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}left|\sin(\theta)\right|

\sin(\theta), & \text{if } 0 \le\theta<\pi \\

-\sin(\theta), & \text{if } \pi \le \theta<2\pi.

\end{cases}

</math>

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<math display="block">h_{E_\alpha}(\theta)=\begin{cases}\cos\left(\frac{\theta}{\alpha}\right),&\text{for }|\theta|\le\frac 1 2 \alpha\pi;\\0,&\text{otherwise}.\end{cases}</math>

The indicator of the [[Barnes G-function]] can be calculated easily from its asymptotic expression (which roughly [[Barnes_G-function#Asymptotic_expansion|says]] that <math>

\log G(z+1)\sim \frac{z^2}{2}\log z</math>):

:<math>h_G(\theta)=\frac{\log(G(re^{i\theta}))}{r^2\log(r)} = \frac12\cos(2\theta).</math>

==Further properties of the indicator==

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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>.