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
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
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
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
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
^ abLevin, B. Ya. (1996). Lectures on Entire Functions. Amer. Math. Soc. ISBN0821802828.
^ abcdLevin, B. Ya. (1964). Distribution of Zeros of Entire Functions. Amer. Math. Soc. ISBN978-0-8218-4505-9.
^Cartwright, M. L. (1962). Integral Functions. Cambridge Univ. Press. ISBN052104586X.
References
Boas, R. P. (1954). Entire Functions. Academic Press. ISBN0121081508.
Volkovyskii, L. I.; Lunts, G. L.; Aramanovich, I. G. (2011). A collection of problems on complex analysis. Dover Publications. ISBN978-0486669137.
Markushevich, A. I.; Silverman, R. A. (1965). Theory of functions of a complex variable, Vol. II. Prentice-Hall Inc. ASINB003ZWIKFC.