Associated function

of a complex variable

A function which is obtained in some manner from a given function $f(z)$ with the aid of some fixed function $F(z)$. For example, if

$$f(z)=\sum _{k=0}^{\mathrm{\infty }}{a}_k{z}^k$$

is an entire function and if

$$F(z)=\sum _{k=0}^{\mathrm{\infty }}{b}_k{z}^k$$

is a fixed entire function with $b_k\ne 0$, $k\ge 0$, then

$$\gamma (z)=\sum _{k=0}^{\mathrm{\infty }}\frac{a_k}{b_k}{z}^{-(k+1)}$$

is a function which is associated to $f(z)$ by means of the function $F(z)$; it is assumed that the series converges in some neighbourhood $|z|>R$. The function $f(z)$ is then represented in terms of $\gamma (z)$ by the formula

$$f(z)=\frac{1}{2\pi i}\underset{|t|=R_1>R}{\int }\gamma (t)F(zt)dt.$$

In particular, if

$$f(z)=\sum _{k=0}^{\mathrm{\infty }}\frac{a^k}{k!}{z}^k$$

is an entire function of exponential type and $F(z)=e^z$, then

$$\gamma (z)=\sum _{k=0}^{\mathrm{\infty }}{a}_k{z}^{-(k+1)}$$

is the Borel-associated function of $f(z)$( cf. Borel transform).