Nachbin's theorem

This article provides a brief review of growth rates, including the idea of a function of exponential type. Classification of growth rates based on type help provide a finer tool than big O or Landau notation, since a number of theorems about the analytic structure of the bounded function and its integral transforms can be stated. A function f ( z ) {\displaystyle f(z)} defined on the complex plane is said to be of exponential type if there exist constants M {\displaystyle M} and α {\displaystyle \alpha } such that in the limit of r → ∞ {\displaystyle r\to \infty } . Here, the complex variable z {\displaystyle z} was written as z = r e i θ {\displaystyle z=re^{i\theta }} to emphasize that the limit must hold in all directions θ {\displaystyle \theta } . Letting α {\displaystyle \alpha } stand for the infimum of all such α {\displaystyle \alpha } , one then says that the function f {\displaystyle f} is of exponential type α {\displaystyle \alpha } . For example, let f ( z ) = sin ⁡ ( π z ) {\displaystyle f(z)=\sin(\pi z)} . Then one says that sin ⁡ ( π z ) {\displaystyle \sin(\pi z)} is of exponential type π {\displaystyle \pi } , since π {\displaystyle \pi } is the smallest number that bounds the growth of sin ⁡ ( π z ) {\displaystyle \sin(\pi z)} along the imaginary axis. So, for this example, Carlson's theorem cannot apply, as it requires functions of exponential type less than π {\displaystyle \pi } .