Integral function of order alpha

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Discussion Overview

The discussion revolves around the concept of an integral function of order \(\alpha\) within the context of complex analytic functions. Participants explore the meaning of this terminology and its implications in complex analysis.

Discussion Character

  • Conceptual clarification
  • Technical explanation

Main Points Raised

  • One participant inquires about the meaning of a function being described as an integral function of order \(\alpha\), noting that \(\alpha\) is a real constant.
  • Another participant suggests that an integral function likely refers to an analytic function from \(\mathbb{C}\) to \(\mathbb{C}\), as opposed to an analytic function defined on a punctured complex plane.
  • A third participant provides a link to a resource that may clarify the concept of function order.
  • The initial poster expresses satisfaction with the resource, noting that they were previously unfamiliar with the term "integral function" being synonymous with "entire function."

Areas of Agreement / Disagreement

Participants generally agree on the connection between integral functions and analytic functions, but the specific implications of the order \(\alpha\) remain unclear and are not fully resolved.

Contextual Notes

The discussion does not clarify the specific mathematical properties or definitions associated with the order of an integral function, leaving some assumptions and definitions unaddressed.

jostpuur
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What does it mean when a function is said to be an integral function of order [itex]\alpha[/itex]? [itex]\alpha[/itex] is some real constant. I encountered this terminology in the context of complex analytic functions.
 
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I just realized that a complex function being integral very probably means that it is analytic function [itex]\mathbb{C}\to\mathbb{C}[/itex], and in particular not only an analytic function [itex]\mathbb{C}\backslash\{z_1,\ldots, z_n\}\to\mathbb{C}[/itex]. I already knew a name for these functions in Finnish, but didn't realize immediately that this is how it gets translated into English.

I'm still wondering about what the order of an integral function means.
 
Thanks, that appears to be precisely what I was after. The text I was reading deals with similar expressions when talking about order.

Looking at the Mathworld, I noticed that I would have recognized the term "entire function", but did not know that "integral function" is synonymous with it.
 

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