complex function
A complex function is a function f from a subset A of ℂ to ℂ.
For every z=x+iy∈A(x,y∈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:
f(z)=u(x,y)+iv(x,y) | (1) |
The functions u and v are called the real part and
the imaginary part of the complex function f,
respectively. Conversely, any two functions u(x,y) and
v(x,y) defined in some subset of ℝ2 determine via
(1) a complex function f.
If f(z) especially is defined as a polynomial
of z, then both u(x,y) and v(x,y) are polynomials of x and
y with real coefficients.
Following are the notations for u and v that are used most commonly (the parentheses around f(z) may be omitted):
u(x,y)=Re(f(z))=ℜ(f(z)) |
v(x,y)=Im(f(z))=ℑ(f(z)) |
The of mathematics concerning differentiable complex functions is called function theory or complex analysis.
Title | complex function |
Canonical name | ComplexFunction |
Date of creation | 2014-02-23 10:20:21 |
Last modified on | 2014-02-23 10:20:21 |
Owner | Wkbj79 (1863) |
Last modified by | pahio (2872) |
Numerical id | 12 |
Author | Wkbj79 (2872) |
Entry type | Definition |
Classification | msc 30A99 |
Classification | msc 03E20 |
Related topic | RealFunction |
Related topic | Meromorphic |
Related topic | Holomorphic |
Related topic | Entire |
Related topic | IndexOfSpecialFunctions |
Related topic | ValuesOfComplexCosine |
Defines | real part |
Defines | imaginary part |
Defines | function theory |
Defines | complex analysis |