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

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

is an entire function and if

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

is a fixed entire function with $b _ {k} \neq 0$, $k \geq 0$, then

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

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

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

In particular, if

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

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

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

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