Biholomorphic illustration

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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.


  • 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 ;
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

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.


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)

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.


  • 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

Individual evidence

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