# Exponential type > Mediated Wiki article. Canonical URL: https://mediated.wiki/source/Exponential_type > Markdown URL: https://mediated.wiki/source/Exponential_type.md > Source: https://en.wikipedia.org/wiki/Exponential_type > Source revision: 1330991469 > License: Creative Commons Attribution-ShareAlike 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/) Type of complex function with growth bounded by an exponential function For exponential types in type theory and programming languages, see [Function type](/source/Function_type). The graph of the function in gray is e − π z 2 {\displaystyle e^{-\pi z^{2}}} , 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 2 π {\displaystyle 2\pi } . In [complex analysis](/source/Complex_analysis), a branch of [mathematics](/source/Mathematics), a [holomorphic function](/source/Holomorphic_function) is said to be of **exponential type C** if its [growth is bounded](/source/Bounded_growth) by the [exponential function](/source/Exponential_function) e C | z | {\displaystyle e^{C|z|}} for some [real-valued](/source/Real_number) constant C {\displaystyle C} as | z | → ∞ {\displaystyle |z|\to \infty } . When a function is bounded in this way, it is then possible to express it as certain kinds of convergent summations over a series of other complex functions, as well as understanding when it is possible to apply techniques such as [Borel summation](/source/Borel_summation), or, for example, to apply the [Mellin transform](/source/Mellin_transform), or to perform approximations using the [Euler–Maclaurin formula](/source/Euler%E2%80%93Maclaurin_formula). The general case is handled by [Nachbin's theorem](/source/Nachbin's_theorem), which defines the analogous notion of **Ψ {\displaystyle \Psi } -type** for a general function Ψ ( z ) {\displaystyle \Psi (z)} as opposed to e z {\displaystyle e^{z}} . ## Basic idea A function f ( z ) {\displaystyle f(z)} defined on the [complex plane](/source/Complex_plane) is said to be of exponential type if there exist real-valued constants M {\displaystyle M} and τ {\displaystyle \tau } such that - | f ( r e i θ ) | ≤ M e τ r {\displaystyle \left|f\left(re^{i\theta }\right)\right|\leq Me^{\tau r}} in the limit of r → ∞ {\displaystyle r\to \infty } . Here, the [complex variable](/source/Complex_variable) z {\displaystyle z} was written as z = r e i θ {\displaystyle z=re^{i\theta }} to emphasize that the limit must hold in all directions θ {\displaystyle \theta } . Letting τ {\displaystyle \tau } stand for the [infimum](/source/Infimum) of all such τ {\displaystyle \tau } , one then says that the function f {\displaystyle f} is of *exponential type τ {\displaystyle \tau }*. For example, let f ( z ) = sin ⁡ ( π z ) {\displaystyle f(z)=\sin(\pi z)} . Then one says that sin ⁡ ( π z ) {\displaystyle \sin(\pi z)} is of exponential type π {\displaystyle \pi } , since π {\displaystyle \pi } is the smallest number that bounds the growth of sin ⁡ ( π z ) {\displaystyle \sin(\pi z)} along the imaginary axis. So, for this example, [Carlson's theorem](/source/Carlson's_theorem) cannot apply, as it requires functions of exponential type less than π {\displaystyle \pi } . Similarly, the [Euler–Maclaurin formula](/source/Euler%E2%80%93Maclaurin_formula) cannot be applied either, as it, too, expresses a theorem ultimately anchored in the theory of [finite differences](/source/Finite_difference). ## Formal definition A [holomorphic function](/source/Holomorphic_function) F ( z ) {\displaystyle F(z)} is said to be of **exponential type** σ > 0 {\displaystyle \sigma >0} if for every ε > 0 {\displaystyle \varepsilon >0} there exists a real-valued constant A ε {\displaystyle A_{\varepsilon }} such that - | F ( z ) | ≤ A ε e ( σ + ε ) | z | {\displaystyle |F(z)|\leq A_{\varepsilon }e^{(\sigma +\varepsilon )|z|}} for | z | → ∞ {\displaystyle |z|\to \infty } where z ∈ C {\displaystyle z\in \mathbb {C} } . We say F ( z ) {\displaystyle F(z)} is of exponential type if F ( z ) {\displaystyle F(z)} is of exponential type σ {\displaystyle \sigma } for some σ > 0 {\displaystyle \sigma >0} . The number - τ ( F ) = σ = lim sup | z | → ∞ | z | − 1 log ⁡ | F ( z ) | {\displaystyle \tau (F)=\sigma =\displaystyle \limsup _{|z|\rightarrow \infty }|z|^{-1}\log |F(z)|} Behavior of F ( z ) = ∑ n = 1 ∞ z 10 n ! ( 10 n ! ) ! . {\displaystyle F(z)=\sum _{n=1}^{\infty }{\frac {z^{10^{n!}}}{(10^{n!})!}}.} The ratio of log ⁡ | F ( z ) | {\displaystyle \log |F(z)|} to | z | {\displaystyle |z|} at a given radius is maximized on the real axis. is the exponential type of F ( z ) {\displaystyle F(z)} . The [limit superior](/source/Limit_superior) here means the limit of the [supremum](/source/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 {\displaystyle r} does not have a limit as r {\displaystyle r} goes to infinity. For example, for the function - F ( z ) = ∑ n = 1 ∞ z 10 n ! ( 10 n ! ) ! {\displaystyle F(z)=\sum _{n=1}^{\infty }{\frac {z^{10^{n!}}}{(10^{n!})!}}} the value of - ( max | z | = r log ⁡ | F ( z ) | ) / r {\displaystyle (\max _{|z|=r}\log |F(z)|)/r} at r = 10 n ! − 1 {\displaystyle r=10^{n!-1}} is dominated by the n−1st term and this goes to zero as n {\displaystyle n} goes to infinity,[1] but F ( z ) {\displaystyle F(z)} is nevertheless of exponential type 1, as can be seen by looking at the points z = 10 n ! {\displaystyle z=10^{n!}} . ## Exponential type with respect to a symmetric convex body [Stein (1957)](#CITEREFStein1957) has given a generalization of exponential type for [entire functions](/source/Entire_function) of [several complex variables](/source/Several_complex_variables). Suppose K {\displaystyle K} is a [convex](/source/Convex_set), [compact](/source/Compact_element), and [symmetric](/source/Symmetric) subset of R n {\displaystyle \mathbb {R} ^{n}} . It is known that for every such K {\displaystyle K} there is an associated [norm](/source/Norm_(mathematics)) ‖ ⋅ ‖ K {\displaystyle \|\cdot \|_{K}} with the property that - K = { x ∈ R n : ‖ x ‖ K ≤ 1 } . {\displaystyle K=\{x\in \mathbb {R} ^{n}:\|x\|_{K}\leq 1\}.} In other words, K {\displaystyle K} is the unit ball in R n {\displaystyle \mathbb {R} ^{n}} with respect to ‖ ⋅ ‖ K {\displaystyle \|\cdot \|_{K}} . The set - K ∗ = { y ∈ R n : x ⋅ y ≤ 1 for all x ∈ K } {\displaystyle K^{*}=\{y\in \mathbb {R} ^{n}:x\cdot y\leq 1{\text{ for all }}x\in {K}\}} is called the [polar set](/source/Polar_set) and is also a [convex](/source/Convex_set), [compact](/source/Compact_element), and [symmetric](/source/Symmetric) subset of R n {\displaystyle \mathbb {R} ^{n}} . Furthermore, we can write - ‖ x ‖ K = sup y ∈ K ∗ | x ⋅ y | . {\displaystyle \|x\|_{K}=\displaystyle \sup _{y\in K^{*}}|x\cdot y|.} We extend ‖ ⋅ ‖ K {\displaystyle \|\cdot \|_{K}} from R n {\displaystyle \mathbb {R} ^{n}} to C n {\displaystyle \mathbb {C} ^{n}} by - ‖ z ‖ K = sup y ∈ K ∗ | z ⋅ y | . {\displaystyle \|z\|_{K}=\displaystyle \sup _{y\in K^{*}}|z\cdot y|.} An entire function F ( z ) {\displaystyle F(z)} of n {\displaystyle n} -complex variables is said to be of exponential type with respect to K {\displaystyle K} if for every ε > 0 {\displaystyle \varepsilon >0} there exists a real-valued constant A ε {\displaystyle A_{\varepsilon }} such that - | F ( z ) | < A ε e 2 π ( 1 + ε ) ‖ z ‖ K {\displaystyle |F(z)|