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Edersee - Alte Brücke Scheid-Bringhausen
Bild: "Edersee - Alte Brücke Scheid-Bringhausen"

Mathematical Functions

Functions

f(x) = e-x2sin(x)

f(x) = e-x2sin(2x)

f(x) = e-x2sin(8x)

f(x) = e-x2cos(x)

f(x) = e-x2cos(2x)

f(x) = e-x2cos(8x)

f(x) = sin(x)cos(x)

f(x) = sin(x)cos(2x)

f(x) = sin(x)cos(3x)

f(x) = sin(x)cos(12x)

f(x) = sin(2x)cos(x)

f(x) = sin(3x)cos(x)

f(x) = sin(12x)cos(x)

Baumstümpfe aus dem Edersee
Bild: "Alte Baumstümpfe aus dem Edersee"

f(x) = sin(x)/x

f(x) = sin(4*x)/x

f(x) = sin(16*x)/x

f(x) = sin(1/x)

Hammerberg
Bild: "Durchbruch Hammerberg"

Möbius Transformation

The Möbius transformation is defined by the following formular:

Möbius Transformation

with complex parameter a,b,c,d and one complex variable z.

If ad-bc = 0 and c≠0; the result of the transformation will be the constant value a/c.

Möbius Transformation degenerated

This implies the following condition for the parameter a,b,c,d of the Möbius transformation:

Möbius Condition

With this condition we have the Möbius transformation definded.

As next step we have some examples with Möbius transformations with real values. We will have the real valued parameter a,b,c,d and the real valued variable x.

Möbius Transformation with Real Values

Asymtotic behavior for x → ∞ or x → -∞. For these limits the following constant y value will be approached:

Asymptotic behavior of the Möbius transformation

If the denominator is zero the function has a pole (c ≠ 0); with the following x value:

Pole of the Möbius transformation.

Values of the functions at x = 0.

Value of the Möbius transformation at x=0.

x value for y=0

x value of the Möbius transformation at y=0.

Examples:

Parameter settings of the examples and the resulting values of the Möbius transformation.

a b c d Determinant Asymptote y Value Pole x Value y=0 at x y Value at x=0
1 1 1 -1 -2 1 1 -1 -1
1 1 -1 1 2 -1 1 -1 1
1 -1 1 1 2 1 -1 1 -1
-1 1 1 1 -2 -1 -1 1 1

Graphs of the Möbius transforms for real parameter

y = 1/(1+x2)

y = 2x/(1+x2)

y = (1-x2)/(1+x2)

y = 1/(1-x2)

y = x/(1-x2)

Bootssteg
Bild: "Bootssteg"
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27. August 2017 Version 1.0
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