Exponential type

In complex analysis, a branch of mathematics, a holomorphic function is said to be of exponential type C if its growth is bounded by the exponential function eC|z| for some real-valued constant C as |z| → ∞. When a function is bounded in this way, it is then possible to express it as certain kinds of convergent summations over a series of other complex functions, as well as understanding when it is possible to apply techniques such as Borel summation, or, for example, to apply the Mellin transform, or to perform approximations using the Euler–Maclaurin formula. The general case is handled by Nachbin's theorem, which defines the analogous notion of Ψ-type for a general function Ψ(z) as opposed to ez. In complex analysis, a branch of mathematics, a holomorphic function is said to be of exponential type C if its growth is bounded by the exponential function eC|z| for some real-valued constant C as |z| → ∞. When a function is bounded in this way, it is then possible to express it as certain kinds of convergent summations over a series of other complex functions, as well as understanding when it is possible to apply techniques such as Borel summation, or, for example, to apply the Mellin transform, or to perform approximations using the Euler–Maclaurin formula. The general case is handled by Nachbin's theorem, which defines the analogous notion of Ψ-type for a general function Ψ(z) as opposed to ez. A function f(z) defined on the complex plane is said to be of exponential type if there exist real-valued constants M and τ such that in the limit of r → ∞ {displaystyle r o infty } . Here, the complex variable z was written as z = r e i θ {displaystyle z=re^{i heta }} to emphasize that the limit must hold in all directions θ. Letting τ stand for the infimum of all such τ, one then says that the function f is of exponential type τ. 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, since π 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. Similarly, the Euler–Maclaurin formula cannot be applied either, as it, too, expresses a theorem ultimately anchored in the theory of finite differences. A holomorphic function F ( z ) {displaystyle F(z)} is said to be of exponential type σ > 0 {displaystyle sigma >0} if for every ε > 0 {displaystyle varepsilon >0} there exists a real-valued constant A ε {displaystyle A_{varepsilon }} such that for | z | → ∞ {displaystyle |z| o infty } where z ∈ C {displaystyle zin mathbb {C} } .We say F ( z ) {displaystyle F(z)} is of exponential type if F ( z ) {displaystyle F(z)} is of exponential type σ {displaystyle sigma } for some σ > 0 {displaystyle sigma >0} . The number is the exponential type of F ( z ) {displaystyle F(z)} . The limit superior here means the limit of the supremum of the ratio outside a given radius as the radius goes to infinity. This is also the limit superior of the maximum of the ratio at a given radius as the radius goes to infinity. The limit superior may exist even if the maximum at radius r does not have a limit as r goes to infinity. For example, for the function