icon-Funktion Math Tutorial: Domain colouring plots of complex functions

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Complex functions

General

The function theory investigates functions of complex variable functions so complex numbers whose values ​​range are also complex numbers. The complex numbers are an extension of the real numbers in the two-dimensional space. Many computational rules of real numbers can be applied to complex numbers. Was justified, the theory of complex functions mainly by Augustin-Louis Cauchy, Bernhard Riemann, and Karl Weierstrass.

Domain Colouring

The color wheel method is a method to graphically represent complex functions. Complex functions represent the two-dimensional complex plane in turn, the real and imaginary values. The color circle method used amount r = |f(z)| and angle φ the complex function value f(z) around the display color of the function value set. According to r and φ the function value is selected the color from the color wheel. The amount defines the saturation and modulo is mapped to intervals . The first interval is 0 .. 1 then follow the intervals ( 1 .. e] , (e. .. e 2 ] , (e 2 ... e 3 ], etc. the color is defined by the angle and in 6 color zones starting with split blue from 0° to 60° and ending with green from 300° to 360°. the method is designed to that the function values ​​are close together are also displayed similar color. mapping the sums on intervals of the power of e corresponds to a logarithmic representation.

Colour Wheel

A compilation of a color wheel can be put together from different points depending on which state of affairs is to be visualized. The basis for the color circle the perception of similar colors. Leaving subjects with normal color pattern according to the sensation on similarity sort, which hues are usually brought in the same order. Beginning and end of the series are around so similar that the series can be closed to form a circle.

Farbkreis

Gauss Plane

The complex numbers are two-dimensional and can be used as vectors in the Gaussian plane of numbers represent. On the horizontal axis (Re) of the real part and on the vertical axis is applied (Im) of the imaginary part of the complex number. Vectors may also be similar to either the complex number in Cartesian coordinates (x, y) or polar coordinates (r, φ) can be expressed. In the color circle method polar coordinates are used and the color wheel is placed on the manner interval Gaussian-number plane.

Gaußsche-Zahlenebene

Hue

The hue is selected according to the angle.

Colour-Wheel-hue

Angle offset for the color wheel:

φ0=

Lightness

The brightness is determined according to the following diagram.

Colour-Wheel-lightness lightness

In the interval [0,0.5) applies val = a1 * k + b1

In the interval [0.5,1) applies val = a2 * k + b2

It is: min ≤ val ≤ max

Setting the brightness gradient:

a1= b1=
a2= b2=
max= min=

Saturation

The saturation is determined according to the following diagram.

Colour-Wheel-saturation saturation

In the interval [0,0.5) applies sat = a1 * k + b1

In the interval [0.5,1) applies sat = a2 * k + b2

It is: min ≤ sat ≤ max

Setting the saturation gradient:

a1= b1=
a2= b2=
max= min=

Selecting a color scheme

Links

Domain coloring bei Wikipedia more ...neue_Seite

Color wheel graphs of complex functions bei Wikipedia more ...neue_Seite

Complex numbers more ...neue_Seite

Program code at Wikimedia more ...neue_Seite

Representation of complex functions

Linear complex function

fz = z = x+iy

withzC

Lineare-Komplexe-Funktion
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Real part of f(z):

Refz = x

Imaginary part of f(z):

Imfz = y

Amount of f(z):

fz = x2+y2

Argument φ of f(z):

φfz = atanxy

Square complex function

fz = z2

mitzC

quadratic-complex-Function
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Real part of f(z):

Refz = x2-y2

Imaginary part of f(z):

Imfz = 2xy

Amount of f(z):

fz = x2+y2

Argument φ of f(z):

φfz = atan2xyx2-y2

Fractional rational complex function

fz = 1z-a

withzCunda=α+iβ

rational-complex-Function
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Real part of f(z):

Refz = x-αx-α2-y-β2

Imaginary part of f(z):

Imfz = -y-βx-α2-y-β2

Amount of f(z):

fz = 1x-α2-y-β2

Argument φ of f(z):

φfz = atany-βα-x

Fractional linear function

fz = z+az+b

mitzCunda=α+iβ;b=γ+iδ

fraction-linear-complex-Function
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Real part of f(z):

Refz = x+αx+γ+y+βy+δx+γ2+y+δ2

Imaginary part of f(z):

Imfz = y+βx+γ-y+δx+αx+γ2+y+δ2

fz = 1z-a + 1z-b

withzCunda=α+iβ;b=γ+iδ

fraction-rational-complex-Function
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Die Funktion f ist eine Zusammensetzung der vorherigen f=f1+f2 und so ergeben sich Real- und Imaginärteil aus der Addition der einzelnen Funktionen f1 und f2.

Real part of f(z):

Ref1+f2 = Ref1+Ref2

Imaginary part of f(z):

Imf1+f2 = Imf1+Imf2

Complex e-function

fz = ez

withzC

Complex-e-Function
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Real part of f(z):

Refz = excosy

Imaginary part of f(z):

Imfz = exsiny

Amount of f(z):

fz = ex

Argument φ of f(z):

φfz = y+2nπ

exp(z) / (z-a)

fz = ezz-a

mitzCunda=α+iβ

e-Function-rational
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The function f is a product of two functions f=f1 * f2 and then the real and imaginary result follows:

Real part of f(z):

Ref1*f2 = Ref1Ref2-Imf1Imf2

Imaginary part of f(z):

Imf1*f2 = Ref1Imf2+Ref2Imf1

exp(z)/(z-a) + (z+b)/(z+c)

fz = ezz-a + z+bz+c

mitzCunda=α+iβ;b=γ+iδ;c=ε+iζ

e-Function-rational-1
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γ=
δ=
ε=
ζ=

The function f is a product of two functions f=f1 * f2 and then the real and imaginary result follows:

Real part of f(z):

Ref1*f2 = Ref1Ref2-Imf1Imf2

Imaginary part of f(z):

Imf1*f2 = Ref1Imf2+Ref2Imf1

Die Funktion f ist außerdem die Summe zweier Funktionen f=f1+f2 und so ergeben sich Real- und Imaginärteil aus der Addition der einzelnen Summanden f1 und f2.

Real part of f(z):

Ref1+f2 = Ref1+Ref2

Imaginary part of f(z):

Imf1+f2 = Imf1+Imf2

Complex sine function

fz = sinz

mitzC

Complex-sin-Function
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Real part of f(z):

Refz = sinxcoshy

Imaginary part of f(z):

Imfz = cosxsinhy

Amount of f(z):

fz = sin2x+sinh2y

Argument φ of f(z):

φfz = atancotxtanhy

x2 + i y2

fz = x2+iy2

play
x-min=
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Real part of f(z):

Refz = x2

Imaginary part of f(z):

Imfz = y2

Amount of f(z):

fz = x2+y2

Argument φ of f(z):

φfz = atany2x2