
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 for some
real-valued constant
as
. 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
-type for a general function
as opposed to
.
Basic idea
A function
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
and
such that
:
in the limit of
. Here, the
complex variable was written as
to emphasize that the limit must hold in all directions
. Letting
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
, one then says that the function
is of ''exponential type
''.
For example, let
. Then one says that
is of exponential type
, since
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
. 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 ...
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
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
does not have a limit as
goes to infinity. For example, for the function
:
the value of
:
at
is dominated by the
term so we have the asymptotic expressions:
:
and this goes to zero as
goes to infinity, but
is nevertheless of exponential type 1, as can be seen by looking at the points
.
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
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
. It is known that for every such
there is an associated
norm with the property that
:
In other words,
is the unit ball in
with respect to
. The set
:
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
. Furthermore, we can write
:
We extend
from
to
by
:
An entire function
of
-complex variables is said to be of exponential type with respect to
if for every
there exists a real-valued constant
such that
: