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Sketch of the Cynefin framework, by Edwin Stoop. The Cynefin framework (/ k ə ˈ n ɛ v ɪ n / kuh-NEV-in) [1] is a conceptual framework used to aid decision-making. [2] Created in 1999 by Dave Snowden when he worked for IBM Global Services, it has been described as a "sense-making device".
In probability theory and statistics, complex random variables are a generalization of real-valued random variables to complex numbers, i.e. the possible values a complex random variable may take are complex numbers. [1] Complex random variables can always be considered as pairs of real random variables: their real and imaginary parts.
As a differentiable function of a complex variable is equal to the sum function given by 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.
The study of functions of a complex variable is known as complex analysis and has enormous practical use in applied mathematics as well as in other branches of mathematics. Often, the most natural proofs for statements in real analysis or even number theory employ techniques from complex analysis (see prime number theorem for an example).
The standard complex normal random variable or standard complex Gaussian random variable is a complex random variable whose real and imaginary parts are independent normally distributed random variables with mean zero and variance /. [3]: p. 494 [4]: pp. 501 Formally,
A complex variable or value is usually represented as a pair of floating-point numbers. Languages that support a complex data type usually provide special syntax for building such values, and extend the basic arithmetic operations ('+', '−', '×', '÷') to act on them .
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematics that investigates functions of complex numbers.It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics, including hydrodynamics, thermodynamics, and electrical engineering.
A complex-valued function of several real variables may be defined by relaxing, in the definition of the real-valued functions, the restriction of the codomain to the real numbers, and allowing complex values. If f(x 1, …, x n) is such a complex valued function, it may be decomposed as