A function ${\displaystyle f(z)}$  defined on the complex plane is said to be of exponential type if there exist real-valued constants ${\displaystyle M}$  and ${\displaystyle \tau }$  such that

${\displaystyle \left|f\left(re^{i\theta }\right)\right|\leq Me^{\tau r}}$ 

in the limit of ${\displaystyle r\to \infty }$ . Here, the complex variable ${\displaystyle z}$  was written as ${\displaystyle z=re^{i\theta }}$  to emphasize that the limit must hold in all directions ${\displaystyle \theta }$ . Letting ${\displaystyle \tau }$  stand for the infimum of all such ${\displaystyle \tau }$ , one then says that the function ${\displaystyle f}$  is of exponential type ${\displaystyle \tau }$ .

For example, let ${\displaystyle f(z)=\sin(\pi z)}$ . Then one says that ${\displaystyle \sin(\pi z)}$  is of exponential type ${\displaystyle \pi }$ , since ${\displaystyle \pi }$  is the smallest number that bounds the growth of ${\displaystyle \sin(\pi z)}$  along the imaginary axis. So, for this example, Carlson's theorem cannot apply, as it requires functions of exponential type less than ${\displaystyle \pi }$ . Similarly, the Euler–Maclaurin formula cannot be applied either, as it, too, expresses a theorem ultimately anchored in the theory of finite differences.

A holomorphic function ${\displaystyle F(z)}$  is said to be of exponential type ${\displaystyle \sigma >0}$  if for every ${\displaystyle \varepsilon >0}$  there exists a real-valued constant ${\displaystyle A_{\varepsilon }}$  such that

${\displaystyle |F(z)|\leq A_{\varepsilon }e^{(\sigma +\varepsilon )|z|}}$ 

for ${\displaystyle |z|\to \infty }$  where ${\displaystyle z\in \mathbb {C} }$ . We say ${\displaystyle F(z)}$  is of exponential type if ${\displaystyle F(z)}$  is of exponential type ${\displaystyle \sigma }$  for some ${\displaystyle \sigma >0}$ . The number

${\displaystyle \tau (F)=\sigma =\displaystyle \limsup _{|z|\rightarrow \infty }|z|^{-1}\log |F(z)|}$ 

is the exponential type of ${\displaystyle F(z)}$ . 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 ${\displaystyle r}$  does not have a limit as ${\displaystyle r}$  goes to infinity. For example, for the function

${\displaystyle F(z)=\sum _{n=1}^{\infty }{\frac {z^{10^{n!}}}{(10^{n!})!}}}$ 

the value of

${\displaystyle (\max _{|z|=r}\log |F(z)|)/r}$ 

at ${\displaystyle r=10^{n!-1}}$  is dominated by the ${\displaystyle n-1^{\text{st}}}$  term so we have the asymptotic expressions:

${\displaystyle {\begin{aligned}\left(\max _{|z|=10^{n!-1}}\log |F(z)|\right)/10^{n!-1}&\sim \left(\log {\frac {(10^{n!-1})^{10^{(n-1)!}}}{(10^{(n-1)!})!}}\right)/10^{n!-1}\\&\sim (\log 10)\left[(n!-1)10^{(n-1)!}-10^{(n-1)!}(n-1)!\right]/10^{n!-1}\\&\sim (\log 10)(n!-1-(n-1)!)/10^{n!-1-(n-1)!}\\\end{aligned}}}$ 

and this goes to zero as ${\displaystyle n}$  goes to infinity,^[1] but ${\displaystyle F(z)}$  is nevertheless of exponential type 1, as can be seen by looking at the points ${\displaystyle z=10^{n!}}$ .

Stein (1957) has given a generalization of exponential type for entire functions of several complex variables. Suppose ${\displaystyle K}$  is a convex, compact, and symmetric subset of ${\displaystyle \mathbb {R} ^{n}}$ . It is known that for every such ${\displaystyle K}$  there is an associated norm ${\displaystyle \|\cdot \|_{K}}$  with the property that

${\displaystyle K=\{x\in \mathbb {R} ^{n}:\|x\|_{K}\leq 1\}.}$ 

In other words, ${\displaystyle K}$  is the unit ball in ${\displaystyle \mathbb {R} ^{n}}$  with respect to ${\displaystyle \|\cdot \|_{K}}$ . The set

${\displaystyle K^{*}=\{y\in \mathbb {R} ^{n}:x\cdot y\leq 1{\text{ for all }}x\in {K}\}}$ 

is called the polar set and is also a convex, compact, and symmetric subset of ${\displaystyle \mathbb {R} ^{n}}$ . Furthermore, we can write

${\displaystyle \|x\|_{K}=\displaystyle \sup _{y\in K^{*}}|x\cdot y|.}$ 

We extend ${\displaystyle \|\cdot \|_{K}}$  from ${\displaystyle \mathbb {R} ^{n}}$  to ${\displaystyle \mathbb {C} ^{n}}$  by

${\displaystyle \|z\|_{K}=\displaystyle \sup _{y\in K^{*}}|z\cdot y|.}$ 

An entire function ${\displaystyle F(z)}$  of ${\displaystyle n}$ -complex variables is said to be of exponential type with respect to ${\displaystyle K}$  if for every ${\displaystyle \varepsilon >0}$  there exists a real-valued constant ${\displaystyle A_{\varepsilon }}$  such that

${\displaystyle |F(z)|<A_{\varepsilon }e^{2\pi (1+\varepsilon )\|z\|_{K}}}$ 

for all ${\displaystyle z\in \mathbb {C} ^{n}}$ .

Collections of functions of exponential type ${\displaystyle \tau }$  can form a complete uniform space, namely a Fréchet space, by the topology induced by the countable family of norms

${\displaystyle \|f\|_{n}=\sup _{z\in \mathbb {C} }\exp \left[-\left(\tau +{\frac {1}{n}}\right)|z|\right]|f(z)|.}$ 

• Paley–Wiener theorem
• Paley–Wiener space

• Stein, E.M. (1957), "Functions of exponential type", Ann. of Math., 2, 65: 582–592, doi:10.2307/1970066, JSTOR 1970066, MR 0085342