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Holomorphic function

A holomorphic function is a complex function that is complex-differentiable on an open set. This article explains the definition, key consequences, examples, history, and how it differs from real differentiability.

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Overview

In mathematics, a holomorphic function is a function defined on an open subset of the complex plane that is complex-differentiable at every point of its domain. More precisely, a function f of a complex variable z is holomorphic on a region if the complex derivative f'(z) exists at every point of that region. The requirement that the derivative exist in the complex sense is far stronger than ordinary differentiability of real-variable functions and imposes a rigid structure on such functions.

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Definition and basic characteristics

Complex differentiability is defined in terms of a limit analogous to the real derivative but respecting the two-dimensional nature of complex numbers. A function that is complex-differentiable throughout an open set is often called holomorphic or, in older texts, regular. Equivalently, a function is holomorphic on a region precisely when it can be represented locally by a convergent power series (that is, it is analytic). At a practical level the Cauchy–Riemann equations give necessary and sufficient conditions for differentiability when the function is expressed in terms of real and imaginary parts.

Key properties

  • Analyticity: holomorphic functions are equal to their Taylor series around each point in their domain.
  • Infinite differentiability: they have derivatives of all orders wherever they are holomorphic.
  • Cauchy integral theorems: contour integrals of holomorphic functions satisfy strong constraints, and the Cauchy integral formula expresses derivatives by integrals.
  • Maximum modulus principle and identity theorem: the values of a holomorphic function are highly constrained by behavior on small sets.
  • Conformal mapping: where the derivative is nonzero, holomorphic maps are angle-preserving and locally invertible.

Examples and uses

Common examples of holomorphic functions include polynomials, the exponential function, trigonometric functions when extended to the complex plane, and any power series inside its radius of convergence. Rational functions are holomorphic away from their poles. Entire functions are holomorphic on the whole complex plane. Holomorphic and related techniques are widely used in complex analysis, mathematical physics, electrical engineering, fluid dynamics, and in evaluating real integrals via residue calculus.

History and development

The formal study of complex-differentiable functions developed in the 19th century with contributions from Augustin Cauchy, Bernhard Riemann, Karl Weierstrass and others. Cauchy's integral theorem and integral formula were among the first deep results showing that complex differentiability implies powerful integral relations. Subsequent work clarified the equivalence between complex differentiability and local representability by power series.

Holomorphic should not be conflated with mere differentiability of the function's real and imaginary parts: a function may have partial derivatives without being holomorphic. Related terms include meromorphic (holomorphic except at isolated poles), harmonic (solutions of Laplace's equation often given by real parts of holomorphic functions), and analytic (often used interchangeably with holomorphic in this context). For contrast with functions on the real line (real numbers), complex differentiability enforces both directional consistency and integrability properties that have no straightforward analogue in one variable.

For further background and formal statements one may consult standard references in complex analysis or introductory texts in function theory. Many resources explain proofs of the Cauchy integral formula, classification of singularities, and applications such as residue calculus and conformal mapping in more detail.

A rectangular grid is transformed into its image using the holomorphic function f

Definitions

Let {\displaystyle U\subseteq \mathbb {C} }an open subset of the complex plane and z_{0}\in Ua point of this subset. A function f\colon U\to {\mathbb {C}}is called complex differentiable at the point z_{0}if the limit

{\displaystyle \lim _{h\to 0}{\frac {f(z_{0}+h)-f(z_{0})}{h}}}

exists. It is then called f'(z_{0}).

The function fis called holomorphic at the point z_{0}if there z_{0}exists a neighborhood of in which fis complex differentiable. If is holomorphic fon all of , then Ucalled fholomorphic. Further, if U={\mathbb {C}}, then is fcalled an entire function.

Notes

Relationship between complex and real differentiability

\mathbb {C} is naturally a two-dimensional real vector space with canonical basis \{1,i\}and so a function f\colon U\to {\mathbb {C}}on an open set {\displaystyle U\subseteq \mathbb {C} }also be examined for its total differentiability in the sense of multidimensional real analysis. As is well known, f(total) differentiable in z_{0}, if an \mathbb {R} -linear mapping Lexists such that.

f(z_{0}+h)=f(z_{0})+L(h)+r(h)

where ris a function with

{\displaystyle \lim _{h\to 0}{\frac {r(h)}{|h|}}=0}

is. Now we see that the function is complex differentiable in z_{0} fexactly if it is totally differentiable there and Leven \mathbb {C} is -linear. The latter is a strong condition. It means that the representation matrix of Lwith respect to the canonical basis \{1,i\}has the form

{\begin{pmatrix}a&-b\\b&a\end{pmatrix}}

has.

Cauchy-Riemann differential equations

→ Main article: Cauchy-Riemann partial differential equations

If we now decompose a function f\left(x+iy\right)=u\left(x,y\right)+i\,v\left(x,y\right)into its real and imaginary parts with real functions u,v, then the total derivative has Las representation matrix the Jacobi matrix

{\displaystyle {\begin{pmatrix}{\frac {\partial u}{\partial x}}&{\frac {\partial u}{\partial y}}\\{\frac {\partial v}{\partial x}}&{\frac {\partial v}{\partial y}}\end{pmatrix}}.}

Consequently, the function f is complex differentiable if and only if it is real differentiable and for u,vthe Cauchy-Riemann differential equations

{\frac {\partial u}{\partial x}}={\frac {\partial v}{\partial y}}

\displaystyle {\frac {\partial u}{\partial y}}=-{\frac {\partial v}{\partial x}}

are fulfilled.

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