Schlicht Functions
The normalizations
- a0 = 0 and a1 = 1
mean that
- f(0) = 0 and f '(0) = 1;
this can always be assured by a linear fractional transformation: starting with an arbitrary injective holomorphic function g defined on the open unit disk and setting
Such functions g are of interest because they appear in the Riemann mapping theorem.
A family of schlicht functions are the rotated Koebe functions
with α a complex number of absolute value 1. If f is a schlicht function and |an| = n for some n ≥ 2, then f is a rotated Koebe function.
The condition of de Branges' theorem is not sufficient to show the function is schlicht, as the function
shows: it is holomorphic on the unit disc and satisfies |an|≤n for all n, but it is not injective since f(−1/2 + z) = f(−1/2 − z).
Read more about this topic: De Branges's Theorem
Famous quotes containing the word functions:
“Nobody is so constituted as to be able to live everywhere and anywhere; and he who has great duties to perform, which lay claim to all his strength, has, in this respect, a very limited choice. The influence of climate upon the bodily functions ... extends so far, that a blunder in the choice of locality and climate is able not only to alienate a man from his actual duty, but also to withhold it from him altogether, so that he never even comes face to face with it.”
—Friedrich Nietzsche (1844–1900)