49] A corollary is the minimum modulus principle: if f is analytic and non-vanishing in D, continuous on \overline{D}, then the infimum of |f(z)| on \overline{D} is attained on \partial D, unless f is constant. This follows by applying the maximum modulus principle to $1/f, which is analytic in D since f has no zeros.^[52] The principle has applications to uniqueness: if two functions f and g are analytic in D and continuous on \overline{D} with f = g on \partial D, then f \equiv g in D, as |f - g| attains its maximum (zero) on the boundary, implying f - g \equiv 0.^[51] This is a local version of Liouville's theorem, which follows similarly for the entire plane \mathbb{C}.^[53]

Comparisons

Analyticity and Differentiability

In complex analysis, a function f: D \to \mathbb{C}, where D \subset \mathbb{C} is a domain, is said to be complex differentiable at a point z_0 \in D if the limit

f'(z_0) = \lim_{h \to 0} \frac{f(z_0 + h) - f(z_0)}{h}

exists, with the limit taken over complex increments h \in \mathbb{C}.^[47] This condition is stricter than real differentiability, as it requires the difference quotient to approach the same value regardless of the direction of approach in the complex plane. In contrast, a function is analytic (or holomorphic) in an open set U \subset \mathbb{C} if it is complex differentiable at every point in U. Analyticity thus demands differentiability throughout an entire neighborhood, not merely at isolated points, and this uniform behavior leads to powerful global properties. For instance, non-analytic functions may satisfy the Cauchy-Riemann equations at a single point but fail them nearby, preventing broader differentiability.^[47] A classic example of a function that is complex differentiable at a point but nowhere analytic is f(z) = |z|^2 = z \bar{z} for z \in \mathbb{C}, with f(0) = 0. At z = 0, the difference quotient simplifies to \frac{|h|^2}{h} = \bar{h}, which approaches 0 as h \to 0, so f'(0) = 0. However, for any z_0 \neq 0, the limit \lim_{h \to 0} \frac{|z_0 + h|^2 - |z_0|^2}{h} = \lim_{h \to 0} \frac{2 \operatorname{Re}(\bar{z_0} h) + |h|^2}{h} depends on the direction of h and does not exist, so f is not differentiable at any nonzero point and hence not analytic in any open disk containing 0.^[54] Continuous functions that are nowhere complex differentiable also exist, providing a complex analog to the real-variable Weierstrass function, which is continuous everywhere but differentiable nowhere on \mathbb{R}. A simple example is the complex conjugate f(z) = \bar{z}, which is continuous on all of \mathbb{C} but fails to be complex differentiable at any point, as the difference quotient \frac{\bar{z_0 + h} - \bar{z_0}}{h} = \frac{\bar{h}}{h} oscillates and has no limit unless h approaches along the real axis specifically.^[54] More pathological constructions, such as certain lacunary series or functions built via the Baire category theorem adapted to the complex plane, yield continuous functions nowhere complex differentiable, underscoring that complex differentiability is a rare property among continuous functions. The key implication of isolated differentiability is that it does not confer the structural benefits of analyticity, such as representation by a convergent power series in a neighborhood or infinite complex differentiability. Only when differentiability holds throughout an open set does the function admit a Taylor series expansion around each point, with all higher derivatives existing and the series converging to the function locally.^[55] Thus, functions differentiable merely at isolated points lack these expansive analytic properties and remain "locally pathological" despite the pointwise derivative.^[47]

Real versus Complex Analytic Functions

A real analytic function is a function f: \mathbb{R} \to \mathbb{R} that is infinitely differentiable and equals its Taylor series in some neighborhood of every point in its domain.^[56] This means that for each point a in the domain, there exists a power series \sum_{n=0}^{\infty} c_n (x - a)^n with real coefficients that converges to f(x) on an open interval around a.^[57] Classic examples include polynomials, the exponential function \exp(x), and the cosine function \cos(x), all of which have Taylor series that converge to the function everywhere on the real line.^[56] In contrast, the function defined by f(x) = e^{-1/x^2} for x > 0 and f(x) = 0 for x \leq 0 is infinitely differentiable (smooth) at every point, including at x = 0 where all derivatives vanish, but it is not real analytic at x = 0 because its Taylor series there is identically zero, which does not equal f(x) in any neighborhood.^[58] Complex analytic functions, also known as holomorphic functions, are defined on open subsets of the complex plane and satisfy the Cauchy-Riemann equations, leading to local power series representations with complex coefficients. When restricted to the real line within their domain, complex analytic functions are always real analytic, as their power series expansions yield real analytic behavior on real subsets.^[59] However, the converse does not hold: while a real analytic function on the real line always extends locally to a holomorphic function in some complex neighborhood of each point, it may not admit a holomorphic extension to the entire complex plane. For instance, the function f(x) = \frac{1}{1 + x^2} is real analytic everywhere on \mathbb{R}, with a convergent Taylor series at every real point, but any attempt to extend it holomorphically encounters poles at x = \pm i in the complex plane, preventing a global holomorphic extension.^[59] This one-way implication highlights a key rigidity in complex analytic functions compared to their real counterparts. Real analytic functions can exhibit "natural boundaries" along the real line where they remain analytic but cannot be extended holomorphically without nearby singularities in the complex plane, as seen in the example above where the poles at \pm i act as barriers. In complex analysis, such singularities enforce strict constraints, ensuring that analyticity propagates more rigidly across the plane, whereas real analytic functions allow greater flexibility without such enforced complex obstructions.^[59]

Extensions

Several Complex Variables

In several complex variables, the notion of analyticity generalizes to functions f: U \subset \mathbb{C}^n \to \mathbb{C}, where n \geq 2, defined on an open set U. Such a function is called holomorphic if it is complex differentiable with respect to each variable z_j separately, holding the others fixed, at every point in U. Equivalently, in terms of Wirtinger derivatives, f is holomorphic if \frac{\partial f}{\partial \bar{z}_j} = 0 for each j = 1, \dots, n, which extends the multivariable Cauchy-Riemann equations from the single-variable case.^[60]^[61] This condition ensures that f is infinitely differentiable and satisfies the necessary partial differential relations for complex differentiability in \mathbb{C}^n.^[62] Holomorphic functions in several variables admit local representations as power series, analogous to the Taylor expansion in one variable, but adapted to multiple dimensions. Specifically, around any point a = (a_1, \dots, a_n) \in U, f can be expanded as

f(z) = \sum_{\alpha \in \mathbb{N}^n} c_\alpha (z - a)^\alpha,

where \alpha = (\alpha_1, \dots, \alpha_n) is a multi-index, z^\alpha = z_1^{\alpha_1} \cdots z_n^{\alpha_n}, and the series converges uniformly on compact subsets of polydisks centered at a. These polydisks, defined as \{ z \in \mathbb{C}^n : |z_j - a_j| < r_j \} for radii r_j > 0, serve as natural domains for such expansions, highlighting how convergence in multiple variables depends on the product structure of \mathbb{C}^n.^[60]^[61] Unlike in one variable, where the radius of convergence determines a disk, the multivariable series may converge in more complex Reinhardt domains but still locally in polydisks.^[62] A key distinction from the single-variable theory arises in analytic continuation: holomorphic functions in \mathbb{C}^n for n \geq 2 exhibit greater flexibility. Hartogs' theorem states that if K \subset U is compact and U \setminus K is connected, then any holomorphic function on U \setminus K extends holomorphically to all of U, allowing extension across compact sets with holes— a phenomenon impossible in one variable due to potential essential singularities.^[63] This less rigid continuation underscores the higher dimensionality's role in removing isolated singularities.^[61] Domains of holomorphy further illustrate these properties, characterizing maximal regions for holomorphic extensions. A domain U \subset \mathbb{C}^n is a domain of holomorphy if there exists a holomorphic function on U that cannot be extended holomorphically to any larger open set containing U. Such domains coincide precisely with pseudoconvex domains, where pseudoconvexity is defined via the existence of a plurisubharmonic exhaustion function or, for smooth boundaries, nonnegative Levi form on the boundary.^[60] In pseudoconvex domains, holomorphic functions achieve their full extent without extendability, providing the natural settings for the theory in multiple variables.^[64]

Applications

Analytic functions play a pivotal role in physics through conformal mappings, which are analytic except at isolated poles and preserve angles, enabling the transformation of complex domains while maintaining harmonic properties essential for physical potentials. In fluid dynamics, the Joukowski transform, a specific conformal mapping given by z \mapsto z + \frac{1}{z}, models the flow around airfoils by mapping the flow past a circle to that around an airfoil shape, facilitating the design of streamlined bodies in aerodynamics.^[65] In electrostatics, potential theory leverages the fact that the real part of an analytic function is harmonic, allowing the electrostatic potential to be represented as the real part of a complex potential whose imaginary part serves as its harmonic conjugate, thus simplifying the solution of Laplace's equation in two dimensions.^[66] In engineering applications, analytic functions underpin signal processing via the z-transform, which extends the discrete-time Fourier transform and represents discrete signals as Laurent series expansions in the complex plane, enabling analysis of system stability and frequency response through pole-zero configurations on the unit circle.^[67] In control theory, entire functions—analytic everywhere in the complex plane—arise in stability analysis of systems with time delays, where quasi-polynomials model the characteristic equations, and criteria like generalized Kharitonov theorems assess Hurwitz stability by ensuring all roots lie in the left half-plane.^[68] Beyond engineering, analytic functions are central to other mathematical domains; in number theory, the Riemann zeta function, initially defined as a Dirichlet series for \Re(s) > 1, undergoes analytic continuation to a meromorphic function on the entire complex plane with a single pole at s=1, enabling the study of prime distribution via its non-trivial zeros.^[69] In complex dynamics, iterations of analytic maps, such as quadratic polynomials f_c(z) = z^2 + c, generate Julia sets as the boundaries of the sets of points with bounded orbits, revealing fractal structures that classify dynamical behavior and connectivity in the complex plane.^[70] Recent developments since 2000 highlight the role of analytic functions in quantum field theory, particularly through holomorphic bundles in string theory, where vector bundles over Calabi-Yau manifolds support the computation of Yukawa couplings as integrals of bundle-valued forms, bridging geometry and particle physics in heterotic compactifications.^[71]

References

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[PDF] 18.04 S18 Topic 2: Analytic functions - MIT OpenCourseWare
A function ( ) is analytic if it has a complex derivative ′( ). In general, the rules for computing derivatives will be familiar to you from single variable ...
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[PDF] IV.2. Power Series Representations of Analytic Functions
Nov 7, 2023 · In this section, we finally prove the fact that an analytic function of a com- plex variable (that is, a continuously differentiable function of ...
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[PDF] Lesson 14. Cauchy-Riemann equations If f(z) = u(z) + iv(z) is
Theorem. If u and v have continuous partial deriva- tives and satisfy the Cauchy-Riemann equations in a domain D, then f = u + iv is analytic in D.
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[PDF] Lecture 21: Power series expansions of analytic functions
The function z−1 is analytic on C\{0} ⊃ D|z0|(z0). • For any branch of logz, its power series expansion at z0 is log(z0) +. ∞. ∑ k=1. (−1)k−1 k. (z − z0) ...
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[PDF] Basic Properties of Analytic Functions
The Cauchy's Theorem and Cauchy's Formula are the linchpin of complex variables. Theorem 2.7 (Cauchy's Theorem). Let f be analytic in a domain D and let γ be a.
[6]
[PDF] 2 Complex Functions and the Cauchy-Riemann Equations
Here are some basic properties of analytic functions, which are easy conse- quences of the Cauchy-Riemann equations: Theorem: Let f(z) be an analytic function.
[7]
Analytic Functions of Several Complex Variables - SpringerLink
May 9, 2011 · One of the main topics will be the proof of the theorem that any meromorphic function on ℂn can be written as a quotient of two entire functions ...
[8]
Studies on a new K-symbol analytic functions generated by a ... - NIH
A subfield of complex analysis called geometric function theory investigates the geometric features and behavior of analytic functions in the complex plane.
[9]
D'Alembert's proof of the fundamental theorem of algebra
In the fragment, d'Alembert showed that algebraic operations on complex numbers, including taking rational and complex powers, produced complex numbers. Then, ...
[10]
Leonhard Euler (1707 - 1783) - Biography - MacTutor
He published his full theory of logarithms of complex numbers in 1751. Analytic functions of a complex variable were investigated by Euler in a number of ...
[11]
STUDIES IN THE HISTORY OF COMPLEX FUNCTION THEORY II
One sees, for instance, that the Cauchy Integral Theorem was already present in Cauchy's first work from the year 1814 (but only for the case of rectangles with ...
[12]
Bernhard Riemann - Biography
### Summary of Riemann's Work in the 1850s on Riemann Surfaces and Multi-Valued Functions
[13]
Karl Weierstrass (1815 - 1897) - Biography - MacTutor
He also studied entire functions, the notion of uniform convergence and functions defined by infinite products. His effort are summed up in [2] as follows ...<|control11|><|separator|>
[14]
[PDF] Weierstrass and Hadamard Factorization of Entire Functions
The infinite product expansion of the sine function comes from the partial fraction expansion of its logarithmic derivative (namely the trigonometric cosecant.
[15]
Augustin-Louis Cauchy (1789 - 1857) - Biography - MacTutor
His mathematical output remained strong and in 1814 he published the memoir on definite integrals that later became the basis of his theory of complex functions ...
[16]
CAUCHY, AUGUSTIN-LOUIS (b. Paris, France, 21 August 1789
integration path, Cauchy handled his residue theorem as a much more powerful tool than ... Cauchy published (1812) the first comprehensive treatise on ...
[17]
The Riemann Mapping Theorem | SpringerLink
Oct 12, 2021 · In his Göttingen dissertation of 1851 Bernhard Riemann enunciated (p. 40 of his Werke, 1953 ed.) and attempted to prove the famous theorem ...Missing: reference | Show results with:reference
[18]
[PDF] 1 Notes on Weierstrass Uniformization - Brown Math
The Weierstrass function is the function that the expression in Equation. 4 wants to be. Here is the definition. P(z) = 1 z2 + ∑ λ6=0.
[19]
Picard theorem - Encyclopedia of Mathematics
Jun 6, 2020 · Picard's theorem on the behaviour of an analytic function f(z) of a complex variable z near an essential singular point a is a result in classical function ...
[20]
Picard's Great Theorem -- from Wolfram MathWorld
Every analytic function assumes every complex value, with possibly one exception, infinitely often in any neighborhood of an essential singularity.
[21]
Kiyoshi Oka (1901 - 1978) - Biography - MacTutor
He became interested in unsolved problems in the theory of functions of several complex variables while working in Paris. The reason that his work took this ...Missing: contributions | Show results with:contributions
[22]
Mathematics of Professor Oka - a landscape in his mind -
His idea does not stay within the realm of several complex variables, but extends far beyond, contributing to the develop- ment of whole mathematics.
[23]
[PDF] Section 2.24. Analytic Functions
Jan 4, 2020 · Definition. A function f of complex variable z is analytic at a point z0 if it has a derivative at each point in some neighborhood of z0.
[24]
[PDF] Complex Analysis I, Christopher Bishop 2024 - Stony Brook University
Section 4.2: Equivalence of Analytic and Holomorphic. Page 17. A complex ... Thm: it There is an analytic function on D = {z : |z| < 1} that does not ...
[25]
[PDF] notes for math 520: complex analysis
Analytic functions. Definition 2.3. A function f : Ω → C (here Ω is open) is differentiable at a ∈ Ω if the derivative.<|control11|><|separator|>
[26]
[PDF] Chapter 13: Complex Numbers - Arizona Math
A function that is analytic at every point in the complex plane is called entire. Polynomials of a complex variable are entire. For instance, f (z)=3z − 7z2 + ...
[27]
[PDF] Chapter 13: Complex Numbers - Arizona Math
The hyperbolic sine and cosine, as well as the sine and cosine, are entire. We have the following relations cosh(iz) = cos(z), sinh(iz) = i sin(z) ...
[28]
[PDF] Math 113 (Spring 2024) Yum-Tong Siu
We can use the power series expansions of the sine and cosine functions in a real variable to define the sine and cosine functions as entire functions on C.
[29]
[PDF] Analytic Functions
A function is analytic at a point if its derivative exists there. It's analytic in a region if its derivative exists at every point in that region.Missing: analysis | Show results with:analysis
[30]
[PDF] Complex Numbers Analytic Functions and Singularities
Sep 6, 2011 · For example, f (z) = sinz=z is analytic at z = 0 if we define f (0) = 1, so z = 0 is a removable singularity. 1. pole: limz→o jf (z)j = 1. Then ...
[31]
Aktosun, Spring 2025, Math 5322, Supplementary Problems 1
... z0; for example f(z)=|z|2 is not analytic at z=0 but it is differentiable at z=0. For a function f(z) defined on a domain W, relate the following to each other:.
[32]
[PDF] Complex Derivatives and Cauchy-Riemann Equations Complex ...
Complex Derivatives and Cauchy-Riemann Equations. Complex Derivative. Let z = x + iy be the coordinate of the complex plane. C and let f(z) be a complex ...
[33]
[PDF] More on the Cauchy-Riemann Equations - Trinity University
The multivariate chain rule can be used to express the C-R equations in terms of polar coordinates. If x = r cos θ, y = r sinθ, then. ∂u. ∂r.
[34]
[PDF] 18.04 Complex analysis with applications - MIT Mathematics
4.6 Cauchy-Riemann equations. The Cauchy-Riemann equations are our first consequence of the fact that the limit defining f(z) must be the same no matter ...
[35]
[PDF] Polar Cauchy-Riemann Eqns
Cauchy-Riemann in polar coordinates. Suppose f is a complex valued function that is differentiable at a point z0 of the complex plane. The idea here is to ...<|control11|><|separator|>
[36]
[PDF] John B. Conway - Functions of One Complex Variable
This book is intended as a textbook for a first course in the theory of functions of one complex variable for students who are mathematically mature enough to ...
[37]
[PDF] Ahlfors, Complex Analysis
Complex Analysis has successfully maintained its place as the standard elementary text on functions of one complex variable. There is, never-.
[38]
[PDF] 18.04 S18 Topic 7: Taylor and Laurent series - MIT OpenCourseWare
When a complex function has an isolated singularity at a point we will replace Taylor series by Laurent series. Not surprisingly we will derive these series ...<|control11|><|separator|>
[39]
[PDF] Properties of Analytic Functions
Our first task is to show that if a function is analytic in a disc, then it can be represented by a power series which is convergent in that disc. The results ...
[40]
[PDF] 1. Review: Complex numbers, Analytic functions - OSU Math
Complex numbers form a field, with similar properties to real numbers. A function of a complex variable is defined on a subset of C, written as z = x + iy.
[41]
[PDF] Notes on Analytic Functions - Northwestern Math Department
Roughly, an analytic function is one which can be expressed as a power series, although the power series needed to do so may change from point to point:.
[42]
[PDF] Complex Analysis Math 147—Winter 2008 - UCI Mathematics
Mar 14, 2008 · Corollary 25.3 A power series converges to an analytic function inside the circle of con- vergence. 25.3 The Identity Theorem. Theorem 25.4 Let ...
[43]
[PDF] Review of complex analysis in one variable
To prove the identity theorem, take the difference of the two functions and consider the zero set. D. 2.1.5. Riemann extension theorem. Theorem 170.2.21 ...
[44]
[PDF] Analytic Continuation and Γ - UC Berkeley math
Nov 23, 2015 · The uniqueness property requires the domains of the two analytic continua- tions to be the same. It is not generally true that if F1 : D1 → C ...
[45]
[PDF] ANALYTIC CONTINUATION - UCI Mathematics
ANALYTIC CONTINUATION. Note: f. ∗. (t) determines all data for an analytic continuation. It is unique: its dif- ference from another function suiting (4.1) ...
[46]
Liouville's Boundedness Theorem -- from Wolfram MathWorld
Liouville's Boundedness Theorem. A bounded entire function in the complex plane C is constant. The fundamental theorem of algebra follows as a simple corollary.
[47]
[PDF] Complex Analysis Notes
Theorem 2.27 (Liouville's Theorem). If f is entire and bounded, then f is constant. Proof. Show f0 = 0 on any z0 ∈ C by Cauchy's inequality. D. Corollary ...<|control11|><|separator|>
[48]
[PDF] Periodic functions - HKUST Math Department
Hence f is a bounded entire function. Thus f is constant by Liouville's theorem. Theorem 5.4.3. The sum of the residues of an elliptic function is zero ...
[49]
[PDF] Section 4.54. Maximum Modulus Principle
Apr 26, 2020 · Note. Another version of the Maximum Modulus Theorem is the following, a proof of which is given in my online class notes for Complex Analysis ( ...
[50]
[PDF] Lecture 28: The maximum modulus theorem
Maximum modulus theorem. Assume f(z) is analytic on E, and continuous on E, where E is a bounded, connected, open set. Then the maximum of |f(z)|.
[51]
[PDF] Chapter 3: The maximum modulus principle
Dec 3, 2003 · Let f,g: G → C be analytic. If the equation f(z) = g(z) has infinitely many solutions z ∈ K, then f ≡ g. Proof. Choose ...
[52]
[PDF] The Maximum Principle - Trinity University
The modulus of a nonconstant analytic function on a domain cannot attain an absolute maximum value there. Proof. Write f (z0) = Meiθ and let g(z) = e−iθf ...
[53]
[PDF] 18.117 Lecture Notes 1
Jun 27, 2005 · Proof. Trivial consequence of the definition of holomorphic. Proof of Maximum Modulus Principle. Assume f(a) is positive (we can do this by ...
[54]
[PDF] Chapter 13: Complex Numbers - Arizona Math
A function that is analytic at every point in the complex plane is called entire. Polynomials of a complex variable are entire. For instance, f (z)=3z − 7z.Missing: non- | Show results with:non-
[55]
[PDF] Honors Real and Complex Analysis Every differentiable function of a ...
Math 55b: Honors Real and Complex Analysis. Every differentiable function of a complex variable is analytic (outline). Let R be a rectangle with sides ...
[56]
Real Analytic Function -- from Wolfram MathWorld
A real function is said to be analytic if it possesses derivatives of all orders and agrees with its Taylor series in a neighborhood of every point.
[57]
Real analytic function - Encyclopedia of Mathematics
Jan 14, 2021 · An analytic function is infinitely differentiable and its power expansion coincides with the Taylor series.Properties · Taylor series · Closure properties · Implicit and inverse function...
[58]
[PDF] Smooth versus Analytic functions
Dec 6, 2009 · We see that g and ˜g are competely different and only equal each other at a single point. So we've shown that g is not analytic.<|separator|>
[59]
[PDF] Tasty Bits of Several Complex Variables
May 20, 2025 · ... overview of differential forms, some basic algebra, measure theory ... Kiyoshi Oka . The set O(𝑈) really is a commutative ring under ...
[60]
[PDF] Several Complex Variables
For any n ≥ 1, the holomorphy or complex differentiability of a function on a domain in Cn implies its analyticity: a holomorphic function has local ...
[61]
[PDF] Chapter 2 Elementary properties of holomorphic functions in several ...
In the previous section we defined a holomorphic function as a function which is continuously differen- tiable and satisfies the Cauchy–Riemann equations in ...
[62]
[PDF] Lecture notes on several complex variables - Harold P. Boas
The discovery of Hartogs shows too that holomorphic functions of several variables never have isolated singularities and never have isolated zeroes, in contrast ...<|separator|>
[63]
[PDF] Plurisubharmonic Functions and Pseudoconvex Domains
Jun 8, 2018 · In general, pseudoconvex domains are an important tool in the study of domains of holomorphy; however they require some upbuilding from more ...
[64]
[PDF] Joukowski Airfoils
One of the more important potential flow results obtained using conformal mapping are the solutions of the potential flows past a family of airfoil shapes ...
[65]
[PDF] Lesson 35. Potential theory, Electrostatic fields - Purdue Math
It is convenient to use the analytic complex potential. F(z) = Φ(x, y) + iΨ(x, y) where z = x + iy and Ψ is a harmonic conjugate of Φ. For a complex potential, ...
[66]
[PDF] A brief introduction to Z-transforms
Oct 5, 2004 · Since the above Laurent series is an expansion in the powers of the variable z, the relationship between types of signals and possible ROCs of ...
[67]
[PDF] Maximizing Stability Degree of Control Systems under Interval ...
A generalized Kharitonov theorem for quasi-polynomials and entire functions occurring in systems with multiple and distributed delays, Proceedings of SPIE ...
[68]
[PDF] Introduction to Analytic Number Theory The Riemann zeta function ...
It follows that ζ also extends to a meromorphic function on C, which is regular except for a simple pole at s = 1, and that this analytic continuation of ζ has.
[69]
[PDF] AN INTRODUCTION TO JULIA AND FATOU SETS In this note, we ...
Holomorphic dynamics is the study of the iterates of a holomorphic map f on a complex man- ifold. Classically, this manifold is one of the complex plane C, the ...
[70]
[PDF] Holomorphic Yukawa Couplings in Heterotic String Theory - arXiv
Apr 3, 2022 · We develop a method for calculating holomorphic Yukawa couplings for such mod- els, by relating bundle-valued forms on the Calabi-Yau manifold ...