For exponential types in type theory and programming languages, see Function type.
The graph of the function in gray is , the Gaussian restricted to the real axis. The Gaussian does not have exponential type, but the functions in red and blue are one sided approximations that have exponential type .
A function f(z) defined on the complex plane is said to be of exponential type if there exist real-valued constants M and τ such that
in the limit of . Here, the complex variablez was written as to emphasize that the limit must hold in all directions θ. Letting τ stand for the infimum of all such τ, one then says that the function f is of exponential type τ.
For example, let . Then one says that is of exponential type π, since π is the smallest number that bounds the growth of along the imaginary axis. So, for this example, Carlson's theorem cannot apply, as it requires functions of exponential type less than π. Similarly, the Euler–Maclaurin formula cannot be applied either, as it, too, expresses an theorem ultimately anchored in the theory of finite differences.
A holomorphic function is said to be of exponential type if for every there exists a real-valued constant such that
for where . We say is of exponential type if is of exponential type for some . The number
is the exponential type of . The limit superior here means the limit of the supremum of the ratio outside a given radius as the radius goes to infinity. This is also the limit superior of the maximum of the ratio at a given radius as the radius goes to infinity. The limit superior may exist even if the maximum at radius r does not have a limit as r goes to infinity. For example, for the function
the value of
at is asymptotic to and thus goes to zero as n goes to infinity, but F(z) is nevertheless of exponential type 1, as seen by looking at the points .
Exponential type with respect to a symmetric convex body