Indicator function (complex analysis)

<h2 id="definition">Definition</h2>
Let us consider an entire function 
 
 
 
 f
 :
 
 C
 
 →
 
 C
 
 
 
 {\displaystyle f:\mathbb {C} \to \mathbb {C} }
 
. Supposing, that its <a href="/facts/Entire_function/XNluKzu7">growth order</a> is 
 
 
 
 ρ
 
 
 {\displaystyle \rho }
 
, the indicator function of 
 
 
 
 f
 
 
 {\displaystyle f}
 
 is defined to be<a class="footnote-ref" id="fnref:1" href="#fn:1">1</a><a class="footnote-ref" id="fnref:2" href="#fn:2">2</a>

h
          
            f
          
        
        (
        θ
        )
        =
        
          lim sup
          
            r
            →
            ∞
          
        
        
          
            
              log
              ⁡
              
                |
              
              f
              (
              r
              
                e
                
                  i
                  θ
                
              
              )
              
                |
              
            
            
              r
              
                ρ
              
            
          
        
        .
      
    
    {\displaystyle h_{f}(\theta )=\limsup _{r\to \infty }{\frac {\log |f(re^{i\theta })|}{r^{\rho }}}.}

The indicator function can be also defined for functions which are not entire but analytic inside an angle 
 
 
 
 D
 =
 {
 z
 =
 r
 
 e
 
 i
 θ
 
 
 :
 α
 <
 θ
 <
 β
 }
 
 
 {\displaystyle D=\{z=re^{i\theta }:\alpha <\theta <\beta \}}
 
.

<h2 id="basic-properties">Basic properties</h2>
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:<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a>: 51–52

h
          
            f
            g
          
        
        (
        θ
        )
        ≤
        
          h
          
            f
          
        
        (
        θ
        )
        +
        
          h
          
            g
          
        
        (
        θ
        )
        .
      
    
    {\displaystyle h_{fg}(\theta )\leq h_{f}(\theta )+h_{g}(\theta ).}

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

h
          
            f
            +
            g
          
        
        (
        θ
        )
        ≤
        max
        {
        
          h
          
            f
          
        
        (
        θ
        )
        ,
        
          h
          
            g
          
        
        (
        θ
        )
        }
        .
      
    
    {\displaystyle h_{f+g}(\theta )\leq \max\{h_{f}(\theta ),h_{g}(\theta )\}.}

<h2 id="examples">Examples</h2>
Elementary calculations show that, if 
 
 
 
 f
 (
 z
 )
 =
 
 e
 
 (
 A
 +
 i
 B
 )
 
 z
 
 ρ
 
 
 
 
 
 
 {\displaystyle f(z)=e^{(A+iB)z^{\rho }}}
 
, then 
 
 
 
 
 |
 
 f
 (
 r
 
 e
 
 i
 θ
 
 
 )
 
 |
 
 =
 
 e
 
 A
 
 r
 
 ρ
 
 
 cos
 ⁡
 (
 ρ
 θ
 )
 −
 B
 
 r
 
 ρ
 
 
 sin
 ⁡
 (
 ρ
 θ
 )
 
 
 
 
 {\displaystyle |f(re^{i\theta })|=e^{Ar^{\rho }\cos(\rho \theta )-Br^{\rho }\sin(\rho \theta )}}
 
. Thus,<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a>: 52

h
          
            f
          
        
        (
        θ
        )
        =
        A
        cos
        ⁡
        (
        ρ
        θ
        )
        −
        B
        sin
        ⁡
        (
        ρ
        θ
        )
        .
      
    
    {\displaystyle h_{f}(\theta )=A\cos(\rho \theta )-B\sin(\rho \theta ).}

In particular,

h
          
            exp
          
        
        (
        θ
        )
        =
        cos
        ⁡
        (
        θ
        )
        .
      
    
    {\displaystyle h_{\exp }(\theta )=\cos(\theta ).}

Since the complex sine and cosine functions are <a href="/facts/Sine_and_cosine/0UtQ97vu">expressible</a> in terms of the exponential, it follows from the above result that

h
 
 sin
 
 
 (
 θ
 )
 =
 
 h
 
 cos
 
 
 (
 θ
 )
 =
 
 
 {
 
 
 
 sin
 ⁡
 (
 θ
 )
 ,
 
 
 
 if 
 
 0
 ≤
 θ
 <
 π
 
 
 
 
 −
 sin
 ⁡
 (
 θ
 )
 ,
 
 
 
 if 
 
 π
 ≤
 θ
 <
 2
 π
 .
 
 
 
 
 
 
 
 
 {\displaystyle h_{\sin }(\theta )=h_{\cos }(\theta )={\begin{cases}\sin(\theta ),&{\text{if }}0\leq \theta <\pi \\-\sin(\theta ),&{\text{if }}\pi \leq \theta <2\pi .\end{cases}}}

Another easily deducible indicator function is that of the <a href="/facts/Reciprocal_Gamma_function/qToKveVM">reciprocal Gamma function</a>. However, this function is of infinite type (and of order 
 
 
 
 ρ
 =
 1
 
 
 {\displaystyle \rho =1}
 
), therefore one needs to define the indicator function to be

h
          
            1
            
              /
            
            Γ
          
        
        (
        θ
        )
        =
        
          lim sup
          
            r
            →
            ∞
          
        
        
          
            
              log
              ⁡
              
                |
              
              1
              
                /
              
              Γ
              (
              r
              
                e
                
                  i
                  θ
                
              
              )
              
                |
              
            
            
              r
              log
              ⁡
              r
            
          
        
        .
      
    
    {\displaystyle h_{1/\Gamma }(\theta )=\limsup _{r\to \infty }{\frac {\log |1/\Gamma (re^{i\theta })|}{r\log r}}.}

<a href="/facts/Stirling%27s_approximation/GfxV3BUH">Stirling's approximation</a> of the Gamma function then yields, that

h
          
            1
            
              /
            
            Γ
          
        
        (
        θ
        )
        =
        −
        cos
        ⁡
        (
        θ
        )
        .
      
    
    {\displaystyle h_{1/\Gamma }(\theta )=-\cos(\theta ).}

Another example is that of the <a href="/facts/Mittag-Leffler_function/qWjX1gP2">Mittag-Leffler function</a> 
 
 
 
 
 E
 
 α
 
 
 
 
 {\displaystyle E_{\alpha }}
 
. This function is of order 
 
 
 
 ρ
 =
 1
 
 /
 
 α
 
 
 {\displaystyle \rho =1/\alpha }
 
, and<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a>: 50 

 
 
 
 
 h
 
 
 E
 
 α
 
 
 
 
 (
 θ
 )
 =
 
 
 {
 
 
 
 cos
 ⁡
 
 (
 
 
 θ
 α
 
 
 )
 
 ,
 
 
 
 for 
 
 
 |
 
 θ
 
 |
 
 ≤
 
 
 1
 2
 
 
 α
 π
 ;
 
 
 
 
 0
 ,
 
 
 
 otherwise
 
 .
 
 
 
 
 
 
 
 
 {\displaystyle h_{E_{\alpha }}(\theta )={\begin{cases}\cos \left({\frac {\theta }{\alpha }}\right),&{\text{for }}|\theta |\leq {\frac {1}{2}}\alpha \pi ;\\0,&{\text{otherwise}}.\end{cases}}}

The indicator of the <a href="/facts/Barnes_G-function/4bmYxJrE">Barnes G-function</a> can be calculated easily from its asymptotic expression (which roughly <a href="/facts/Barnes_G-function/4bmYxJrE">says</a> that 
 
 
 
 log
 ⁡
 G
 (
 z
 +
 1
 )
 ∼
 
 
 
 z
 
 2
 
 
 2
 
 
 log
 ⁡
 z
 
 
 {\displaystyle \log G(z+1)\sim {\frac {z^{2}}{2}}\log z}
 
):

h
          
            G
          
        
        (
        θ
        )
        =
        
          
            
              log
              ⁡
              (
              G
              (
              r
              
                e
                
                  i
                  θ
                
              
              )
              )
            
            
              
                r
                
                  2
                
              
              log
              ⁡
              (
              r
              )
            
          
        
        =
        
          
            1
            2
          
        
        cos
        ⁡
        (
        2
        θ
        )
        .
      
    
    {\displaystyle h_{G}(\theta )={\frac {\log(G(re^{i\theta }))}{r^{2}\log(r)}}={\frac {1}{2}}\cos(2\theta ).}

<h2 id="further-properties-of-the-indicator">Further properties of the indicator</h2>
Those 
 
 
 
 h
 
 
 {\displaystyle h}
 
 indicator functions which are of the form

h
        (
        θ
        )
        =
        A
        cos
        ⁡
        (
        ρ
        θ
        )
        +
        B
        sin
        ⁡
        (
        ρ
        θ
        )
      
    
    {\displaystyle h(\theta )=A\cos(\rho \theta )+B\sin(\rho \theta )}

are called 
 
 
 
 ρ
 
 
 {\displaystyle \rho }
 
-trigonometrically convex (
 
 
 
 A
 
 
 {\displaystyle A}
 
 and 
 
 
 
 B
 
 
 {\displaystyle B}
 
 are real constants). If 
 
 
 
 ρ
 =
 1
 
 
 {\displaystyle \rho =1}
 
, we simply say, that 
 
 
 
 h
 
 
 {\displaystyle h}
 
 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 
 
 
 
 (
 α
 ,
 β
 )
 
 
 {\displaystyle (\alpha ,\beta )}
 
:<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a>: 55–57 <a class="footnote-ref" id="fnref:7" href="#fn:7">7</a>: 54–61 

<ul><li>If 
 
 
 
 h
 (
 
 θ
 
 1
 
 
 )
 =
 −
 ∞
 
 
 {\displaystyle h(\theta _{1})=-\infty }
 
 for a 
 
 
 
 
 θ
 
 1
 
 
 ∈
 (
 α
 ,
 β
 )
 
 
 {\displaystyle \theta _{1}\in (\alpha ,\beta )}
 
, then 
 
 
 
 h
 =
 −
 ∞
 
 
 {\displaystyle h=-\infty }
 
 everywhere in 
 
 
 
 (
 α
 ,
 β
 )
 
 
 {\displaystyle (\alpha ,\beta )}
 
.</li>
<li>If 
 
 
 
 h
 
 
 {\displaystyle h}
 
 is bounded on 
 
 
 
 (
 α
 ,
 β
 )
 
 
 {\displaystyle (\alpha ,\beta )}
 
, then it is continuous on this interval. Moreover, 
 
 
 
 h
 
 
 {\displaystyle h}
 
 satisfies a <a href="/facts/Lipschitz_condition/Hw20EPEU">Lipschitz condition</a> on 
 
 
 
 (
 α
 ,
 β
 )
 
 
 {\displaystyle (\alpha ,\beta )}
 
.</li>
<li>If 
 
 
 
 h
 
 
 {\displaystyle h}
 
 is bounded on 
 
 
 
 (
 α
 ,
 β
 )
 
 
 {\displaystyle (\alpha ,\beta )}
 
, then it has both left-hand-side and right-hand-side derivative at every point in the interval 
 
 
 
 (
 α
 ,
 β
 )
 
 
 {\displaystyle (\alpha ,\beta )}
 
. 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.</li>
<li>If 
 
 
 
 h
 
 
 {\displaystyle h}
 
 is bounded on 
 
 
 
 (
 α
 ,
 β
 )
 
 
 {\displaystyle (\alpha ,\beta )}
 
, then it has a derivative at all points, except possibly on a countable set.</li>
<li>If 
 
 
 
 h
 
 
 {\displaystyle h}
 
 is 
 
 
 
 ρ
 
 
 {\displaystyle \rho }
 
-trigonometrically convex on 
 
 
 
 [
 α
 ,
 β
 ]
 
 
 {\displaystyle [\alpha ,\beta ]}
 
, then 
 
 
 
 h
 (
 θ
 )
 +
 h
 (
 θ
 +
 π
 
 /
 
 ρ
 )
 ≥
 0
 
 
 {\displaystyle h(\theta )+h(\theta +\pi /\rho )\geq 0}
 
, whenever 
 
 
 
 α
 ≤
 θ
 <
 θ
 +
 π
 
 /
 
 ρ
 ≤
 β
 
 
 {\displaystyle \alpha \leq \theta <\theta +\pi /\rho \leq \beta }
 
.</li></ul>
<h2 id="notes">Notes</h2>

<ul><li>Boas, R. P. (1954). Entire Functions. Academic Press. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0121081508.</li>
<li>Volkovyskii, L. I.; Lunts, G. L.; Aramanovich, I. G. (2011). A collection of problems on complex analysis. Dover Publications. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0486669137.</li>
<li>Markushevich, A. I.; Silverman, R. A. (1965). Theory of functions of a complex variable, Vol. II. Prentice-Hall Inc. <a href="/facts/ASIN_(identifier)/twP2MtLk">ASIN</a> <a href="https://www.amazon.com/dp/B003ZWIKFC">B003ZWIKFC</a>.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Levin, B. Ya. (1996). Lectures on Entire Functions. Amer. Math. Soc. ISBN 0821802828. <a href="0821802828" target="_blank">0821802828</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Levin, B. Ya. (1964). Distribution of Zeros of Entire Functions. Amer. Math. Soc. ISBN 978-0-8218-4505-9. <a href="978-0-8218-4505-9" target="_blank">978-0-8218-4505-9</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Levin, B. Ya. (1964). Distribution of Zeros of Entire Functions. Amer. Math. Soc. ISBN 978-0-8218-4505-9. <a href="978-0-8218-4505-9" target="_blank">978-0-8218-4505-9</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Levin, B. Ya. (1964). Distribution of Zeros of Entire Functions. Amer. Math. Soc. ISBN 978-0-8218-4505-9. <a href="978-0-8218-4505-9" target="_blank">978-0-8218-4505-9</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Cartwright, M. L. (1962). Integral Functions. Cambridge Univ. Press. ISBN 052104586X. <a href="052104586X" target="_blank">052104586X</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Levin, B. Ya. (1996). Lectures on Entire Functions. Amer. Math. Soc. ISBN 0821802828. <a href="0821802828" target="_blank">0821802828</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">Levin, B. Ya. (1964). Distribution of Zeros of Entire Functions. Amer. Math. Soc. ISBN 978-0-8218-4505-9. <a href="978-0-8218-4505-9" target="_blank">978-0-8218-4505-9</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
</ol>

Indicator function (complex analysis) open-in-new

Indicator function (complex analysis)