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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
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Cross-references: differentiable, variables, real functions, real, complex, subset, function
There are 100 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 7981 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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