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[parent] complex function (Definition)

A complex function is a function $ f$ from a subset $ A$ of $ \mathbb{C}$ to $ \mathbb{C}$.

For every $ z = x+iy\in A\,\,\,(x,\,y \in \mathbb{R})$ the complex value $ f(z)$ can be split into its real and imaginary parts $ u$ and $ v$, respectively, which can be considered as real functions of two real variables:

$\displaystyle f(z) = u(x,\,y)+iv(x,\,y)$
The functions $ u$ and $ v$ are called the real part and the imaginary part of the complex function $ f$, respectively. Following are the notations for $ u$ and $ v$ that are used most commonly (the parentheses around $ f(z)$ may be omitted):

$\displaystyle u(x,\,y) =$   Re$\displaystyle \left(f(z)\right) = \Re\left(f(z)\right)$
$\displaystyle v(x,\,y) =$   Im$\displaystyle \left(f(z)\right) = \Im\left(f(z)\right)$

The branch of mathematics concerning differentiable complex functions is called function theory or complex analysis.



"complex function" is owned by Wkbj79. [ full author list (2) | owner history (1) ]
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See Also: real function, meromorphic, holomorphic, entire function, index of special functions, values of complex cosine

Also defines:  real part, imaginary part, function theory, complex analysis

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Attachments:
sine integral (Definition) by pahio
antiderivative of complex function (Definition) by Wkbj79
topic entry on complex analysis (Topic) by pahio
logarithmic integral (Definition) by pahio
difference quotient (Definition) by pahio
Fresnel integrals (Definition) by pahio
hyperbolic sine integral (Definition) by pahio
antiholomorphic (Definition) by pahio
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Cross-references: differentiable, variables, real functions, real, complex, subset, function
There are 112 references to this entry.

This is version 6 of complex function, born on 2004-09-24, modified 2007-05-30.
Object id is 6226, canonical name is ComplexFunction.
Accessed 10491 times total.

Classification:
AMS MSC: 03E20 (Mathematical logic and foundations :: Set theory :: Other classical set theory )
 30A99 (Functions of a complex variable :: General properties :: Miscellaneous)
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