Exponential type

 ( Complex Analysis ) 

In complex analysis, a branch of mathematics, a holomorphic function is said to be of exponential type C if its bounded growth by the exponential function $e^\{C|z|\}$ for some real number constant $C$ as $|z|\backslash to\backslash infty$. 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, or, for example, to apply the Mellin transform, or to perform approximations using the Euler–Maclaurin formula. The general case is handled by Nachbin's theorem, which defines the analogous notion of $\backslash Psi$-type for a general function $\backslash Psi(z)$ as opposed to $e^z$.

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

$\backslash left|f\backslash left(re^\{i\backslash theta\}\backslash right)\backslash right|\; \backslash le\; Me^\{\backslash tau\; r\}$

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

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

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

$|F(z)|\backslash leq\; A\_\backslash varepsilon\; e^\{(\backslash sigma+\backslash varepsilon)|z|\}$

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

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

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

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

the value of

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

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

$\backslash begin\{align\}$

\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{align}

and this goes to zero as $n$ goes to infinity,In fact, even $(\backslash max\_\{|z|=r\}\backslash log\; \backslash log|F(z)|)\; /(\backslash log\; r)$ goes to zero at $r=10^\{n!-1\}$ as $n$ goes to infinity. but $F(z)$ is nevertheless of exponential type 1, as can be seen by looking at the points $z=10^\{n!\}$.

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Exponential type with respect to a symmetric convex body

has given a generalization of exponential type for [[entire function]]s of several complex variables.
     

Suppose $K$ is a convex set, compact element, and symmetric subset of $\backslash mathbb\{R\}^n$. It is known that for every such $K$ there is an associated norm $\backslash |\backslash cdot\backslash |\_K$ with the property that

$K=\backslash \{x\backslash in\backslash mathbb\{R\}^n\; :\; \backslash |x\backslash |\_K\; \backslash leq1\backslash \}.$

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

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

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

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

We extend $\backslash |\backslash cdot\backslash |\_K$ from $\backslash mathbb\{R\}^n$ to $\backslash mathbb\{C\}^n$ by

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

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

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

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

$\backslash |f\backslash |\_n\; =\; \backslash sup\_\{z\; \backslash in\; \backslash mathbb\{C\}\}\; \backslash exp\; \backslash left-\backslash left(\backslash tau|f(z)|.$

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See also

• Paley–Wiener theorem
• Paley–Wiener space

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