# 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]).

## Individual evidence

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