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

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

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).