Indicator function (complex analysis): Difference between revisions

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{{Short description|Notion from the theory of entire functions}}

{{Orphan|date=June 2024}}

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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==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:<ref name="Levin2" />{{rp|pp=51-5251–52}}

<math display="block">h_{fg}(\theta)\le h_f(\theta)+h_g(\theta).</math>

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<math display="block">h_{\exp}(\theta) = \cos(\theta).</math>

Since the complex sine and cosine functions are [[Sine_and_cosineSine and cosine#Complex_argumentsComplex arguments|expressible]] in terms of the exponential, it follows from the above result that

:<math>

h_{\sin}(\theta)=h_{\cos}(\theta)=\begin{cases}

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are called <math>\rho</math>-trigonometrically convex (<math>A</math> and <math>B</math> are real constants). If <math>\rho = 1</math>, we simply say, that <math>h</math> 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 {{nowrap|<math>(\alpha,\beta)</math>:}}<ref name="Levin" />{{rp|pp=55-5755–57}}<ref name="Levin2" />{{rp|pp=54-6154–61}}

* If <math>h(\theta_1)=-\infty</math> for a <math>\theta_1\in(\alpha,\beta)</math>, then <math>h = -\infty</math> everywhere in <math>(\alpha,\beta)</math>.