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 $$\limsup _ {k \rightarrow \infty } {| a _ {k} | } ^ {1/k} < \infty .$$

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