Indicator function (complex analysis)

In the field of mathematics known as complex analysis, the indicator function of an entire function indicates the rate of growth of the function in different directions.

Definition

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Let us consider an entire function  . Supposing, that its growth order is  , the indicator function of   is defined to be[1][2]  

The indicator function can be also defined for functions which are not entire but analytic inside an angle  .

Basic properties

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By the very definition of the indicator function, we have that the indicator of the product of two functions does not exceed the sum of the indicators:[2]: 51–52   

Similarly, the indicator of the sum of two functions does not exceed the larger of the two indicators:  

Examples

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Elementary calculations show that, if  , then  . Thus,[2]: 52   

In particular,  

Since the complex sine and cosine functions are expressible in terms of the exponential, it follows from the above result that

 

Another easily deducible indicator function is that of the reciprocal Gamma function. However, this function is of infinite type (and of order  ), therefore one needs to define the indicator function to be  

Stirling's approximation of the Gamma function then yields, that  

Another example is that of the Mittag-Leffler function  . This function is of order  , and[3]: 50 

 

The indicator of the Barnes G-function can be calculated easily from its asymptotic expression (which roughly says that  ):

 

Further properties of the indicator

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Those   indicator functions which are of the form   are called  -trigonometrically convex (  and   are real constants). If  , we simply say, that   is trigonometrically convex.

Such indicators have some special properties. For example, the following statements are all true for an indicator function that is trigonometrically convex at least on an interval  :[1]: 55–57 [2]: 54–61 

  • If   for a  , then   everywhere in  .
  • If   is bounded on  , then it is continuous on this interval. Moreover,   satisfies a Lipschitz condition on  .
  • If   is bounded on  , then it has both left-hand-side and right-hand-side derivative at every point in the interval  . Moreover, the left-hand-side derivative is not greater than the right-hand-side derivative. It also holds true, that the right-hand-side derivative is continuous from the right, while the left-hand-side derivative is continuous from the left.
  • If   is bounded on  , then it has a derivative at all points, except possibly on a countable set.
  • If   is  -trigonometrically convex on  , then  , whenever  .

Notes

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  1. ^ a b Levin, B. Ya. (1996). Lectures on Entire Functions. Amer. Math. Soc. ISBN 0821802828.
  2. ^ a b c d Levin, B. Ya. (1964). Distribution of Zeros of Entire Functions. Amer. Math. Soc. ISBN 978-0-8218-4505-9.
  3. ^ Cartwright, M. L. (1962). Integral Functions. Cambridge Univ. Press. ISBN 052104586X.

References

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  • Boas, R. P. (1954). Entire Functions. Academic Press. ISBN 0121081508.
  • Volkovyskii, L. I.; Lunts, G. L.; Aramanovich, I. G. (2011). A collection of problems on complex analysis. Dover Publications. ISBN 978-0486669137.
  • Markushevich, A. I.; Silverman, R. A. (1965). Theory of functions of a complex variable, Vol. II. Prentice-Hall Inc. ASIN B003ZWIKFC.