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| {{redirect3|Exponential type|For exponential types in type theory and programming languages, see [[Function type]]}}
| | Alyson Meagher is the title her mothers and fathers gave her but she doesn't like when individuals use her full name. Distributing production has been his profession for some time. Doing ballet is some thing she would by no means give up. Her family lives in Ohio.<br><br>my site :: tarot card readings ([http://kard.dk/?p=24252 listen to this podcast]) |
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| In [[mathematics]], in the area of [[complex analysis]], '''Nachbin's theorem''' (named after [[Leopoldo Nachbin]]) is commonly used to establish a bound on the growth rates for an [[analytic function]]. This article will provide a brief review of growth rates, including the idea of a '''function of exponential type'''. Classification of growth rates based on type help provide a finer tool than [[big O notation|big O]] or [[Landau notation]], since a number of theorems about the analytic structure of the bounded function and its [[integral transform]]s can be stated. In particular, Nachbin's theorem may be used to give the domain of convergence of the '''generalized Borel transform''', given below.
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| ==Exponential type==
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| {{Main|Exponential type}}
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| A function ''f''(''z'') defined on the [[complex plane]] is said to be of exponential type if there exist constants ''M'' and τ such that
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| :<math>|f(re^{i\theta})|\le Me^{\tau r}</math>
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| in the limit of <math>r\to\infty</math>. Here, the [[complex variable]] ''z'' was written as <math>z=re^{i\theta}</math> to emphasize that the limit must hold in all directions θ. Letting τ stand for the [[infimum]] of all such τ, one then says that the function ''f'' is of ''exponential type τ''.
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| For example, let <math>f(z)=\sin(\pi z)</math>. Then one says that <math>\sin(\pi z)</math> is of exponential type π, since π is the smallest number that bounds the growth of <math>\sin(\pi z)</math> along the imaginary axis. So, for this example, [[Carlson's theorem]] cannot apply, as it requires functions of exponential type less than π.
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| ==Ψ type==
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| Bounding may be defined for other functions besides the exponential function. In general, a function <math>\Psi(t)</math> is a '''comparison function''' if it has a series
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| :<math>\Psi(t)=\sum_{n=0}^\infty \Psi_n t^n</math>
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| with <math>\Psi_n>0</math> for all ''n'', and
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| :<math>\lim_{n\to\infty} \frac{\Psi_{n+1}}{\Psi_n} = 0.</math>
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| Note that comparison functions are necessarily [[entire function|entire]], which follows from the [[ratio test]]. If <math>\Psi(t)</math> is such a comparison function, one then says that ''f'' is of Ψ-type if there exist constants ''M'' and ''τ'' such that
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| :<math>\left|f\left(re^{i\theta}\right)\right| \le M\Psi(\tau r)</math>
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| as <math>r\to \infty</math>. If τ is the infimum of all such ''τ'' one says that ''f'' is of Ψ-type ''τ''.
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| ==Nachbin's theorem==
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| Nachbin's theorem states that a function ''f''(''z'') with the series
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| :<math>f(z)=\sum_{n=0}^\infty f_n z^n</math>
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| is of Ψ-type τ if and only if
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| :<math>\limsup_{n\to\infty} \left| \frac{f_n}{\Psi_n} \right|^{1/n} = \tau.</math>
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| ==Borel transform==
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| Nachbin's theorem has immediate applications in [[Cauchy's integral formula|Cauchy theorem]]-like situations, and for [[integral transforms]]. For example, the '''generalized Borel transform''' is given by
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| :<math>F(w)=\sum_{n=0}^\infty \frac{f_n}{\Psi_n w^{n+1}}.</math>
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| If ''f'' is of Ψ-type ''τ'', then the exterior of the domain of convergence of <math>F(w)</math>, and all of its singular points, are contained within the disk
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| :<math>|w| \le \tau.</math> | |
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| Furthermore, one has
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| :<math>f(z)=\frac{1}{2\pi i} \oint_\gamma \Psi (zw) F(w)\, dw</math> | |
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| where the [[contour of integration]] γ encircles the disk <math>|w| \le \tau</math>. This generalizes the usual '''Borel transform''' for exponential type, where <math>\Psi(t)=e^t</math>. The integral form for the generalized Borel transform follows as well. Let <math>\alpha(t)</math> be a function whose first derivative is bounded on the interval <math>[0,\infty)</math>, so that
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| :<math>\frac{1}{\Psi_n} = \int_0^\infty t^n\, d\alpha(t)</math> | |
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| where <math>d\alpha(t)=\alpha^{\prime}(t)\,dt</math>. Then the integral form of the generalized Borel transform is
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| :<math>F(w)=\frac{1}{w} \int_0^\infty f \left(\frac{t}{w}\right) \, d\alpha(t).</math>
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| The ordinary Borel transform is regained by setting <math>\alpha(t)=e^{-t}</math>. Note that the integral form of the Borel transform is just the [[Laplace transform]].
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| ==Nachbin resummation==
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| Nachbin resummation (generalized Borel transform) can be used to sum divergent series that escape to the usual [[Borel resummation]] or even to solve (asymptotically) integral equations of the form:
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| :<math> g(s)=s\int_0^\infty K(st) f(t)\,dt </math>
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| where ''f''(''t'') may or may not be of exponential growth and the kernel ''K''(''u'') has a [[Mellin transform]]. The solution, pointed out by L. Nachbin himself, can be obtained as <math> f(x)= \sum_{n=0}^\infty \frac{a_n}{M(n+1)}x^n </math> with <math> g(s)= \sum_{n=0}^\infty a_n s^{-n} </math> and ''M''(''n'') is the Mellin transform of ''K''(''u''). an example of this is the Gram series <math> \pi (x) \approx \sum_{n=1}^{\infty} \frac{\log^{n}(x)}{n\cdot n!\zeta (n+1)} </math>
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| ==Fréchet space==
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| Collections of functions of exponential type <math>\tau</math> can form a [[complete space|complete]] [[uniform space]], namely a [[Fréchet space]], by the [[topological space|topology]] induced by the countable family of [[norm (mathematics)|norm]]s
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| :<math> \|f\|_{n} = \sup_{z \in \mathbb{C}} \exp \left[-\left(\tau + \frac{1}{n}\right)|z|\right]|f(z)| </math>
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| ==See also==
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| * [[Divergent series]]
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| * [[Borel summation]]
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| * [[Euler summation]]
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| * [[Cesàro summation]]
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| * [[Lambert summation]]
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| * [[Nachbin resummation]]
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| * [[Phragmén–Lindelöf principle]]
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| * [[Abelian and tauberian theorems]]
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| * [[Van Wijngaarden transformation]]
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| ==References==
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| * L. Nachbin, "An extension of the notion of integral functions of the finite exponential type", ''Anais Acad. Brasil. Ciencias.'' '''16''' (1944) 143–147.
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| * Ralph P. Boas, Jr. and R. Creighton Buck, ''Polynomial Expansions of Analytic Functions (Second Printing Corrected)'', (1964) Academic Press Inc., Publishers New York, Springer-Verlag, Berlin. Library of Congress Card Number 63-23263. ''(Provides a statement and proof of Nachbin's theorem, as well as a general review of this topic.)''
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| * {{springer|author=A.F. Leont'ev|id=F/f041990|title=Function of exponential type}}
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| * {{springer|author=A.F. Leont'ev|id=B/b017190|title= Borel transform}}
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| * Garcia J. Borel Resummation & the Solution of Integral Equations '' Prespacetime Journal '' nº 4 Vol 4. 2013 http://prespacetime.com/index.php/pst/issue/view/42/showToc
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| [[Category:Integral transforms]]
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| [[Category:Theorems in complex analysis]]
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| [[Category:Summability methods]]
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Alyson Meagher is the title her mothers and fathers gave her but she doesn't like when individuals use her full name. Distributing production has been his profession for some time. Doing ballet is some thing she would by no means give up. Her family lives in Ohio.
my site :: tarot card readings (listen to this podcast)