Convergence area

A region of convergence is in the Analysis , a branch of mathematics , a sequence of functions , or (more often) function number assigned and (often also in the sense of denoting a inclusion maximum) set of points in the domain in which the function number converges pointwise . Convergence areas are areas , i.e. open , contiguous subsets of convergence areas. The terms convergence area and area generalize the terms “convergence interval” or “convergence circle disc ” from elementary, real analysis and elementary function theory . For historical reasons, convergence criteria for function sequences and series are sometimes referred to as (generalized) Cauchy-Hadamard formulas . The classical Cauchy-Hadamard theorem formulates such criteria for complex power series.

Frequently used function series

The series considered in the following are always to be understood as complex series, i.e. their coefficients are complex, the independent variable is complex, the terms of the series are defined on a subset of functions and their areas of convergence are subsets of . Of course, the series themselves only represent functions if their maximum convergence area is not empty. ${\displaystyle \mathbb {C} }$ ${\displaystyle \mathbb {C} }$

• For a power series the maximum area of ​​convergence is an open disc around the point of development , the radius of which is called the radius of convergence or (for ) its maximum area of ​​convergence , then it has no area of ​​convergence. ${\displaystyle z_{0}}$ ${\displaystyle \rho \in \mathbb {R} _{+}\cup \lbrace \infty \rbrace }$ ${\displaystyle \rho =0}$ ${\displaystyle \lbrace z_{0}\rbrace }$
• For a Laurent series the maximum convergence area is an open annulus around the development point or there is no convergence area.
• For a Dirichlet the maximum convergence area is a "right" half-plane in the complex plane by is given. The number is called the convergence abscissa of the Dirichlet series. Also in the case one speaks of a (formal) Dirichlet series with this convergence abscissa, however this does not converge at any point of , therefore it has no convergence domains and its only and maximum convergence domain is the empty set. ${\displaystyle H}$ ${\displaystyle H=\lbrace z\in \mathbb {C} ,\operatorname {Re} (z)>\sigma _{0}\rbrace }$ ${\displaystyle \sigma _{0}\in \mathbb {R} \cup \lbrace -\infty \rbrace }$ ${\displaystyle \sigma _{0}=+\infty }$ ${\displaystyle \mathbb {C} }$

If there is a convergence area at all, the following applies in all three cases:

• There is exactly one maximum convergence area ( the convergence area).
• The series converges compactly on each convergence domain .
• The maximum convergence area is a subset of the closed envelope of the maximum convergence area and therefore
• the maximum convergence area is exactly the interior of the maximum convergence area.
• The series diverges at every point that is not in the closed hull of the maximum convergence region.
• There are series that converge at some, but not all, points that lie on the edge of the maximum convergence area. The convergence in such an edge point can also be absolute, without it being possible to infer the convergence behavior in other edge points directly.

Generalization for metric spaces

Let be a metric space and a Banach space . Let there be a sequence of continuous functions . Then ${\displaystyle (M,d)}$ ${\displaystyle (E,\|\cdot \|)}$ ${\displaystyle f_{n}\colon M\to E}$

• the series converges in the point if the sequence of partial sums , which is a sequence of points in the domain , converges. ${\displaystyle \textstyle \sum _{n=0}^{\infty }f_{n}}$ ${\displaystyle x\in M}$ ${\displaystyle \textstyle S_{k}(x):=\sum _{n=0}^{k}f_{n}(x)}$ ${\displaystyle E}$
• the series converges absolutely at the point if the number series converges over the norms of the summands . ${\displaystyle \textstyle \sum _{n=0}^{\infty }f_{n}}$ ${\displaystyle x\in M}$ ${\displaystyle \textstyle \sum _{n=0}^{\infty }\|f_{n}(x)\|}$

Any set of points where there is convergence is called a convergence region . Every connected component of the interior of the set of all points in which the sequence converges is a maximum convergence region . ${\displaystyle x\in M}$

Note: In the edge points of a convergence area or a convergence area, there does not have to be an absolute convergence, the corresponding series can even be divergent in the value area.

Cauchy-Hadamard's classic theorem

The following statements about the convergence areas of complex power series were (essentially) initially formulated by Augustin Louis Cauchy in 1821, but were hardly noticed ( Bernhard Riemann used them in his lecture notes in 1856) until they were rediscovered by Jacques Hadamard . He published them in 1888. Therefore they (and some modern generalizations) are called the Cauchy-Hadamard formula or theorem . Formulated in a modern way, but without generalizations to anything other than power series, the Cauchy-Hadamard theorem says:

Be , and with for each , d. H. the function series is a complex power series . Then: ${\displaystyle M=\mathbb {C} }$ ${\displaystyle E=\mathbb {C} }$ ${\displaystyle f_{n}(x)=c_{n}\cdot x^{n}}$ ${\displaystyle c_{n}\in \mathbb {C} }$ ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle \textstyle \sum _{n=0}^{\infty }f_{n}(x)=\sum _{n=0}^{\infty }c_{n}x^{n}}$

1. The open circular disk around the zero point with radius belongs to the maximum convergence range, if is fulfilled for all but finitely many . ${\displaystyle B(0,r)}$ ${\displaystyle r>0}$ ${\displaystyle \left|c_{n}\right|\cdot r^{n}<1}$ ${\displaystyle n\in \mathbb {N} }$
2. The complement of the closed circular disk does not intersect the maximum convergence area if it holds for infinitely many . ${\displaystyle B(0,R)}$ ${\displaystyle \left|c_{n}\right|\cdot R^{n}>1}$ ${\displaystyle n\in \mathbb {N} }$
3. There is a radius at which the two aforementioned statements “meet”. The radius of convergence is used if the limes superior exists as a real number, i.e. in the actual sense, and is not 0. If the limes is superior 0, then the radius of convergence , if the limes is superior , then is the radius of convergence . The maximum convergence range of the power series contains the open circular disk around 0 with a radius . In the case this is the empty set, otherwise the maximum convergence area. ${\displaystyle \textstyle \rho =(\limsup _{n\to \infty }{\sqrt[{n}]{|c_{n}|}})^{-1}}$ ${\displaystyle \rho =+\infty }$ ${\displaystyle +\infty }$ ${\displaystyle \rho =0}$ ${\displaystyle \rho }$ ${\displaystyle \rho =0}$
4. The power series converges at all points whose distance from zero is smaller than the radius of convergence . In addition, it diverges in all points whose distance is greater . No general statement can be made about the convergence in points whose distance from the zero point is exact (i.e. the circular line with this radius). ${\displaystyle \rho }$ ${\displaystyle \rho }$ ${\displaystyle \rho }$

The last statement applies analogously to the edge points of the maximum convergence areas of Laurent and Dirichlet series. Their maximum convergence area can also be calculated using suitable limites superiores .

Major and minor criterion

→ Main article : Weierstrasse's majorant criterion

The following convergence criteria were originally formulated for power series and the classical form of Cauchy-Hadamard's theorem is based on them. In the formulation given here, however, they also apply more generally under the conditions formulated above in the section #Generalization for metric spaces .

1. (Majorante) If there is a convergent series with positive real terms and a domain with for all and all but finitely many , then it is a subset of a maximal convergence domain. The convergence is on absolute, uniform and compact, so the limit function defined by the series on is on continuous, if this is true for all but finitely many partial sums. ${\displaystyle \textstyle \sum _{n=0}^{\infty }a_{n}}$ ${\displaystyle G\subseteq M}$ ${\displaystyle \|f_{n}(x)\|\leq a_{n}}$ ${\displaystyle x\in G}$ ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle G}$ ${\displaystyle G}$ ${\displaystyle G}$ ${\displaystyle F}$ ${\displaystyle G}$
2. (Minorante) If there is a divergent series with positive real terms and if the inequality applies to all and to all but finitely many in a domain , then the complement of the maximum convergence area is contained as a subset. ${\displaystyle \textstyle \sum _{n=0}^{\infty }b_{n}}$ ${\displaystyle H\subseteq M}$ ${\displaystyle \|f_{n}(x)\|>b_{n}}$ ${\displaystyle x\in H}$ ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle H}$
3. (Limitation) If the majorant criterion is fulfilled in one area and all partial sums of the function series are continuously up and the majorant criterion is also fulfilled for an edge point (possibly after continuous continuation of the continuous partial sums), then the function series also converges in uniformly and the limit function is continuously or continuously continuable on and for the limit function or its continuation applies ${\displaystyle G}$ ${\displaystyle G}$ ${\displaystyle x_{0}\in \partial G}$ ${\displaystyle G}$ ${\displaystyle \!\,\lbrace x_{0}\rbrace \cup G}$ ${\displaystyle F}$ ${\displaystyle \lbrace x_{0}\rbrace \cup G}$

${\displaystyle F(x_{0})=\lim _{N\to \infty }\lim _{x\to x_{0} \atop x\in G}\sum _{n=0}^{N}f_{n}(x)=\lim _{x\to x_{0} \atop x\in G}\lim _{N\to \infty }\sum _{n=0}^{N}f_{n}(x)}$

Examples

• The power series of the natural exponential function converges absolutely everywhere, so its radius of convergence is The convergence on is absolute, compact and locally uniform , but not uniform. ${\displaystyle \textstyle \exp(z):=\sum _{n=0}^{\infty }{\frac {z^{n}}{n!}},z\in \mathbb {C} }$ ${\displaystyle \rho =+\infty .}$ ${\displaystyle \mathbb {C} }$
• The formal power series converges absolutely against in the interior of the unit disk . For their maximum convergence area is the set of complex numbers ( ), otherwise exactly this unit circle ( ). ${\displaystyle \textstyle \sum _{n=0}^{\infty }{\frac {a(a-1)\dots (a-n+1)}{n!}}z^{n},z\in \mathbb {C} }$ ${\displaystyle (1+z)^{a}}$ ${\displaystyle a\in \mathbb {N} _{0}}$ ${\displaystyle \rho =+\infty }$ ${\displaystyle \rho =1}$
• The formal Dirichlet series of the Riemann zeta function has the convergence abscissa . For the edge point of the maximum convergence area , this Dirichlet series is the divergent harmonic series . ${\displaystyle \textstyle \zeta (s)=\sum _{n=1}^{\infty }n^{-s},s\in \mathbb {C} }$ ${\displaystyle \sigma _{0}=1}$ ${\displaystyle s=1}$ ${\displaystyle G=\lbrace s\in \mathbb {C} |\operatorname {Re} (s)>1\rbrace }$

literature

Textbooks

• Heinrich Behnke , Friedrich Sommer : Theory of the analytical functions of a complex variable . 3rd edition study edition. Springer, Berlin et al. 1976, ISBN 3-540-07768-5 .
• Harro Heuser : Functional Analysis. Theory and application . 3rd, revised edition. Teubner, Stuttgart 1992, ISBN 3-519-22206-X . - Table of contents .
• Harro Heuser: Textbook of Analysis . 14th, updated edition. tape 2 . Vieweg and Teubner, Wiesbaden 2008, ISBN 978-3-8351-0208-8 . - Table of contents .

On the history of Cauchy-Hadamard's theorem

• Umberto Bottazzini : The Higher Calculus. A History of Real and Complex Analysis from Euler to Weierstrass . Translated by Warren van Egmond. Springer, New York NY et al. 1986, ISBN 0-387-96302-2 (Italian).
• Jacques Hadamard : Sur le rayon de convergence des séries ordonnées suivant les puissances d'une variable . In: Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences . tape 106 , 1888, ISSN  0001-4036 , p. 259-262 (French). Digitized .

Individual evidence

1. ^ Augustin Louis Cauchy : Cours d'analysis de l'école Royale polytechnique. Lot 1: Analysis algébrique. Debure, Paris 1821, digitized .
2. ↑ Detlef Laugwitz, Cauchy-Hadamard's formula in Riemann's estate, Mathematical Semester Reports, Volume 40, 1993, pp. 115-120
3. ↑ Detlef Laugwitz, Erwin Neuenschwander, Riemann and the Cauchy-Hadamard formula for the convergence of power series, Historia Mathematica, Volume 21, 1994, pp. 64-70
4. ↑ Bottazzini: The Higher Calculus. 1986, p. 115 ff.
5. ^ Hadamard: Sur le rayon de convergence des séries ordonnées suivant les puissances d'une variable. 1888.