"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."
The study of complex functions, including their behavior and properties.
Complex Numbers: Understanding complex numbers and their properties, such as modulus, argument, and polar form, is crucial before delving deeper into complex analysis.
Functions of a Complex Variable: In complex analysis, functions of a complex variable are the central objects of study. It includes topics such as complex differentiability, analyticity, and meromorphic functions.
Power Series: Power series can be used to represent complex functions, and it plays a crucial role in complex analysis. It also covers topics like convergence and divergence of power series.
Contour Integration: Contour integration is an essential tool in complex analysis, and it helps in evaluating complex integrals numerically. It involves calculating integrals over curves in the complex plane.
Cauchy's Theorem: Cauchy's theorem is one of the fundamental results in complex analysis. It states that the line integral of a complex function around a simple closed curve is zero, provided the function is analytic within the curve.
Residues: Residues are another important concept in complex analysis, and it helps to evaluate definite integrals using contour integration.
Conformal Mappings: Conformal mappings are one-to-one holomorphic functions, which preserve angles locally. It includes topics like stereographic projection and conformal symmetry.
Taylor and Laurent Series: Taylor and Laurent series are used to represent complex functions around a point. They are similar to power series, but they can represent more general functions.
Harmonic Functions: Harmonic functions are real-valued functions that satisfy Laplace's equation. It includes topics like the Dirichlet problem and Poisson's equation.
Singularities: Singularities are points where a complex function fails to be analytic. It includes removable singularities, poles, and essential singularities.
Analytic Continuation: Analytic continuation is the process of extending an analytic function to a larger domain. It plays a central role in many areas of mathematics, including number theory and quantum field theory.
Riemann Surfaces: Riemann surfaces are complex manifolds of dimension one, that is, they are two-dimensional surfaces that can be covered by complex coordinates. It includes topics like the Riemann mapping theorem and uniformization.
Special Functions: Special functions are functions that appear frequently in mathematical physics, including Bessel functions, Legendre polynomials, and hypergeometric functions.
Complex Dynamics: Complex dynamics is the study of how complex functions behave under iteration. It includes topics such as Julia and Mandelbrot sets, Fatou sets, and complex dynamics in one dimension.
Applications: Complex analysis has numerous applications in various fields of science and engineering, including fluid dynamics, electromagnetism, quantum physics, and signal processing.
Complex functions: The study of functions whose input and output are complex numbers.
Complex integration: The study of techniques for integrating complex functions over complex domains.
Analytic functions: The study of functions that can be expressed locally by means of convergent power series.
Cauchy-Riemann equations: A system of partial differential equations that describe the conditions under which a function is analytic.
Complex plane geometry: The study of the geometry of complex numbers and their relationships to functions.
Singularities: Points where a function becomes undefined or infinite, but the behavior of the function in the vicinity of these points can still be understood.
Complex dynamics: The study of the behavior of complex functions under iteration, including chaotic behavior.
Conformal mapping: The study of functions that preserve angles, and their applications in geometry and physics.
Potential theory: The study of harmonic functions and their relationship to the solution of certain classes of partial differential equations.
Special functions: A wide range of functions, including the gamma function, hypergeometric functions, and elliptic functions, that arise frequently in complex analysis and other fields of mathematics.
"It is helpful in many branches of mathematics, including algebraic geometry, number theory, analytic combinatorics, 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."
"Complex analysis is particularly concerned with analytic functions of a complex variable (that is, holomorphic functions)."
"Complex analysis, traditionally known as the theory of functions of a complex variable…"
"Complex analysis… investigates functions of complex numbers."
"It is helpful in many branches of mathematics, including algebraic geometry…"
"…including the branches of hydrodynamics…"
"…engineering fields such as nuclear, aerospace, mechanical and electrical engineering."
"…analytic functions of a complex variable…"
"It is helpful in many branches of mathematics…"
"It is helpful in many branches of mathematics, including… applied mathematics."
"…use of complex analysis also has applications in engineering fields…"
"By extension, use of complex analysis also has applications in engineering fields…"
"…the theory of functions of a complex variable…"
"It is helpful in many branches of mathematics, including… number theory…"
"…hydrodynamics, thermodynamics, quantum mechanics, and twistor theory."
"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…"
"Complex analysis is particularly concerned with…"