Biholomorphic illustration
In function theory , a biholomorphic or simple mapping is a bijective holomorphic mapping with a holomorphic inverse mapping . Sometimes, however, a simple mapping is also understood as an injective (not necessarily bijective), holomorphic mapping.
properties
- A bijective holomorphic mapping is always biholomorphic. In the one-dimensional case this follows directly from the theorem about implicit functions , in higher dimensions from Osgood's theorem .
- In the one-dimensional case, a biholomorphic map is a conformal map . Conversely, a bijective, continuously differentiable mapping of areas of the complex plane , which is conformal and orientation- preserving and whose derivation does not disappear, is also biholomorphic.
- A biholomorphic map is actually .
One-dimensional examples
The linear function
- (with as complex numbers) results
- for and a shift ( translation )
- for real and positive a centric stretching with the stretching factor and stretching center ;
- for and a turn . If you use polar coordinates for the number , the “point” can be identified by (see Fig. 1). Because
is obtained with the Euler formula
- .
If is set, is and thus
- .
The point (with the argument ( radian measure ) φ _{z} and the amount r = r _{z} ) thus merges into the point (with φ _{a} + φ _{z} ) and the amount r _{z} , which is a rotation. _{}_{}_{}_{}_{}
- for and a twist stretch .
For example, the rotational stretching converts the point into the image point . The image points of two further points, which can form a triangle with the first, can also be calculated so that the image triangle can be drawn and this rotational stretching can thus be easily illustrated.
- for and a twist extension with displacement.
inversion
The image
is called inversion or mirroring of a circle . With it, the inside of the circle with radius = 1 (so-called unit circle) is mapped onto the outside, the outside onto the inside, the edge of the circle merges into itself. 1 and −1 are mapped to 1 and −1, these are the two fixed points of the inversion.
Square function
With the square function
is not zero if z is not zero. If one chooses the definition and target area in such a way that the zero is not included and the restriction is bijective, then one obtains a biholomorphic map. One can for example
Select the right half-plane as the definition area and the plane slotted along the negative real axis as the target area.
From w = u + iv = (x + iy) ^{2} = x ^{2} - y ^{2} + (2xy) i, the comparison of the coefficients for the real and imaginary parts results
- u = x ^{2} - y ^{2} and v = 2xy.
The hyperbolas lying symmetrically to the x-axis (see Fig. 2)
x ^{2} - y ^{2} = const merge into vertical parallels u = const. The hyperbolas 2xy = const, which lie symmetrically to the first bisector, merge into horizontal parallels.
literature
- Klaus Fritzsche, Hans Grauert : From Holomorphic Functions to Complex Manifolds. Springer-Verlag, New York NY 2002, ISBN 0-387-95395-7 ( Graduate Texts in Mathematics 213).
- Otto Forster : Riemann surfaces. Springer, Berlin a. a. 1977, ISBN 3-540-08034-1 ( Heidelberger Taschenbücher 184), (English: Lectures on Riemann Surfaces. Corrected 2nd printing. Ibid 1991, ISBN 3-540-90617-7 [ Graduate Texts in Mathematics , 81]).
Web links
- Biholomorphic mapping . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).
Individual evidence
- ↑ Klas Diederich, Reinhold Remmert : Function theory I. Springer, Berlin 1972.