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In
complex analysis Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebraic ...
, a branch of
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, a
holomorphic function In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex de ...
is said to be of exponential type C if its growth is bounded by the exponential function e^ for some real-valued constant C as , 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, 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 \Psi-type for a general function \Psi(z) as opposed to e^z.


Basic idea

A function f(z) defined on the
complex plane In mathematics, the complex plane is the plane (geometry), plane formed by the complex numbers, with a Cartesian coordinate system such that the horizontal -axis, called the real axis, is formed by the real numbers, and the vertical -axis, call ...
is said to be of exponential type if there exist real-valued constants M and \tau such that :\left, f\left(re^\right)\ \le Me^ in the limit of r\to\infty. Here, the complex variable z was written as z=re^ to emphasize that the limit must hold in all directions \theta. Letting \tau stand for the
infimum In mathematics, the infimum (abbreviated inf; : infima) of a subset S of a partially ordered set P is the greatest element in P that is less than or equal to each element of S, if such an element exists. If the infimum of S exists, it is unique ...
of all such \tau, one then says that the function f is of ''exponential type \tau''. For example, let f(z)=\sin(\pi z). Then one says that \sin(\pi z) is of exponential type \pi, since \pi is the smallest number that bounds the growth of \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 \pi. Similarly, the Euler–Maclaurin formula cannot be applied either, as it, too, expresses a theorem ultimately anchored in the theory of
finite difference A finite difference is a mathematical expression of the form . Finite differences (or the associated difference quotients) are often used as approximations of derivatives, such as in numerical differentiation. The difference operator, commonly d ...
s.


Formal definition

A
holomorphic function In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex de ...
F(z) is said to be of exponential type \sigma>0 if for every \varepsilon>0 there exists a real-valued constant A_\varepsilon such that :, F(z), \leq A_\varepsilon e^ for , z, \to\infty where z\in\mathbb. We say F(z) is of exponential type if F(z) is of exponential type \sigma for some \sigma>0. The number :\tau(F)=\sigma=\displaystyle\limsup_, z, ^\log, F(z), is the exponential type of F(z). The
limit superior In mathematics, the limit inferior and limit superior of a sequence can be thought of as limiting (that is, eventual and extreme) bounds on the sequence. They can be thought of in a similar fashion for a function (see limit of a function). For ...
here means the limit of the
supremum In mathematics, the infimum (abbreviated inf; : infima) of a subset S of a partially ordered set P is the greatest element in P that is less than or equal to each element of S, if such an element exists. If the infimum of S exists, it is unique, ...
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)=\sum_^\infty\frac the value of : (\max_ \log, F(z), ) / r at r=10^ is dominated by the n-1^\text term so we have the asymptotic expressions: :\begin \left(\max_ \log, F(z), \right) / 10^&\sim\left(\log\frac\right)/10^\\ &\sim(\log 10)\left n!-1)10^-10^(n-1)!\right10^\\ &\sim(\log 10)(n!-1-(n-1)!)/10^\\ \end and this goes to zero 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^.


Exponential type with respect to a symmetric convex body

has given a generalization of exponential type for
entire function In complex analysis, an entire function, also called an integral function, is a complex-valued function that is holomorphic on the whole complex plane. Typical examples of entire functions are polynomials and the exponential function, and any ...
s of
several complex variables The theory of functions of several complex variables is the branch of mathematics dealing with functions defined on the complex coordinate space \mathbb C^n, that is, -tuples of complex numbers. The name of the field dealing with the properties ...
. Suppose K is a
convex Convex or convexity may refer to: Science and technology * Convex lens, in optics Mathematics * Convex set, containing the whole line segment that joins points ** Convex polygon, a polygon which encloses a convex set of points ** Convex polytop ...
,
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact, a type of agreement used by U.S. states * Blood compact, an ancient ritual of the Philippines * Compact government, a t ...
, and
symmetric Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is invariant under some transformations ...
subset of \mathbb^n. It is known that for every such K there is an associated norm \, \cdot\, _K with the property that : K=\. In other words, K is the unit ball in \mathbb^ with respect to \, \cdot\, _K. The set :K^=\ is called the
polar set In functional and convex analysis, and related disciplines of mathematics, the polar set A^ is a special convex set associated to any subset A of a vector space X, lying in the dual space X^. The bipolar of a subset is the polar of A^\circ, but ...
and is also a
convex Convex or convexity may refer to: Science and technology * Convex lens, in optics Mathematics * Convex set, containing the whole line segment that joins points ** Convex polygon, a polygon which encloses a convex set of points ** Convex polytop ...
,
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact, a type of agreement used by U.S. states * Blood compact, an ancient ritual of the Philippines * Compact government, a t ...
, and
symmetric Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is invariant under some transformations ...
subset of \mathbb^n. Furthermore, we can write :\, x\, _K = \displaystyle\sup_, x\cdot y, . We extend \, \cdot\, _K from \mathbb^n to \mathbb^n by :\, z\, _K = \displaystyle\sup_, z\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 \varepsilon>0 there exists a real-valued constant A_\varepsilon such that :, F(z), for all z\in\mathbb^.


Fréchet space

Collections of functions of exponential type \tau can form a complete
uniform space In the mathematical field of topology, a uniform space is a topological space, set with additional mathematical structure, structure that is used to define ''uniform property, uniform properties'', such as complete space, completeness, uniform con ...
, namely a Fréchet space, by the
topology Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...
induced by the countable family of norms : \, f\, _n = \sup_ \exp \left z, \rightf(z), .


See also

* Paley–Wiener theorem * Paley–Wiener space


References

* {{citation , last = Stein , first = E.M. , author-link = Elias M. Stein , title = Functions of exponential type , journal = Ann. of Math. , series = 2 , volume = 65 , year = 1957 , issue = 3 , pages = 582–592 , mr = 0085342 , jstor = 1970066 , doi = 10.2307/1970066 Complex analysis