Function of exponential type

An entire function $ f ( z) $ satisfying the condition

$$ | f ( z) | < A e ^ {a | z| } ,\ \ | z | < \infty ,\ A , a < \infty . $$

If $ f ( z) $ is represented by a series

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

then

$$ \overline{\lim\limits}\; _ {k \rightarrow \infty } {| a _ {k} | } ^ {1/k} < \infty . $$

The simplest examples of functions of exponential type are $ e ^ {cx} $, $ \sin \alpha z $, $ \cos \beta z $, and $ \sum _ {k=} 1 ^ {n} A _ {k} e ^ {a _ {k} z } $.

A function of exponential type has an integral representation

$$ f ( z) = \frac{1}{2 \pi i } \int\limits _ { C } \gamma ( t) e ^ {zt} d t , $$

where $ \gamma ( t) $ is the function associated with $ f ( z) $ in the sense of Borel (see Borel transform) and $ C $ is a closed contour enclosing all the singularities of $ \gamma ( t) $.

References

[1]	B.Ya. Levin, "Distribution of zeros of entire functions" , Amer. Math. Soc. (1964) (Translated from Russian) MR0156975 Zbl 0152.06703

References

[a1]	R.P. Boas, "Entire functions" , Acad. Press (1954) MR0068627 Zbl 0058.30201