Complex-Valued Function

A Complex-Valued Function is a mathematical function that takes complex numbers as inputs and returns complex numbers as outputs.

• Context:
  • It can (typically) be represented as [math]\displaystyle{ f(z) = u(x, y) + iv(x, y) }[/math], where [math]\displaystyle{ z = x + iy }[/math] is a complex number and [math]\displaystyle{ u(x, y) }[/math] and [math]\displaystyle{ v(x, y) }[/math] are real-valued functions of the real variables [math]\displaystyle{ x }[/math] and [math]\displaystyle{ y }[/math].
  • It can (often) be studied for its holomorphic (analytic) properties, meaning that it is complex differentiable in an open domain, leading to powerful results in complex analysis, such as the Cauchy-Riemann equations.
  • It can range from simple functions like [math]\displaystyle{ f(z) = z^n }[/math], where [math]\displaystyle{ n }[/math] is an integer, to more complex functions involving transcendental elements like [math]\displaystyle{ f(z) = e^z }[/math].
  • It can include special types of complex functions such as Meromorphic Functions, which are holomorphic except at isolated poles, and Entire Functions, which are holomorphic over the entire complex plane.
  • It can have applications in solving physical problems where waveforms, electromagnetic fields, or quantum states are naturally described by complex functions.
  • It can be explored through various transformations, such as the Fourier Transform or Laplace Transform, which are integral transforms that convert complex functions into different representations.
  • It can exhibit significant properties, such as conformality in the case of holomorphic functions, where angles between curves are preserved under the function's mapping.
  • ...
• Example(s):
  • Polynomial Functions of a complex variable, such as [math]\displaystyle{ f(z) = z^2 + 1 }[/math], which are simple examples of complex functions.
  • Exponential Functions, like [math]\displaystyle{ f(z) = e^z }[/math], which exhibit periodic behavior in the complex plane due to their relation to trigonometric functions via Euler's formula.
  • Logarithmic Functions, such as [math]\displaystyle{ f(z) = \log(z) }[/math], which are multi-valued and require branch cuts for a well-defined single-valued function.
  • Trigonometric Functions, like [math]\displaystyle{ f(z) = \sin(z) }[/math], which are periodic and have applications in complex analysis, particularly in Fourier analysis.
  • Rational Functions, like [math]\displaystyle{ f(z) = \frac{P(z)}{Q(z)} }[/math], where [math]\displaystyle{ P(z) }[/math] and [math]\displaystyle{ Q(z) }[/math] are polynomials and [math]\displaystyle{ Q(z) }[/math] is non-zero, providing examples of functions with poles and zeros.
  • Entire Functions, like [math]\displaystyle{ f(z) = \sin(z) }[/math] and [math]\displaystyle{ f(z) = e^z }[/math], which are holomorphic across the entire complex plane.
  • Meromorphic Functions, such as the Riemann Zeta Function, which are holomorphic except at isolated poles.
  • ...
• Counter-Example(s):
  • Real Function, which maps real numbers to real numbers and does not involve complex numbers.
  • Vector-Valued Function, which maps elements from a domain to a vector space rather than to complex numbers.
  • Distribution (Mathematics), which generalizes functions to objects like the Dirac delta function and is not necessarily defined in terms of complex numbers.
• See: Holomorphic Function, Meromorphic Function, Entire Function, Complex Plane, Cauchy-Riemann Equations, Complex Analysis, Fourier Transform, Laplace Transform, Numeric-Output Function, Metric Function, Complex Function.

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References

2013

• http://en.wikipedia.org/wiki/Complex-valued_function
  • In mathematics, a complex-valued function (sometimes referred to as complex function) is a function whose values are complex numbers. In other words, it is a function that assigns a complex number to each member of its domain. This domain does not necessarily have any structure related to complex numbers. Most important uses of such functions in complex analysis and in functional analysis are explicated below.

    A vector space and a commutative algebra of functions over complex numbers can be defined in the same way as for real-valued functions. Also, any complex-valued function f on an arbitrary set X can be considered as an ordered pair of two real-valued functions: (Ref, Imf) or, alternatively, as a real-valued function φ on X × {0, 1} (the disjoint union of two copies of X) such that for any x: :[math]\displaystyle{ \mathrm{Re } f(x) = F(x, 0) }[/math] :[math]\displaystyle{ \mathrm{Im} f(x) = F(x, 1) }[/math] Some properties of complex-valued functions (such as measurability and continuity) are nothing more than corresponding properties of real-valued functions.

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