complex analysis

(noun)

theory of functions of a complex variable; a branch of mathematical analysis that investigates functions of complex numbers

Related Terms

Examples of complex analysis in the following topics:

  • Applications of Taylor Series

    • An analytic function is uniquely extended to a holomorphic function on an open disk in the complex plane.
    • This makes the machinery of complex analysis available.
    • This result is of fundamental importance in many fields of mathematics (for example, in complex analysis), physics and engineering.

  • Hyperbolic Functions

    • In complex analysis, the hyperbolic functions arise as the imaginary parts of sine and cosine.
    • When considered defined by a complex variable, the hyperbolic functions are rational functions of exponentials, and are hence meromorphic.

  • Differentiation and Rates of Change in the Natural and Social Sciences

    • Derivatives and their generalizations appear in many fields of mathematics, such as complex analysis, functional analysis, differential geometry, measure theory and abstract algebra.

  • Power Series

    • These power series arise primarily in real and complex analysis, but also occur in combinatorics (under the name of generating functions) and in electrical engineering (under the name of the $Z$-transform).

  • Series

    • Unlike finite summations, infinite series need tools from mathematical analysis, specifically the notion of limits, to be fully understood and manipulated.
    • For any sequence of rational numbers, real numbers, complex numbers, functions thereof, etc., the associated series is defined as the ordered formal sum:

  • Summing an Infinite Series

    • Unlike finite summations, infinite series need tools from mathematical analysis, and specifically the notion of limits, to be fully understood and manipulated.
    • For any infinite sequence of real or complex numbers, the associated series is defined as the ordered formal sum

  • Newton's Method

    • In numerical analysis, Newton's method (also known as the Newton–Raphson method), named after Isaac Newton and Joseph Raphson, is a method for finding successively better approximations to the roots (or zeroes) of a real-valued function.
    • The method can also be extended to complex functions and to systems of equations.

  • Further Transcendental Functions

    • Formally, an analytic function $ƒ(z)$ of the real or complex variables $z_1, \cdots ,z_n$ is transcendental if $z_1, \cdots ,z_n$, $ƒ(z)$ are algebraically independent, i.e., if $ƒ$ is transcendental over the field $C(z_1, \cdots ,z_n)$.
    • In dimensional analysis, transcendental functions are notable because they make sense only when their argument is dimensionless (possibly after algebraic reduction).

  • Limit of a Function

    • The limit of a function is a fundamental concept in calculus and analysis concerning the behavior of a function near a particular input.
    • The limit of a function is a fundamental concept in calculus and analysis concerning the behavior of that function near a particular input.

  • Taylor and Maclaurin Series

    • A function that is equal to its Taylor series in an open interval (or a disc in the complex plane) is known as an analytic function.
    • The Taylor series of a real or complex-valued function $f(x)$ that is infinitely differentiable in a neighborhood of a real or complex number $a$ is the power series: