Complex Analysis

Branch of mathematics studying functions of a complex variable

Not to be confused with Complexity theory.

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Mathematical analysis → Complex analysis Complex analysis Complex numbers

Basic theory

Complex functions

Theorems

Geometric function theory

People

Complex analysis, a field traditionally known as the theory of functions of a complex variable, delves into the intricate world of functions that operate on complex numbers. It's a discipline that proves remarkably useful across a broad spectrum of mathematical disciplines, including the sophisticated realms of algebraic geometry and number theory, the structured elegance of analytic combinatorics, and the practical applications of applied mathematics. Its influence extends significantly into the physical sciences, touching upon areas like hydrodynamics, thermodynamics, quantum mechanics, and the esoteric field of twistor theory. Furthermore, the techniques and insights derived from complex analysis find their way into various engineering disciplines, such as nuclear, aerospace, mechanical, and electrical engineering.[1]

The core of complex analysis lies in the study of analytic functions, more specifically, holomorphic functions. This focus stems from a fundamental property: a differentiable function of a complex variable is inherently equivalent to the sum function represented by its Taylor series. This characteristic distinguishes it significantly from its real counterpart. The principles of complex analysis can be elegantly extended to encompass functions of several complex variables, opening up even more complex landscapes for investigation.

In stark contrast to real analysis, which meticulously examines real numbers and functions solely dependent on real variables, complex analysis embraces the richer, multi-dimensional nature of complex numbers.

History

Augustin-Louis Cauchy, one of the architects of complex analysis.

Complex analysis stands as one of the foundational pillars of modern mathematics, with its origins tracing back to the 18th century and even earlier. Esteemed mathematicians such as Euler, Gauss, Riemann, Cauchy, and Weierstrass are intrinsically linked to its development, with countless others contributing throughout the 20th century and beyond. The theory of conformal mappings, a significant branch of complex analysis, has found extensive utility in numerous physical applications and plays a crucial role in analytic number theory. More recently, complex analysis has experienced a resurgence in popularity, propelled by advancements in complex dynamics and the captivating imagery of fractals generated through the iterative application of holomorphic functions. Its impact is also deeply felt in theoretical physics, particularly in string theory, where it is instrumental in analyzing conformal invariants within quantum field theory.

Complex Functions

A complex function is, at its most basic, a mapping from the set of complex numbers to itself. More formally, it's a function whose domain is a subset of the complex numbers, and whose codomain is also the complex numbers. Typically, the domain is assumed to contain a non-empty open subset of the complex plane.

Any complex function can be decomposed into its real and imaginary components. If we consider a complex number $z$ from the domain, and its image $f(z)$ in the range, we can write:

$z = x + iy$ and $f(z) = f(x + iy) = u(x, y) + iv(x, y)$ ,

where $x$ , $y$ , $u(x, y)$ , and $v(x, y)$ are all real numbers. This means that a complex function $f: \mathbb{C} \to \mathbb{C}$ can be viewed as two real-valued functions, $u: \mathbb{R}^2 \to \mathbb{R}$ and $v: \mathbb{R}^2 \to \mathbb{R}$ , each depending on two real variables, $x$ and $y$ .

Alternatively, any complex-valued function $f$ defined on an arbitrary set $X$ can be considered as an ordered pair of two real-valued functions: $( \text{Re } f, \text{Im } f)$ . This perspective allows us to treat complex functions as vector-valued functions mapping into $\mathbb{R}^2$ .

Some properties of complex-valued functions, such as continuity, translate directly to the properties of their vector-valued counterparts in two real variables. However, other concepts, like differentiability, while seemingly a direct generalization, possess vastly different implications in the complex domain.

The crucial distinction lies in complex differentiability. A complex function that is differentiable at a point is not merely "smooth" in the way a real function can be; it is, in fact, analytic. This means that the function can be locally represented by a convergent power series. This strong condition implies that if two complex differentiable functions agree on a small neighborhood, they must agree everywhere on the intersection of their domains, provided those domains are connected. This principle underpins analytic continuation, a powerful technique that allows us to extend the definition of real analytic functions into the complex plane, often creating analytic functions defined on the entire complex plane except for a few "singularities." Many fundamental and special complex functions, such as the complex exponential function, complex logarithm functions, and trigonometric functions, are defined and understood through this process.

Holomorphic Functions

Main article: Holomorphic function

A complex function $f$ defined on an open subset $\Omega$ of the complex plane is called holomorphic on $\Omega$ if it is differentiable at every point within $\Omega$ . The derivative of $f$ at a point $z_0$ is defined as:

$f'(z_0) = \lim_{z \to z_0} \frac{f(z) - f(z_0)}{z - z_0}$

This definition, while appearing similar to the real derivative, carries far more weight in the complex realm. For this limit to exist, the difference quotient must converge to the same complex number irrespective of the path taken by $z$ as it approaches $z_0$ . This stringent requirement leads to profound consequences: holomorphic functions are not just differentiable once, but are infinitely differentiable. Moreover, they possess the property of analyticity, meaning they can be locally represented by a convergent power series. This implies that any holomorphic function can be approximated to arbitrary accuracy by polynomials in a neighborhood of any point in its domain. This is a stark contrast to real functions, where smooth functions can exist that are nowhere real analytic.

Many elementary functions, such as the exponential function, trigonometric functions, and all polynomial functions, when extended to complex arguments, become entire functions – holomorphic on the entire complex plane. Rational functions, being ratios of polynomials, are holomorphic everywhere except at the zeros of the denominator. Functions that are holomorphic everywhere except at a set of isolated points are termed meromorphic. Conversely, functions like $z \mapsto \Re(z)$ (the real part), $z \mapsto |z|$ (the modulus), and $z \mapsto \bar{z}$ (the complex conjugate) are nowhere holomorphic on the complex plane, a fact demonstrable through their failure to satisfy the Cauchy–Riemann equations.

The Cauchy–Riemann conditions establish a critical link between the partial derivatives of the real and imaginary components of a holomorphic function. If $f(z) = f(x + iy) = u(x, y) + iv(x, y)$ is holomorphic on a region $\Omega$ , then for any $z_0 \in \Omega$ , we have:

$\frac{\partial f}{\partial \bar{z}}(z_0) = 0$ , where $\frac{\partial}{\partial \bar{z}} := \frac{1}{2}\left(\frac{\partial}{\partial x} + i\frac{\partial}{\partial y}\right)$ .

In terms of the real and imaginary parts, $u$ and $v$ , this is equivalent to the pair of equations:

$u_x = v_y$ and $u_y = -v_x$ .

However, these conditions alone, without further assumptions on continuity, do not fully characterize holomorphic functions; the Looman–Menchoff theorem provides the necessary additional conditions.

Holomorphic functions exhibit remarkable properties. Picard's theorem states that the range of an entire function can only be one of three possibilities: the entire complex plane $\mathbb{C}$ , the complex plane with a single point removed ( $\mathbb{C} \setminus \{z_0\}$ ), or a single point $\{z_0\}$ . Essentially, if an entire function fails to take on two distinct complex values, it must be a constant function. Furthermore, a holomorphic function on a connected open set is uniquely determined by its values on any smaller open subset within it.

Conformal Map

This section is an excerpt from Conformal map. [edit]

A rectangular grid (top) and its image under a conformal map $f$ (bottom). Notice how pairs of lines initially at 90° still intersect at 90° after the transformation.

In the realm of mathematics, a conformal map is a function that preserves angles between directed curves at a point, while not necessarily preserving lengths.

More formally, consider two open subsets $U$ and $V$ of $\mathbb{R}^n$ . A function $f: U \to V$ is deemed conformal at a point $u_0 \in U$ if it preserves the angles between directed curves passing through $u_0$ , and also preserves orientation. Conformal maps maintain both angles and the shapes of infinitesimally small figures, though their size and curvature might change.

The conformal nature of a transformation can be analyzed through its Jacobian matrix. A transformation is conformal if, at each point, its Jacobian is a positive scalar multiple of a rotation matrix (an orthogonal matrix with determinant one). Some definitions broaden this to include orientation-reversing mappings whose Jacobians are scalar multiples of any orthogonal matrix. [3]

In two dimensions, orientation-preserving conformal maps are precisely the locally invertible complex analytic functions. In higher dimensions, Liouville's theorem significantly restricts the types of conformal mappings to a few specific forms. The concept of conformality can be naturally extended to maps between Riemannian or semi-Riemannian manifolds.

Major Results

Color wheel graph of the function $f(x) = (x^2 - 1)(x - 2 - i)^2 / (x^2 + 2 + 2i)$ . Hue represents the argument, brightness the magnitude.

One of the cornerstones of complex analysis is the line integral. The Cauchy integral theorem states that the line integral of a holomorphic function around a closed path is zero, provided the function is holomorphic within the region enclosed by the path. Cauchy's integral formula further allows for the computation of a holomorphic function's values inside a disk by integrating along its boundary. These path integrals are invaluable for evaluating complex real integrals, where the theory of residues plays a pivotal role, particularly in methods of contour integration. A "pole" or isolated singularity of a function is a point where the function's value becomes unbounded. The residue at such a pole is a key component in calculating line integrals, as formalized by the powerful residue theorem. The peculiar behavior of holomorphic functions near essential singularities is elucidated by Picard's theorem. Functions that possess only poles and no essential singularities are termed meromorphic. Laurent series, the complex analogue of Taylor series, are used to analyze the behavior of functions near singularities by expressing them as infinite sums of known functions.

Liouville's theorem asserts that a bounded function holomorphic on the entire complex plane must be constant. This theorem provides an elegant and concise proof for the fundamental theorem of algebra, which declares that the field of complex numbers is algebraically closed.

If a function is holomorphic throughout a connected domain, its values are completely determined by its values on any smaller subdomain. The extension of the function to the larger domain is known as analytic continuation. This principle is crucial for extending the definitions of functions like the Riemann zeta function, initially defined by infinite sums that converge only on limited domains, to nearly the entire complex plane. In some cases, such as with the natural logarithm, analytic continuation to a non-simply connected domain is not possible directly on the complex plane. However, it can be extended to a holomorphic function on a related surface called a Riemann surface.

These concepts primarily pertain to complex analysis in one variable. The theory of complex analysis in more than one complex dimension is also a vibrant area of study. While analytic properties like power series expansion generalize, many geometric properties, such as conformality, do not carry over as readily. The Riemann mapping theorem, a central result in the one-dimensional theory concerning the conformal equivalence of certain domains in the complex plane, illustrates this divergence by failing dramatically in higher dimensions.

A significant application of certain complex spaces is found in quantum mechanics, where they serve as wave functions.