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

${\displaystyle w=f(z)=mz+n}$ (with as complex numbers) results ${\displaystyle m,n,w,z}$

• for and a shift ( translation ) ${\displaystyle m=1}$ ${\displaystyle n\neq 0}$
• for real and positive a centric stretching with the stretching factor and stretching center ; ${\displaystyle m\neq 1}$ ${\displaystyle m}$ ${\displaystyle {\tfrac {n}{1-m}}}$

Fig. 1 polar coordinates of z

• for and a turn . If you use polar coordinates for the number , the “point” can be identified by (see Fig. 1). Because ${\displaystyle |m|=1}$ ${\displaystyle n=0}$ ${\displaystyle z}$ ${\displaystyle z}$ ${\displaystyle (r,\phi )}$

${\displaystyle z=r_{z}(\cos(\phi )+i\sin(\phi ))}$

is obtained with the Euler formula

${\displaystyle e^{i\phi }=\cos(\phi )+i\sin(\phi )}$

${\displaystyle w=mz=r_{m}e^{i\phi _{m}}\cdot r_{z}e^{i\phi _{z}}}$.

If is set, is and thus ${\displaystyle |m|=1}$ ${\displaystyle r_{m}=1}$

${\displaystyle w=r_{z}e^{i\phi _{m}+\phi _{u}}}$.

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. ${\displaystyle z}$ ${\displaystyle w}$

• for and a twist stretch . ${\displaystyle |m|\neq 1}$ ${\displaystyle n=0}$

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. ${\displaystyle w=(1+i)z}$ ${\displaystyle z=2+3i}$ ${\displaystyle w=(1+i)(2+3i)=-1+5i}$

• for and a twist extension with displacement. ${\displaystyle |m|\neq 1}$ ${\displaystyle n\neq 0}$

inversion

The image

${\displaystyle w=f(z)=1/z}$

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

${\displaystyle w=f(z)=z^{2}}$

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 ${\displaystyle f'}$ ${\displaystyle f}$

${\displaystyle f\colon \{z\in \mathbb {C} \mid \operatorname {Re} z>0\}\to \mathbb {C} \setminus \{z\in \mathbb {R} \mid z\leq 0\}}$

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)

Fig. 2 Hyperbola and parallels

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

1. ↑ Klas Diederich, Reinhold Remmert : Function theory I. Springer, Berlin 1972.