Indicator function (complex analysis): Difference between revisions

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

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.

{{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==

Let us consider an entire function <math>f : \mathbb{C}Complex \to \mathbb{C}Complex</math>. Supposing, that its [[Entire_functionEntire function#Order_and_typeOrder and type|growth order]] is <math>\rho</math>, the indicator function of <math>f</math> is defined to be<ref name="Levin">{{cite book |last1=Levin |first1=B. Ya. |title=Lectures on Entire Functions |date=1996 |publisher=Amer. Math. Soc. |isbn=0821802828}}</ref><ref name="Levin2">{{cite book |last1=Levin |first1=B. Ya. |title=Distribution of Zeros of Entire Functions |date=1964 |publisher=Amer. Math. Soc. |isbn=978-0-8218-4505-9}}</ref>

:<math display="block">h_f(\theta)=\limsup_{r\to\infty}\frac{\log|f(re^{i\theta})|}{r^\rho}.</math>

The indicator function can be also defined for functions which are not entire but analytic inside an angle <math>D = \{z=re^{i\theta}:\alpha<\theta<\beta\}</math>.

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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=51–52}}

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

:<math>h_{fg}(\theta)\le h_f(\theta)+h_g(\theta).</math>

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

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

<math>h_{f+g}(\theta)\le \max\{h_f(\theta),h_g(\theta)\}.</math>

==Examples==

Elementary calculations show, that, if <math>f(z)=e^{(A+iB)z^\rho}</math>, then <math>|f(re^{i\theta})|=e^{Ar^\rho\cos(\rho\theta)-Br^\rho\sin(\rho\theta)}</math>. Thus,<ref name="Levin2" />{{rp|p=52}}

:<math display="block">h_f(\theta) = A\cos(\rho\theta)-B\sin(\rho\theta).</math>

:<math>h_f(\theta)=A\cos(\rho\theta)-B\sin(\rho\theta).</math>

In particular,

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

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

:<math>h_{\exp}(\theta)=\cos(\theta).</math>

:<math>

h_{\sin}(\theta)=h_{\cos}(\theta)= \left|\sin(\theta)\right|

</math>

Another easily deducible indicator function is that of the [[reciprocal Gamma function]]. However, this function is of infinite type (and of order <math>\rho = 1</math>), therefore one needs to define the indicator function to be

:<math display="block">h_{1/\Gamma}(\theta) = \limsup_{r\to\infty}\frac{\log|1/\Gamma(re^{i\theta})|}{r\log r}.</math>

:<math>h_{1/\Gamma}(\theta)=\limsup_{r\to\infty}\frac{\log|1/\Gamma(re^{i\theta})|}{r\log r}.</math>

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

:<math display="block">h_{1/\Gamma}(\theta)=-\cos(\theta).</math>

Another example is that of the [[Mittag-Leffler function]] <math>E_\alpha</math>. This function is of order <math>\rho = 1/\alpha</math>, and<ref name="Cartwright">{{cite book |last1=Cartwright|first1=M. L. |title=Integral Functions |date=1962 |publisher=Cambridge Univ. Press |isbn=052104586X}}</ref>{{rp|p=50}}

:<math>h_{1/\Gamma}(\theta)=-\cos(\theta).</math>

<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==

Those <math>h</math> indicator functions which are of the form

:<math display="block">h(\theta)=A\cos(\rho\theta)+B\sin(\rho\theta)</math>

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|55-57pp=55–57}}<ref name="Levin2" />{{rp|54-61pp=54–61}}:

* 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>.

* If <math>h</math> is bounded on <math>(\alpha,\beta)</math>, then it is continuous on this interval. Moreover, <math>h</math> satisfies a [[Lipschitz condition]] on <math>(\alpha,\beta)</math>.

* If <math>h</math> is bounded on <math>(\alpha,\beta)</math>, then it has both left-hand-side and right-hand-side derivative at every point in the interval <math>(\alpha,\beta)</math>. 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 <math>h</math> is bounded on <math>(\alpha,\beta)</math>, then it has a derivative at all points, except possibly on a countable set.

* If <math>h</math> is <math>\rho</math>-trigonometrically convex on <math>[\alpha,\beta]</math>, then <math>h(\theta)+h(\theta+\pi/\rho) \ge0ge 0</math>, whenever <math>\alpha \le \theta < \theta+\pi/\rho\le\beta</math>.

== Notes ==

{{Reflist}}

==References==

{{refbegin}}

* {{cite book |last1=Boas |first1=R. P. |title=Entire Functions |date=1954 |publisher=Academic Press |isbn=0121081508}}

* {{cite book |last1=Volkovyskii |first1=L. I. |last2=Lunts |first2=G. L. |last3= Aramanovich|first3=I. G.|title=A collection of problems on complex analysis |date=2011 |publisher=Dover Publications |isbn=978-0486669137}}

* {{cite book |last1=Markushevich |first1=A. I. |last2=Silverman |first2=R. A. |title=Theory of functions of a complex variable, Vol. II |date=1965 |publisher=Prentice-Hall Inc.|asin=B003ZWIKFC}}

{{refend}}

[[Category:Complex analysis]]