Indicator function (complex analysis)

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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.

Let us consider an entire function f : C → C {\displaystyle f:\mathbb {C} \to \mathbb {C} } . Supposing, that its growth order is ρ {\displaystyle \rho } , the indicator function of f {\displaystyle f} is defined to be^[1][2] 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 \}} .

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:^{[2]: 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 )\}.}

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,^[2]: 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 expressible in terms of the exponential, it follows from the above result that

h sin ( θ ) = h cos ( θ ) = | sin ⁡ ( θ ) | {\displaystyle h_{\sin }(\theta )=h_{\cos }(\theta )=\left|\sin(\theta )\right|}

Another easily deducible indicator function is that of the reciprocal Gamma function. 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}}.}

Stirling's approximation 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 Mittag-Leffler function E α {\displaystyle E_{\alpha }} . This function is of order ρ = 1 / α {\displaystyle \rho =1/\alpha } , and^[3]: 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 Barnes G-function can be calculated easily from its asymptotic expression (which roughly says 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 ).}

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 )} :^{[1]: 55–57 [2]: 54–61}

• 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 )} .
• 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 Lipschitz condition on ( α , β ) {\displaystyle (\alpha ,\beta )} .
• 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.
• 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.
• 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 } .