Complex-Valued Function

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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.


References

2013