Biholomorphism

In the theory of functions a biholomorphic or simple mapping is a bijective holomorphic map with holomorphic inverse mapping. Sometimes one sees but with a simple illustration and an injective (not necessarily bijective ), holomorphic map.

Properties

• A bijective holomorphic map is always biholomorphic. In the one-dimensional case, this follows directly from the implicit function theorem in higher dimensions from the set of Osgood.
• In the one-dimensional case, a biholomorphic map is a conformal mapping. Conversely, a bijective, continuously differentiable mapping of regions of the complex plane, which is consistent and orientation-preserving and its derivative does not vanish, also biholomorphic.
• A biholomorphic map actually is.

One-dimensional examples

The linear function

• And for a shift ( translation)
• For real and positive a central dilation with the stretch factor stretch and center;

• And for a rotation. Namely one uses polar coordinates for the number, then the "point" are characterized by ( Fig. 1). because

Is obtained with the Euler formula

If it is set, and thus

The point ( with the argument ( in radians )? Z and the amount r = rz ) thus goes to the point ( with øa ? Z ) and the amount rz over, which is a rotation.

• And for a rotation and expansion.

For example, the rotation and expansion causes the point on the image point. The pixels of two further points that can form a triangle with the first, can be well calculated, so that the image triangle can be drawn and therefore this rotation and expansion can be illustrated easily.

• And for a rotation-dilation with postponement.

Inversion

The figure

Called inversion or mirroring circuit. For her the inside of the circle with radius = 1 (so-called unit circle ) on the exterior, the exterior is mapped into the interior, the edge of the circle into itself over itself. 1 and -1 are mapped to 1 and -1, are the two fixed points of the inversion.

Square function

In the square function

Is not zero, when z is not zero. If you choose definition and target area so that the zero is not included, and the restriction of is bijective, we obtain thus a biholomorphic map. Can for example be

Choose so as domain the right half-plane and as the target area along the negative real axis slotted level.

Of w = u iv = ( x iy) = 2 x 2 - y 2 ( 2xy ) i, a comparison of the coefficients in the real and imaginary

The hyperbolas symmetrically located to the x- axis ( see Figure 2)

X2 - y2 = const go into vertical parallels u = const on. The symmetrically lying to the first bisecting line hyperbolas 2xy = const pass into horizontally extending parallels.

Swell

• 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 and others 1977, ISBN 3-540-08034-1 ( paperback Heidelberg 184 ), ( English: .. Lectures on Riemann Surfaces Corrected 2nd printing ibid, 1991, ISBN 3-540-90617-7 ( Graduate Texts in Mathematics 81) ).
• Michiel Hazewinkel (ed.): biholomorphic mapping. In: Encyclopaedia of Mathematics. Springer -Verlag, Berlin 2002, ISBN 1-4020-0609-8 ( online ).