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 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
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 \}} .
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:
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 )\}.}
Examples 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,
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
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