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==

Let us consider an entire function <math>f : \Complex \to \Complex</math>. Supposing, that its 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>.

==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>

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>

==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>

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

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

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 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 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|pp=55–57}}<ref name="Levin2" />{{rp|pp=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) \ge 0</math>, whenever <math>\alpha \le \theta < \theta+\pi/\rho\le\beta</math>.

== Notes == {{Reflist}}

==References== {{refbegin}} * {{cite book |last1=Boas |first1=R. P. |author-link=Ralph P. Boas Jr. |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