Sequences of Analytic Functions

The power series $${a_1}z + {a_2}{z^2} + \cdots + {a_n}{z^n} + \cdots = w$$ which converges not only for z = 0 and for which a₁≠ 0 establishes a conformal one to one mapping of a certain neighbourhood of z = 0 onto a certain neighbourhood of w = 0. Consequently the relationship between z and w can also be represented by the expansion $${b_1}w + {b_2}{w^2} + \cdots + {b_n}{w^n} + \cdots = z,$$a₁b₁ = 1. To compute the second series from the first we set $$\frac{1}{{{a_1} + {a_2}z + {a_3}{z^2} + \cdots + {a_n}{z^{n - 1}} + \cdots }} = \varphi \left( z \right).$$