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
Supposing, that its growth order is
, the indicator function of
The indicator function can be also defined for functions which are not entire but analytic inside an angle
: α < θ < β }
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
Similarly, the indicator of the sum of two functions does not exceed the larger of the two indicators:
( θ ) ≤ max {
Elementary calculations show that, if
cos ( ρ θ ) −
sin ( ρ θ )
cos ( ρ θ ) −
sin ( ρ θ ) .
Since the complex sine and cosine functions are expressible in terms of the exponential, it follows from the above result that Another easily deducible indicator function is that of the reciprocal Gamma function.
However, this function is of infinite type (and of order
), therefore one needs to define the indicator function to be
Stirling's approximation of the Gamma function then yields, that
Another example is that of the Mittag-Leffler function
This function is of order
θ α
α π ;
The indicator of the Barnes G-function can be calculated easily from its asymptotic expression (which roughly says that
indicator functions which are of the form
cos ( ρ θ ) +
sin ( ρ θ )
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 )