Rescaling principles in function theory (Q2782449)

Let \(f\) be a meromorphic function in some domain of the complex plane \(\mathbb C\) and let \(L\) and \(M\) be affine functions. Then \(L\circ f\circ M\) is said to be a rescaling of \(f\). In this survey, the author introduced some cases where for a sequence \(\{f_k\}\), and associated sequences \(\{L_k\}\) and \(\{M_k\}\), the rescaling sequence \(\{L_k\circ f_k\circ M_k\}\) converges to a meromorphic function in \(\mathbb C\). Having information about the sequence \(\{f_k\}\) some conclusions about \(f\) may then be true and vice versa. Here the author surveyed some of these, focussing on Zalcman's lemma and the Wiman-Valiron method, but also including a brief discussion of Poincareé functions, and discussed applications to complex differential equations, value distribution and the Ahlfors theory of covering surfaces.NEWLINENEWLINEFor the entire collection see [Zbl 0980.00021].