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Mathematical analysis → Complex analysis
Complex analysis
Complex numbers
Real number Imaginary number Complex plane Complex conjugate Unit complex number
Complex functions
Complex-valued function Analytic function Holomorphic function Cauchy–Riemann equations Formal power series
Basic theory
Zeros and poles Cauchy's integral theorem Local primitive Cauchy's integral formula Winding number Laurent series Isolated singularity Residue theorem Conformal map Schwarz lemma Harmonic function Laplace's equation
Geometric function theory
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Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebraic geometry, number theory, analytic combinatorics, and applied mathematics, as well as in physics, including the branches of hydrodynamics, thermodynamics, quantum mechanics, and twistor theory. By extension, use of complex analysis also has applications in engineering fields such as nuclear, aerospace, mechanical and electrical engineering.^[1]

As a differentiable function of a complex variable is equal to its Taylor series (that is, it is analytic), complex analysis is particularly concerned with analytic functions of a complex variable, that is, holomorphic functions. The concept can be extended to functions of several complex variables.

History

Complex analysis is one of the classical branches in mathematics, with roots in the 18th century and just prior. Important mathematicians associated with complex numbers include Euler, Gauss, Riemann, Cauchy, Gösta Mittag-Leffler, Weierstrass, and many more in the 20th century. Complex analysis, in particular the theory of conformal mappings, has many physical applications and is also used throughout analytic number theory. In modern times, it has become very popular through a new boost from complex dynamics and the pictures of fractals produced by iterating holomorphic functions. Another important application of complex analysis is in string theory which examines conformal invariants in quantum field theory.

Complex functions

A complex function is a function from complex numbers to complex numbers. In other words, it is a function that has a (not necessarily proper) subset of the complex numbers as a domain and the complex numbers as a codomain. Complex functions are generally assumed to have a domain that contains a nonempty open subset of the complex plane.

For any complex function, the values ${\displaystyle z}$ from the domain and their images ${\displaystyle f(z)}$ in the range may be separated into real and imaginary parts:

${\displaystyle z=x+iy\quad {\text{ and }}\quad f(z)=f(x+iy)=u(x,y)+iv(x,y),}$

where ${\displaystyle x,y,u(x,y),v(x,y)}$ are all real-valued.

In other words, a complex function ${\displaystyle f:\mathbb {C} \to \mathbb {C} }$ may be decomposed into

${\displaystyle u:\mathbb {R} ^{2}\to \mathbb {R} \quad }$ and ${\displaystyle \quad v:\mathbb {R} ^{2}\to \mathbb {R} ,}$

i.e., into two real-valued functions (${\displaystyle u}$, ${\displaystyle v}$) of two real variables (${\displaystyle x}$, ${\displaystyle y}$).

Similarly, any complex-valued function f on an arbitrary set X (is isomorphic to, and therefore, in that sense, it) can be considered as an ordered pair of two real-valued functions: (Re f, Im f) or, alternatively, as a vector-valued function from X into ${\displaystyle \mathbb {R} ^{2}.}$

Some properties of complex-valued functions (such as continuity) are nothing more than the corresponding properties of vector valued functions of two real variables. Other concepts of complex analysis, such as differentiability, are direct generalizations of the similar concepts for real functions, but may have very different properties. In particular, every differentiable complex function is analytic (see next section), and two differentiable functions that are equal in a neighborhood of a point are equal on the intersection of their domain (if the domains are connected). The latter property is the basis of the principle of analytic continuation which allows extending every real analytic function in a unique way for getting a complex analytic function whose domain is the whole complex plane with a finite number of curve arcs removed. Many basic and special complex functions are defined in this way, including the complex exponential function, complex logarithm functions, and trigonometric functions.

Holomorphic functions

Complex functions that are differentiable at every point of an open subset ${\displaystyle \Omega }$ of the complex plane are said to be holomorphic on ${\displaystyle \Omega }$. In the context of complex analysis, the derivative of ${\displaystyle f}$ at ${\displaystyle z_{0}}$ is defined to be^[2]

${\displaystyle f'(z_{0})=\lim _{z\to z_{0}}{\frac {f(z)-f(z_{0})}{z-z_{0}}}.}$

Superficially, this definition is formally analogous to that of the derivative of a real function. However, complex derivatives and differentiable functions behave in significantly different ways compared to their real counterparts. In particular, for this limit to exist, the value of the difference quotient must approach the same complex number, regardless of the manner in which we approach ${\displaystyle z_{0}}$ in the complex plane. Consequently, complex differentiability has much stronger implications than real differentiability. For instance, holomorphic functions are infinitely differentiable, whereas the existence of the nth derivative need not imply the existence of the (n + 1)th derivative for real functions. Furthermore, all holomorphic functions satisfy the stronger condition of analyticity, meaning that the function is, at every point in its domain, locally given by a convergent power series. In essence, this means that functions holomorphic on ${\displaystyle \Omega }$ can be approximated arbitrarily well by polynomials in some neighborhood of every point in ${\displaystyle \Omega }$. This stands in sharp contrast to differentiable real functions; there are infinitely differentiable real functions that are nowhere analytic; see Non-analytic smooth function § A smooth function which is nowhere real analytic.

Most elementary functions, including the exponential function, the trigonometric functions, and all polynomial functions, extended appropriately to complex arguments as functions ${\displaystyle \mathbb {C} \to \mathbb {C} }$, are holomorphic over the entire complex plane, making them entire functions, while rational functions ${\displaystyle p/q}$, where p and q are polynomials, are holomorphic on domains that exclude points where q is zero. Such functions that are holomorphic everywhere except a set of isolated points are known as meromorphic functions. On the other hand, the functions ${\displaystyle z\mapsto \Re (z)}$, ${\displaystyle z\mapsto |z|}$, and ${\displaystyle z\mapsto {\bar {z}}}$ are not holomorphic anywhere on the complex plane, as can be shown by their failure to satisfy the Cauchy–Riemann conditions (see below).

An important property of holomorphic functions is the relationship between the partial derivatives of their real and imaginary components, known as the Cauchy–Riemann conditions. If ${\displaystyle f:\mathbb {C} \to \mathbb {C} }$, defined by ${\displaystyle f(z)=f(x+iy)=u(x,y)+iv(x,y)}$, where ${\displaystyle x,y,u(x,y),v(x,y)\in \mathbb {R} }$, is holomorphic on a region ${\displaystyle \Omega }$, then for all ${\displaystyle z_{0}\in \Omega }$,

${\displaystyle {\frac {\partial f}{\partial {\bar {z}}}}(z_{0})=0,\ {\text{where }}{\frac {\partial }{\partial {\bar {z}}}}\mathrel {:=} {\frac {1}{2}}\left({\frac {\partial }{\partial x}}+i{\frac {\partial }{\partial y}}\right).}$

In terms of the real and imaginary parts of the function, u and v, this is equivalent to the pair of equations ${\displaystyle u_{x}=v_{y}}$ and ${\displaystyle u_{y}=-v_{x}}$Zdroj:https://en.wikipedia.org?pojem=Complex_Analysis
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