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Holomorphic function

In complex analysis, a holomorphic function is a complex-valued function f: \Omega \to \mathbb{C}, where \Omega \subset \mathbb{C} is an open domain, that is complex differentiable at every point in \Omega.^[1] Complex differentiability means that the limit \lim_{h \to 0} \frac{f(z_0 + h) - f(z_0)}{h} exists for every z_0 \in \Omega, where h approaches 0 in the complex plane.^[2] This notion is stronger than real differentiability and implies that holomorphic functions are infinitely differentiable and analytic everywhere in their domain.^[3] Writing f(z) = u(x, y) + iv(x, y) in terms of real and imaginary parts, with z = x + iy, holomorphicity is equivalent to the function satisfying 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}, provided the partial derivatives exist and are continuous in \Omega.^[4] Moreover, the set of holomorphic functions on \Omega forms a ring under pointwise addition and multiplication, and they are closed under composition.^[5] Holomorphic functions exhibit profound rigidity: they cannot be constant on any open set without being constant everywhere in the connected component of the domain, as per the identity theorem.^[2] Key theorems underscore the power of holomorphic functions. Cauchy's integral formula states that if f is holomorphic inside and on a simple closed contour \gamma, and a is inside \gamma, then f(a) = \frac{1}{2\pi i} \oint_\gamma \frac{f(z)}{z - a} \, dz, allowing the function's values to be recovered from boundary integrals.^[6] The maximum modulus principle asserts that a non-constant holomorphic function on a bounded domain attains its maximum modulus on the boundary, implying that interior maxima force constancy.^[7] Additionally, non-constant holomorphic functions are open mappings, sending open sets to open sets, which highlights their conformal (angle-preserving) nature in geometric applications.^[8]

Fundamentals

Definition

In complex analysis, a function f: D \to \mathbb{C}, where D is an open subset of the complex plane \mathbb{C}, is said to be holomorphic if it is complex differentiable at every point a \in D. Complex differentiability at a means that the limit \lim_{z \to a} \frac{f(z) - f(a)}{z - a} exists as a complex number.^[9]^[10] The domain D is an open subset of the complex plane; in complex analysis, a connected open set is often called a domain. A function cannot be holomorphic at an isolated point, as complex differentiability requires the existence of a neighborhood around that point where the function is defined.^[9] The complex derivative is formally defined as f'(z) = \lim_{h \to 0} \frac{f(z + h) - f(z)}{h}, where h is a complex number approaching zero from any direction in the complex plane.^[10] The term "holomorphic" was introduced in 1875 by Charles Briot and Jean-Claude Bouquet, students of Augustin-Louis Cauchy, to describe functions that are entire (or whole) in a local sense; this built on Cauchy's foundational work in the early 19th century, which in turn extended Leonhard Euler's 18th-century explorations of complex quantities.^[11]

Terminology

In complex analysis, the term "holomorphic function" is often used interchangeably with "analytic function" when considering functions of a single complex variable, as the two concepts are equivalent in this context: a function is holomorphic on an open set if and only if it admits a convergent power series expansion (i.e., is analytic) at every point in that set.^[12] However, the terminology carries subtle emphases—"holomorphic" typically stresses complex differentiability throughout an open domain, while "analytic" highlights the local power series representation.^[13] Another synonym, "regular function," appears in older literature, particularly associated with Karl Weierstrass's work around 1870, where it denoted functions expandable in power series without singularities in their domain of definition.^[14] Historically, additional terms like "monogenic" were employed in early texts, originating from Augustin-Louis Cauchy's contributions to complex differentiation, to describe functions possessing a unique derivative independent of direction.^[13] These archaic synonyms, including "monodromic" and "synectic" (also from Cauchy), reflect the evolving nomenclature as the field formalized in the 19th century, gradually standardizing around "holomorphic" in modern usage.^[13] Standard notation denotes a function f as holomorphic on an open set \Omega \subset \mathbb{C} if it is complex differentiable at every point in \Omega. (When \Omega is connected, it is called a domain.)^[15] Equivalently, in terms of Wirtinger derivatives, f is holomorphic on \Omega if \frac{\partial f}{\partial \bar{z}} = 0 throughout \Omega, indicating no dependence on the conjugate variable \bar{z}.^[16] A special case arises when \Omega = \mathbb{C}, the entire complex plane; such a function is termed "entire," encompassing polynomials and exponentials, which extend holomorphy globally without singularities.^[17]

Properties

Differentiability and Analyticity

In complex analysis, a function f(z) = u(x, y) + i v(x, y), where z = x + i y and u, v: \Omega \to \mathbb{R} with \Omega \subset \mathbb{C} open, is holomorphic on \Omega if and only if it is complex differentiable at every point in \Omega, which is equivalent to u and v satisfying the Cauchy-Riemann equations

\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}, \quad \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}

throughout \Omega, provided the partial derivatives exist and are continuous.^[18] This equivalence establishes that holomorphy is a local property determined by the behavior of the real and imaginary parts as solutions to a first-order system of partial differential equations.^[19] When the Cauchy-Riemann equations hold, the complex derivative f'(z) can be expressed in terms of the partial derivatives of u and v:

f'(z) = \frac{\partial u}{\partial x} + i \frac{\partial v}{\partial x} = \frac{\partial v}{\partial y} - i \frac{\partial u}{\partial y}.

This formula aligns the complex derivative with the directional derivative along the real axis, highlighting the directional nature of complex differentiability.^[18] The continuity of the partial derivatives ensures that the limit defining complex differentiability exists uniformly in suitable neighborhoods.^[20] A key consequence of holomorphy is analyticity: every holomorphic function on an open set is analytic, meaning it admits a power series expansion converging to f(z) in some disk around each point in the domain.^[19] Unlike infinitely differentiable real-valued functions, which may not be representable by Taylor series (e.g., certain smooth bump functions), holomorphic functions are infinitely differentiable in the complex sense and their Taylor series converge locally to the function itself.^[18] To see this, holomorphy implies that higher-order complex derivatives exist recursively via the Cauchy-Riemann equations, yielding infinite complex differentiability. A local power series expansion then follows from the uniformity of the derivatives in disks, ensuring convergence by estimates on the remainder terms analogous to real Taylor theorems but strengthened by the rigidity of complex differentiability.^[20] This equivalence underscores the profound difference between real and complex smoothness, where complex holomorphy enforces global analytic structure from local conditions.^[21]

Cauchy's Theorems and Formulas

One of the cornerstone results in complex analysis is Cauchy's integral theorem, which asserts that if a function f is holomorphic throughout a simply connected domain D and \gamma is a simple closed contour in D, then

\begin{equation*} \oint_{\gamma} f(z) \, dz = 0. \end{equation*}

^[22] This theorem highlights the path-independence of integrals of holomorphic functions in such domains, contrasting sharply with real analysis where integrals over closed paths generally do not vanish.^[23] The modern proof of Cauchy's integral theorem relies on Goursat's theorem, formulated in 1883, which eliminates the need to assume continuity of the derivative f' and instead requires only that f be complex differentiable at each point.^[24] Goursat's approach proceeds by considering a triangular contour T within the domain where f is holomorphic; the proof divides T into four smaller triangles, estimates the integrals over these sub-triangles using the definition of the derivative, and shows inductively that the integral over T vanishes as the subdivision refines, without invoking continuity of f'.^[25] This refinement strengthens Cauchy's original 1825 result, which assumed continuous differentiability, and extends the theorem to a broader class of holomorphic functions.^[24] A direct consequence of Cauchy's integral theorem is Cauchy's integral formula, which provides an explicit expression for the value of a holomorphic function inside a contour in terms of its boundary values. Specifically, if f is holomorphic inside and on a simple closed positively oriented contour \gamma, and a is a point interior to \gamma, then \begin{equation*} f(a) = \frac{1}{2\pi i} \oint_{\gamma} \frac{f(z)}{z - a} , dz. \end{equation*}^[26] The derivation outlines from the theorem by considering the function g(z) = f(z)/(z - a) for z \neq a, noting that g has a removable singularity at a (since f is holomorphic there), and applying the theorem to the difference between g and its holomorphic extension, yielding the formula after accounting for the winding number.^[22] This formula implies the mean value property for holomorphic functions: for f holomorphic in a disk of radius r centered at a, \begin{equation*} f(a) = \frac{1}{2\pi} \int_0^{2\pi} f(a + r e^{i\theta}) , d\theta. \end{equation*}^[27] The property follows by parametrizing the circular contour \gamma: z = a + r e^{i\theta}, substituting into the integral formula, and simplifying the resulting expression, which equates the function's value at the center to its average over the circle.^[28] This averaging principle underscores the smoothing effect of holomorphy, analogous to but stronger than properties of harmonic functions. Cauchy's integral formula extends to higher-order derivatives, revealing that all derivatives of a holomorphic function exist and can be expressed via contour integrals. For n \geq 1, \begin{equation*} f^{(n)}(a) = \frac{n!}{2\pi i} \oint_{\gamma} \frac{f(z)}{(z - a)^{n+1}} , dz, \end{equation*} where \gamma encloses a.^[26] This is obtained by formally differentiating the integral formula n times with respect to a under the integral sign, justified by the uniform convergence of the series expansion of f near a, thereby confirming the infinite differentiability of holomorphic functions.^[22]

Power Series and Laurent Series

A holomorphic function f defined on an open domain D \subset \mathbb{C} admits a local power series expansion around any point a \in D. Specifically, there exists a disk \Delta(a, r) \subset D with radius r > 0 such that f is represented by the Taylor series

f(z) = \sum_{n=0}^{\infty} a_n (z - a)^n

for all z \in \Delta(a, r), where the coefficients are given by a_n = \frac{f^{(n)}(a)}{n!}.^[29] This expansion converges uniformly on compact subsets of \Delta(a, r), and the function defined by the series is holomorphic within its disk of convergence.^[29] The radius of convergence r is at least the distance from a to the boundary of D, ensuring the series captures the local behavior within the disk of holomorphy.^[29] The radius of convergence can be precisely estimated using Cauchy's estimates. If |f(z)| \leq M on the circle |z - a| = \rho < r, then the coefficients satisfy |a_n| \leq \frac{M}{\rho^n} for all n \geq 0.^[30] These bounds imply that the series converges absolutely for |z - a| < \rho, and by choosing \rho arbitrarily close to the distance to the nearest singularity or boundary point, the full radius is determined.^[30] Moreover, the coefficients a_n are uniquely determined by the integral formula derived from Cauchy's integral theorem:

a_n = \frac{1}{2\pi i} \int_{\gamma} \frac{f(w)}{(w - a)^{n+1}} \, dw,

where \gamma is a positively oriented circle around a within the domain of holomorphy; this uniqueness follows from the fact that distinct power series agreeing on a set with limit point must be identical.^[30] For functions holomorphic in a punctured disk $0 < |z - a| < R around an isolated singularity at a, the Laurent series provides the appropriate representation:

f(z) = \sum_{n=-\infty}^{\infty} a_n (z - a)^n.

This series converges uniformly on compact annular subsets of the punctured disk, separating the principal part \sum_{n=1}^{\infty} a_{-n} (z - a)^{-n} (capturing the singularity) from the regular holomorphic part.^[31] The coefficients a_n (for both positive and negative n) are again uniquely given by the same integral formula over a suitable contour \gamma encircling a but lying in the domain of holomorphy.^[31] Power and Laurent series enable analytic continuation of holomorphic functions along paths within their domain. If a power series represents f in one disk and the path remains within the region of holomorphy, the series can be re-expanded at successive points along the path to extend the representation, preserving the function's values due to the uniqueness of analytic continuations.^[29] This process highlights the rigid global structure imposed by local series expansions on holomorphic functions.^[29]

Maximum Modulus and Other Principles

One of the fundamental global properties of holomorphic functions is the maximum modulus principle, which states that if f is holomorphic in a bounded domain \Omega \subset \mathbb{C} and continuous up to the boundary \partial \Omega, then the maximum of |f(z)| on the closure \overline{\Omega} is attained on the boundary \partial \Omega, unless f is constant throughout \Omega.^[32] This principle implies that non-constant holomorphic functions cannot achieve their maximum modulus value in the interior of the domain. A related result is the minimum modulus principle, which asserts that if f is holomorphic and non-zero in a bounded domain \Omega with f continuous up to \partial \Omega, then the minimum of |f(z)| on \overline{\Omega} is attained on \partial \Omega, unless f is constant.^[33] Additionally, since the real part \operatorname{Re} f of a holomorphic function f is harmonic, the maximum principle applies to \operatorname{Re} f, stating that if f is holomorphic in \Omega and continuous up to \partial \Omega, then the maximum of \operatorname{Re} f(z) on \overline{\Omega} is on \partial \Omega, unless f is constant.^[34] The open mapping theorem follows as a corollary: if f is a non-constant holomorphic function on a domain \Omega, then f maps open sets in \Omega to open sets in \mathbb{C}.^[35] This highlights the openness-preserving nature of non-constant holomorphic maps. A significant application is Liouville's theorem, which states that every bounded entire function—that is, holomorphic on the whole complex plane \mathbb{C}—must be constant; the proof relies on the maximum modulus principle applied to large disks, showing that the maximum on the boundary bounds the function uniformly, implying constancy via Cauchy's estimates where |f(z)| \leq \max_{| \zeta | = R} |f(\zeta)| for |z| < R.^[36]

Examples

Elementary Examples

Polynomials with complex coefficients provide the simplest examples of holomorphic functions. Any polynomial p(z) = \sum_{k=0}^n c_k z^k, where the c_k are complex constants, is entire, meaning it is holomorphic on the entire complex plane \mathbb{C}.^[37] The exponential function \exp(z) = \sum_{n=0}^\infty \frac{z^n}{n!} is another fundamental entire function, holomorphic everywhere in \mathbb{C} and periodic with period $2\pi i, satisfying \exp(z + 2\pi i) = \exp(z) for all z \in \mathbb{C}.^[38] The trigonometric functions sine and cosine extend naturally to the complex plane via their power series or exponential definitions: \sin(z) = \frac{\exp(iz) - \exp(-iz)}{2i} and \cos(z) = \frac{\exp(iz) + \exp(-iz)}{2}. As linear combinations of entire functions, both \sin(z) and \cos(z) are entire.^[39] Rational functions, formed as ratios of polynomials, are holomorphic on \mathbb{C} except at the poles where the denominator vanishes. For instance, f(z) = \frac{1}{z-1} is holomorphic on \mathbb{C} \setminus \{1\}, with a simple pole at z = 1.^[18] These examples satisfy the Cauchy-Riemann equations, confirming their holomorphicity. For \exp(z), write \exp(x + iy) = e^x \cos y + i e^x \sin y, so u(x,y) = e^x \cos y and v(x,y) = e^x \sin y. The partial derivatives are u_x = e^x \cos y = v_y and u_y = -e^x \sin y = -v_x, verifying the equations everywhere./02:_Analytic_Functions/2.06:_Cauchy-Riemann_Equations)

Non-Holomorphic Functions

Non-holomorphic functions serve as important counterexamples in complex analysis, illustrating cases where complex differentiability fails, often due to violation of the Cauchy-Riemann conditions or the non-existence of the complex derivative limit. These examples highlight the stricter requirements for holomorphy compared to real differentiability. The complex conjugate function f(z) = \bar{z}, where z = x + iy, can be expressed as f(z) = x - iy, so its real part u(x,y) = x and imaginary part v(x,y) = -y. The partial derivatives are \partial u / \partial x = 1, \partial v / \partial y = -1, which are not equal, violating the Cauchy-Riemann condition \partial u / \partial x = \partial v / \partial y everywhere in \mathbb{C}.^[40] Consequently, f(z) = \bar{z} is nowhere holomorphic.^[41] The modulus function f(z) = |z| = \sqrt{x^2 + y^2} is real-valued with imaginary part zero. At the origin, the complex derivative limit \lim_{h \to 0} \frac{|h| - |0|}{h} = \lim_{h \to 0} \frac{|h|}{h} does not exist, as approaching along the real axis yields 1 while along the imaginary axis yields -i.^[42] Thus, |z| is not complex differentiable anywhere, including the origin.^[41] Continuous real-valued functions on an open set in \mathbb{C} are holomorphic only if constant, since setting the imaginary part to zero in the Cauchy-Riemann equations implies \partial u / \partial x = 0 and \partial u / \partial y = 0, forcing u constant.^[43] For instance, f(z) = |z|^2 = x^2 + y^2 is a non-constant real-valued function that satisfies the Cauchy-Riemann equations nowhere, as \partial u / \partial x = 2x \neq 0 = \partial v / \partial y except on the imaginary axis, and similarly for the other equation.^[44] A specific example is f(z) = \bar{z}^2, which satisfies the Cauchy-Riemann equations only at z = 0 but fails elsewhere, and is complex differentiable solely at the origin where the derivative limit exists and equals zero.^[44] This illustrates that isolated differentiability does not imply holomorphy on an open domain.

Multivariable Extensions

Definition in Several Complex Variables

In the context of several complex variables, the definition of a holomorphic function extends the one-variable case to functions defined on open subsets of \mathbb{C}^n. Let \Omega \subset \mathbb{C}^n be an open set, and consider a function f: \Omega \to \mathbb{C}. The function f is holomorphic on \Omega if, for every point a \in \Omega, f is complex differentiable with respect to each variable separately. This means that, fixing all other variables, the partial derivative with respect to each z_j exists in the complex sense, analogous to the single-variable definition. Equivalently, f satisfies the system of Cauchy-Riemann equations in multiple variables, expressed using Wirtinger derivatives: \frac{\partial f}{\partial \bar{z}_j} = 0 for each j = 1, \dots, n.^[45]^[46] The domain \Omega is typically an open set in \mathbb{C}^n, which can be identified with \mathbb{R}^{2n} via the map z_j = x_j + i y_j. Holomorphic functions are often studied on polydisks, which are products of open disks in each complex variable, such as \mathbb{D}^n = \{ (z_1, \dots, z_n) \in \mathbb{C}^n : |z_j| < r_j \ \forall j \}, providing a natural generalization of the unit disk in one variable. This separate differentiability condition ensures that f behaves analytically in each direction, but unlike in one variable, it is not equivalent to mere real differentiability; a function that is continuously real differentiable on \Omega need not satisfy the complex structure conditions unless the Wirtinger derivatives vanish.^[14]^[47] The Wirtinger derivatives formalize this extension. Treating the complex variables z_j = x_j + i y_j and their conjugates \bar{z}_j = x_j - i y_j as independent, the operators are defined as

\frac{\partial}{\partial z_j} = \frac{1}{2} \left( \frac{\partial}{\partial x_j} - i \frac{\partial}{\partial y_j} \right), \quad \frac{\partial}{\partial \bar{z}_j} = \frac{1}{2} \left( \frac{\partial}{\partial x_j} + i \frac{\partial}{\partial y_j} \right).

Holomorphy requires that the anti-holomorphic part \frac{\partial f}{\partial \bar{z}_j} = 0 for all j, mirroring the single-variable Cauchy-Riemann equation but applied componentwise. This framework was introduced in the early 20th century by William F. Osgood and Eugenio Levi-Civita, who laid the foundations for analyzing such functions beyond the one-dimensional setting.^[46]

Properties in Several Variables

In several complex variables, a function holomorphic at a point a = (a_1, \dots, a_n) \in \mathbb{C}^n admits a local power series representation

f(z_1, \dots, z_n) = \sum_{k_1=0}^\infty \cdots \sum_{k_n=0}^\infty c_{k_1 \dots k_n} (z_1 - a_1)^{k_1} \cdots (z_n - a_n)^{k_n},

where the series converges absolutely in some polydisk |z_j - a_j| < r_j for j = 1, \dots, n, with r_j > 0. This representation underscores the analyticity of holomorphic functions, analogous to the one-variable case, but the domain of convergence for global power series is typically a distinguished polydisk or a more general complete Reinhardt domain, reflecting the product structure of \mathbb{C}^n. Unlike in one variable, where convergence domains are annuli or disks, the multi-index nature allows for asymmetric radii in each direction, facilitating expansions over product neighborhoods.^[46] The identity theorem in several variables asserts that if a holomorphic function f on a connected open set U \subset \mathbb{C}^n vanishes on a subset with non-empty interior, then f \equiv 0 on U. Equivalently, the zero set of a non-constant holomorphic function has empty interior, though it may have positive Hausdorff measure and codimension at least 1. This generalizes the one-variable identity theorem, where zeros are isolated, but highlights a key difference: in higher dimensions, zero sets can be complex hypersurfaces of dimension $2n-2, such as \{z_1 \cdot z_2 = 0\} for f(z_1, z_2) = z_1 z_2, without isolating points.^[48] Hartogs' theorem provides a profound extension property absent in one variable: if n \geq 2 and f is holomorphic on \mathbb{C}^n \setminus K for some compact set K, then f extends to a holomorphic function on all of \mathbb{C}^n. This removability of compact singularities contrasts sharply with the one-variable case, where compact sets like \{0\} support non-removable singularities (e.g., $1/z). The theorem implies that isolated singularities or compact zero sets cannot occur for non-constant holomorphic functions in several variables, as the extension would force constancy by the identity theorem.^[48] The Hartogs–Bochner theorem extends this phenomenon to bounded domains: for n \geq 2, if \Omega \subset \mathbb{C}^n is a bounded domain with smooth, connected boundary, then any function continuous on \overline{\Omega} and holomorphic on \Omega extends holomorphically to some neighborhood of \overline{\Omega}. On compact complex manifolds, holomorphic functions are necessarily constant, as the maximum modulus principle forces |f| to be constant on the compact space, implying f is constant; this trivial "extension" to global sections underscores the rigidity of holomorphic bundles over compact bases. While the maximum modulus principle holds in several variables—stating that a non-constant holomorphic function on a domain cannot attain its maximum modulus interiorly—counterexamples arise in related contexts, such as the failure of direct analogues for separately holomorphic or CR functions, where boundary maxima can occur without constancy. Power series expansions in multiple variables converge within polydisks, enabling uniform approximation on such sets but revealing geometric differences from one-variable disks in extension problems.^[49]^[48]

Advanced Extensions

Conformal Mappings

A non-constant holomorphic function f defined on an open domain in the complex plane induces a conformal mapping wherever f'(z) \neq 0, meaning it locally preserves oriented angles between intersecting curves. This angle-preserving property follows from the local behavior of f near a point z_0, where f(z) \approx f(z_0) + f'(z_0)(z - z_0), representing a composition of translation, rotation by \arg(f'(z_0)), and scaling by |f'(z_0))|, none of which alter angles.^[50] The preservation can also be analyzed via the argument principle, which equates the change in argument of f along a closed path to the winding number around zero, ensuring that the mapping maintains angular measures up to a uniform rotation determined by f'.^[51] The Riemann mapping theorem exemplifies the power of conformal mappings in complex analysis: for any simply connected open subset U \subset \mathbb{C} that is not the entire plane, there exists a biholomorphic (hence conformal) map from U onto the open unit disk \mathbb{D} = \{ z \in \mathbb{C} : |z| < 1 \}. This result, first articulated by Bernhard Riemann in his 1851 doctoral dissertation, implies that all such domains are conformally equivalent, providing a canonical model for simply connected regions.^[50]^[52] The automorphism group of the unit disk consists of all biholomorphic self-maps of \mathbb{D}, which take the explicit form

\phi_a(z) = e^{i\theta} \frac{z - a}{1 - \bar{a} z}, \quad |a| < 1, \quad \theta \in \mathbb{R}.

These Möbius transformations preserve the disk and capture its full conformal symmetry group.^[53] Closely related is the Schwarz lemma, which bounds holomorphic self-maps of the disk fixing the origin: if f: \mathbb{D} \to \mathbb{D} is holomorphic with f(0) = 0, then |f(z)| \leq |z| for all z \in \mathbb{D}, and |f'(0)| \leq 1, with equality holding throughout if and only if f(z) = e^{i\theta} z for some real \theta. This lemma quantifies the contraction inherent in non-automorphic maps of the disk.^[54] Riemann's contributions in the 1850s, particularly through his dissertation and subsequent habilitation lecture, pioneered the study of conformal mappings and directly influenced the development of the uniformization theorem, which classifies all Riemann surfaces (beyond simply connected plane domains) as conformally equivalent to the Riemann sphere, the complex plane, or the unit disk. The Riemann mapping theorem serves as the foundational case for uniformization, enabling the conformal modeling of simply connected surfaces in broader geometric and topological applications.^[52]^[55]

Applications in Functional Analysis

In functional analysis, the concept of holomorphy extends to functions between complex Banach spaces, providing tools for studying infinite-dimensional phenomena in operator theory and partial differential equations. A function f: U \to Y, where U is an open subset of a complex Banach space X and Y is another complex Banach space, is said to be holomorphic at a point z \in U if it admits a Fréchet derivative at z. This means there exists a bounded linear operator Df(z): X \to Y such that

f(z + h) = f(z) + Df(z) h + o(\|h\|)

as \|h\| \to 0 in the norm of X. Fréchet differentiability is stronger than Gâteaux differentiability, which only requires the existence of directional derivatives along lines, and the two coincide in finite dimensions but diverge in infinite-dimensional settings, where Fréchet ensures uniform approximation. The foundations of this theory were laid in the 1950s by Einar Hille and Ralph S. Phillips, who developed holomorphic functional calculus and semigroups on Banach spaces to address evolution equations. Their work enabled the analysis of operator semigroups that are holomorphic in sectors of the complex plane, facilitating solutions to abstract Cauchy problems of the form u'(t) = A u(t), where A generates a holomorphic semigroup. A key application arises in partial differential equations, such as the heat equation \partial_t u = \Delta u on a domain, where the solution operator forms a holomorphic semigroup analytic in a right half-plane, providing smoothing and stability properties essential for well-posedness. Entire holomorphic functions on Hilbert spaces, which are holomorphic everywhere on the space, play a prominent role in reproducing kernel Hilbert spaces and Bargmann-Segal transforms. ^[56] Higher-order derivatives of such functions, including compositions, are governed by extensions of Faà di Bruno's formula to Banach-valued maps, expressing the nth derivative as a sum over partitions involving multilinear forms. ^[57] These tools find modern applications in quantum field theory, where infinite-dimensional holomorphy aids in constructing rigorous models via analytic continuation of correlation functions in Fock spaces. ^[58]

References

[1]
[PDF] 1. Holomorphic functions 1.1. Complex-valued functions. At the start ...
A complex-valued function F of a complex variable is holomorphic in a region Ω ⊂ C if F is complex differentiable at each point of Ω. Examples: (i) ...
[2]
[PDF] A rapid review of complex function theory 1 Holomorphic functions
The derivative is (Dzf)(h) = f0(z)h. It is not only R-linear but also C-linear. Complex linearity of the derivative is equivalent to the Cauchy–Riemann equation.
[3]
[PDF] Complex analysis
The proof uses no complex analysis, so we skip it. Theorem 6.7 (Holomorphic implies analytic). If f is holomorphic in Ω, and D is an open disc centered at ...
[4]
[PDF] Short Questions I.1 [5%] State the Cauchy-Riemann equations for
I. 1 [5%] State the Cauchy-Riemann equations for a holomorphic function f(z) = u(x, y) + iv(x, y). Solution. The Cauchy-Riemann equations are ux = vy, uy = −vx.
[5]
[PDF] COMPLEX ANALYSIS Rudi Weikard
5 Basic properties of holomorphic functions. Holomorphic functions are contin- uous on their domain. Sums, differences, and products of holomorphic ...Missing: key | Show results with:key
[6]
[PDF] 13. Cauchys Integral Formula Suppose that f is a holomorphic ...
Theorem 13.6 (Cauchy's Integral Formula). Let γ be a circle with centre a and let f(z) be a holomorphic function on the circle. Then f(a) = 1. 2πi. ∫ γ f(z).
[7]
[PDF] 15 - The Maximum Modulus Principle and the Mean Value Property
Oct 7, 2025 · 2 The maximum modulus principle We saw on October 7 that if the range of a holomorphic function defined on a domain lies on a horizontal line, ...
[8]
[PDF] basic results arising from Cauchy's theorem 1. Maximum modulus ...
Nov 3, 2014 · Maximum modulus principle. 2. Open mapping theorem. 3. Rouché's ... holomorphic function is an open function, in the sense that it maps open.
[9]
[PDF] Introduction to Complex Analysis - excerpts - Mathematics
Jun 2, 2003 · Definition 2.14 A function f is holomorphic (or analytic) at a point z ∈ C if it is. C-differentiable in a neighborhood of z. Example 2.15 The ...
[10]
None
### Summary of Holomorphic Functions from the PDF
[11]
[PDF] A Brief History of Quaternions and the Theory of Holomorphic ...
Mar 19, 2009 · In the late 1700s, Euler made significant contributions to complex analysis, but most of the fundamental results which now form the core of ...
[12]
[PDF] 15 The Riemann zeta function and prime number theorem
Nov 3, 2015 · Theorem 15.5 implies that the terms “holomorphic” and “analytic” can be used interchangeably; modern usage tends to favor the former, but ...
[13]
[PDF] Theory of Complex Functions
... holomorphic function has a natural integral representation and is thereby accessible to the methods of analysis. The. CAUCHY theory was completed by J ...
[14]
[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 ...<|control11|><|separator|>
[15]
[PDF] CHAPTER 1 Basic properties of holomorphic functions Preview of ...
1.1 Holomorphic functions. Later on we will use the terms 'analytic' and 'holomorphic' interchangeably, but for the moment we will distinguish between them.
[16]
[PDF] 2. Holomorphic and Harmonic Functions - UCSD Math
In these terms, the condition above becomes ∂f ∂¯z = 0. In other words, a function f : C -→ C is holomorphic if and only if “it does not depend on ¯z”.
[17]
[PDF] 15 Elliptic curves over C (part 1)
Apr 2, 2015 · Examples of holomorphic functions include polynomials and convergent power series. Functions that admit a power series expansion with a positive ...
[18]
[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-.
[19]
[PDF] Lectures on Complex Anal- ysis - IISc Math
... Cauchy integral formula is that holomorphic functions are analytic. Theorem 9.1.2. Let Ω ⊂ C open, and f : Ω → C a holomorphic function. Then f is analytic.
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[PDF] Guide to Cultivating Complex Analysis
Dec 18, 2020 · We can quite quickly jump to holomorphic functions as solutions of the Cauchy–Riemann equations, for instance. The connection is to understand ...
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[PDF] COMPUTATIONAL COMPLEX ANALYSIS
The functions which are differentiable in this complex sense are called holomorphic functions. This book initiates a basic study of such functions. That is all ...
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[PDF] 2 Cauchy's Theorem and Its - Princeton University
As a corollary to the Cauchy integral formula, we arrive at a second remarkable fact about holomorphic functions, namely their regularity. We also obtain ...
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[PDF] LECTURE-7 1. Theorems of Cauchy and Goursat In the ... - IISc Math
Cauchy's theorem follows immediately from the theorem below, and the fundamental theorem for complex integrals. Theorem 2.1. Let f : D → C be a holomorphic ...
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[PDF] Math 113 (Spring 2024) Yum-Tong Siu 1 Theorem of Cauchy ...
Goursat improved Cauchy's theorem by removing the condition that f′(z) is continuous. It turns out that a holomorphic function is always locally expressible as ...
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[PDF] Cauchy's Theorem(s) - John McCuan
Mar 10, 2022 · Let us prove Goursat's theorem and Cauchy's theorem in a triangle. Proof of Goursat's theorem: Every triangular domain U can be partitioned into.
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[PDF] 18.04 S18 Topic 4: Cauchy's integral formula - MIT OpenCourseWare
If you learn just one theorem this week it should be Cauchy's integral formula! We start with a statement of the theorem for functions. After some examples ...
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[PDF] Math 113 (Spring 2024) Yum-Tong Siu
. Mean Value Property of Holomorphic Functions as Reformulation of Cauchy's. Integral Formula for a Point and a Circle Centered at the Point. Let f(z) be a ...
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[PDF] 14. The mean value and maximum property Definition 14.1. Let h: U
u(a + reiθ)dθ. For the second proof, pick a harmonic conjugate v and let f = u+iv. Then f is holomorphic. Cauchy's integral formula implies that f(a) = 1.
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[PDF] Contents 2 Complex Power Series - Evan Dummit
Indeed, as we will see in the next chapter, in fact every holomorphic function on a region R can be written locally as a power series, and so this method does ...
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[PDF] Contents 4 Applications of Cauchy's Integral Formula - Evan Dummit
◦ Specifically, if f(z) is holomorphic for |z − z0| ≤ r, then the value f(z0) is equal to the average value of f(z) on the circle |z − z0| = r.
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Math 246A, Notes 4: singularities of holomorphic functions
Oct 11, 2016 · We can use Laurent series to analyse an isolated singularity. Suppose that {f: D(z_0,r) \backslash \{z_0\} is holomorphic on a punctured disk ...
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[PDF] 18.117 Lecture Notes 1
Jun 27, 2005 · If ƒ € O(U) and Ref = 0, then ƒ is constant. Proof. Trivial consequence of the definition of holomorphic. Proof of Maximum Modulus Principle.
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[PDF] Section 4.54. Maximum Modulus Principle
Apr 26, 2020 · F. The Minimum Modulus Theorem. Let a function f be continuous on a closed bounded region R, and let it be analytic and not constant ...
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[PDF] Math 115 (2006-2007) Yum-Tong Siu 1 Maximum Principle Let u be ...
Then u is the real part of some holomorphic function f on {|z − a| ≤ ˜r}. ... property, the following maximum principle holds. Maximum Principle. Suppose ...
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[PDF] Open mapping and inverse function theorems. Local analytic ...
Theorem 2 (Open Mapping Theorem). If f is holomorphic and non-constant in a region Ω, then it is open (i.e., f maps open sets to open sets) ...
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[PDF] Lecture 23: Liouville's Theorem, The Fundamental Theorem of Algebra
Bounded entire functions. Liouville's Theorem. Suppose f(z) is an entire function; that is, it is analytic on C. If |f(z)| ≤ M for all z ∈ C, for some M ...
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[PDF] A course on interpolation - IMJ-PRG
An entire function is a function C → C which is analytic. (= holomorphic) in C. Examples are : polynomials, the exponential function ez = X n≥0.<|separator|>
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[PDF] Lectures on Complex Analysis - IISc Math
Feb 1, 2025 · know that the exponential function is holomorphic on the entire plane. Such functions, that are holomorphic on the entire complex plane, are ...
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[PDF] Chapter 2 Complex Analysis
complex sine and cosine functions as sin(z) ≡ eiz − e−iz. 2i and cos(z) ≡ eiz + e−iz. 2 . Being linear combinations of the entire functions exp(±iz), they ...
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[PDF] A Note on Differentiability - UCSD Math
So, f(z)=¯z is real-differentiable but not complex-differentiable. The point of all this is that the algebraic structure placed on the complex plane introduces ...
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[PDF] Notes for Math H185
Jan 31, 2012 · Complex conjugation and absolute value are also not complex differentiable. The condition of complex differentiability in Definition 2.2 is much ...<|separator|>
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[PDF] Chapter 13: Complex Numbers - Arizona Math
If f has a derivative at z = z0. , we say that f is differentiable at z = z0 . Examples: f (z)=¯z is continuous but not differentiable at z = 0. f (z) ...
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[PDF] Complex analysis examples discussion 03
Nov 17, 2020 · ... open mapping theorem: images of opens under non- constant holomorphic functions are open. The Cantor set contains no non-empty open subsets.
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[PDF] math 122b: introduction to theory of complex variables
Lets first consider an example of a non-holomorphic function. Example 1.1. The function f(z) = z is continuous for all z ∈ C. However this function is.
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[PDF] 2.2. The Complex Derivative
The complex derivative of f at z, denoted as f'(z), exists if lim (f(w) - f(z)) / (w - z) exists. If f is differentiable at every point of an open set U, then ...
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[PDF] A short introduction to several complex variables
In other words, a function D → C is holomorphic if it is continuous and holomorphic in each of its variables. Proposition 2.1 (Basic properties of holomorphic ...Missing: key | Show results with:key
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[PDF] Lecture notes on several complex variables - Harold P. Boas
Theorem 8 (Osgood, 1900). If 𝑓(𝑧1,𝑧2) is holomorphic in each variable separately, then there is a dense open subset of the domain of 𝑓 on which ...
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[PDF] A short tutorial on Wirtinger Calculus with applications in quantum ...
Dec 8, 2023 · Similarly, a function is complex- differentiable at a point if its complex derivative exists at that point, and it is holomorphic on an open set ...<|control11|><|separator|>
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[PDF] Tasty Bits of Several Complex Variables
May 20, 2025 · Complex analysis is the study of holomorphic (or complex-analytic) functions. Holomorphic functions are a generalization of polynomials, and ...
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[PDF] Hartogs-Bochner type theorem in projective space - arXiv
We will say that a complex manifold Y is disk-convex if for any compact set L ⊂ Y , there exists a compact set CL such that, for any irreducible analytic subset ...
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[PDF] The Riemann Mapping Theorem - UChicago Math
We call a function biholomorphic if it is a holomorphic bijection. Spaces between which biholomorphisms exist are called biholomorphically equivalent.
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[PDF] Conformal mapping 1. Conformal (angle-preserving) maps
Nov 23, 2014 · Exhibiting the map as a holomorphic map shows that it preserves angles. Sectors with edges elsewhere than the positive real axis can be rotated, ...
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[PDF] History of Riemann Mapping Theorem - Stony Brook University
Riemann to Prym. In his paper of 1851, Riemann claimed that. Two given simply connected plane surfaces can always be mapped onto one another in such a way ...
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[PDF] Chapter 2 Schwarz lemma and automorphisms of the disk
A holomorphic map f : U → V that is one-to-one and onto is called a biholomorphism from U to V . A biholomorphism from. U to U is called an automorphism of U.
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[PDF] 14a. Schwarz' lemma and automorphisms of H
Mar 25, 2021 · [1.1] Theorem: (Schwarz) Let f be holomorphic on the open unit disk D, with f(0) = 0, and |f(z)| ≤ 1 for all z ∈ D. Then |f(z)|≤|z| for all ...
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[PDF] Uniformization of Riemann Surfaces
Apr 5, 2004 · The uniformization theorem states that every simply connected Riemann surface is conformally equivalent to the open unit disk, ...
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Hilbert Spaces of Holomorphic Functions - SpringerLink
A Hilbert space of holomorphic functions on D is a subspace of which is equipped with the structure of a Hilbert space such that the embedding is continuous.
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[PDF] COMPLEX APPROXIMATION AND FIBERS OF BANACH ...
Many of the results of complex holomorphic functions can be translated to holomorphic functions on complex Banach spaces. ... Hille and R. S. Phillips.<|control11|><|separator|>
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Some Aspects of Infinite-Dimensional Holomorphy in Mathematical ...
This chapter discusses some of the aspects of infinite-dimensional holomorphy in mathematical physics. The theory has been very much developed in Banach ...