|
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 ''e''''C''|''z''| for some 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 ''e''''z''. ==Basic idea== A function ''f''(''z'') defined on the complex plane is said to be of exponential type if there exist constants ''M'' and ''τ'' such that : in the limit of . Here, the complex variable ''z'' was written as 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 . Then one says that is of exponential type π, since π is the smallest number that bounds the growth of 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 an theorem ultimately anchored in the theory of finite differences. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「exponential type」の詳細全文を読む スポンサード リンク
|