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Runge's Theorem

In complex analysis, Runge's theorem (also known as Runge's approximation theorem) is named after the German mathematician Carl Runge who first proved it in the year 1885. It states the following:

Denoting by C the set of complex numbers, let K be a compact subset of C and let f be a function which is holomorphic on an open set containing K. If A is a set containing at least one complex number from every bounded connected component of C\K then there exists a sequence of rational functions which converges uniformly to f on K and such that all the poles of the functions are in A.

Note that not every complex number in A needs to be a pole of every rational function of the sequence . We merely know that for all members of that do have poles, those poles lie in A.

One aspect that makes this theorem so powerful is that one can choose the set A arbitrarily. In other words, one can choose any complex numbers from the bounded connected components of C\K and the theorem guarantees the existence of a sequence of rational functions with poles only amongst those chosen numbers.

For the special case in which C\K is a connected set (or equivalently that K is simply-connected), the set A in the theorem will clearly be empty. Since rational functions with no poles are simply polynomials, we get the following corollary: If K is a compact subset of C such that C\K is a connected set, and f is a holomorphic function on K, then there exists a sequence of polynomials that approaches f uniformly on K.

Runge's theorem generalises as follows: if one takes A to be a subset of the Riemann sphere C∪{∞} and requires that A intersect also the unbounded connected component of K (which now contains ∞). That is, in the formulation given above, the rational functions may turn out to have a pole at infinity, while in the more general formulation the pole can be chosen instead anywhere in the unbounded connected component of K.

This is an excerpt from the article Runge's Theorem from the Wikipedia free encyclopedia. A list of authors is available at Wikipedia.

Denoting by C the set of complex numbers, let K be a compact subset of C and let f be a function which is holomorphic on an open set containing K. If A is a set containing at least one complex number from every bounded connected component of C\K then there exists a sequence of rational functions which converges uniformly to f on K and such that all the poles of the functions are in A.

Note that not every complex number in A needs to be a pole of every rational function of the sequence . We merely know that for all members of that do have poles, those poles lie in A.

One aspect that makes this theorem so powerful is that one can choose the set A arbitrarily. In other words, one can choose any complex numbers from the bounded connected components of C\K and the theorem guarantees the existence of a sequence of rational functions with poles only amongst those chosen numbers.

For the special case in which C\K is a connected set (or equivalently that K is simply-connected), the set A in the theorem will clearly be empty. Since rational functions with no poles are simply polynomials, we get the following corollary: If K is a compact subset of C such that C\K is a connected set, and f is a holomorphic function on K, then there exists a sequence of polynomials that approaches f uniformly on K.

Runge's theorem generalises as follows: if one takes A to be a subset of the Riemann sphere C∪{∞} and requires that A intersect also the unbounded connected component of K (which now contains ∞). That is, in the formulation given above, the rational functions may turn out to have a pole at infinity, while in the more general formulation the pole can be chosen instead anywhere in the unbounded connected component of K.

This is an excerpt from the article Runge's Theorem from the Wikipedia free encyclopedia. A list of authors is available at Wikipedia.

The article Runge's Theorem at en.wikipedia.org was accessed 339 times in the last 30 days. (as of: 12/08/2013)

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Runge's theorem - Wikipedia, the free encyclopedia

In complex analysis, Runge's Theorem (also known as Runge's approximation
theorem) is named after the German mathematician Carl Runge who first proved
it ...

en.wikipedia.org/wiki/Runge's_theorem

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Runge's Theorem -- from Wolfram MathWorld

Runge's Theorem. Let K subset= C be compact, let f be analytic on a
neighborhood of K , and let P subset= C^*\K contain at least one point from each
connected ...

mathworld.wolfram.com/RungesTheorem.html

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Application of Runge's theorem - Math StackExchange

Apr 25, 2013 ... Runge's Theorem states: Let $K$ be a compact subset of $\mathbb C$ and let $S\
subset \overline{\mathbb C}\setminus K$, such that $S$ ...

math.stackexchange.com/questions/372552/application-of-runges-theorem

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On Runge's Theorem We spent several lectures studying ...

On Runge's Theorem. We spent several lectures studying approximation of
functions on the real line by poly- nomials. We proved, for example, that on a
closed ...

people.maths.ox.ac.uk/greenbj/papers/runge.pdf

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GENERALIZATION OF RUNGE'S THEOREM TO APPROXIMATION ...

Introduction. According to Runge's Theorem. [4](2), any function. M(z) which is
analytic in a given region R can be expanded in a sequence of rational functions
...

www.ams.org/journals/tran/1952-072-01/S0002-9947-1952-0047137-1/S0002-9947-1952-0047137-1.pdf

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Lecture notes

Runge's Theorem. 13. 1. Partitions of unity. 13. 2. Smeared out Cauchy Integral
Formula. 13. 3. Runge's Theorem. 13. Chapter 3. Applications of Runge's ...

www.uio.no/studier/emner/matnat/math/MAT4800/h12/complex-analysis.pdf

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Runge theorem - Encyclopedia of Mathematics

Feb 7, 2011 ... A theorem on the possibility of polynomially approximating holomorphic functions
, first proved by C. Runge (1885) (cf. also Approximation of ...

www.encyclopediaofmath.org/index.php/Runge_theorem

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Lecture Notes in Complex Analysis

May 9, 2007 ... 2.3 Homology Form of Cauchy's Theorem . ... 3.3 Runge's Theorem . .... The proof
of the following theorem is exactly the same as the proof of ...

www.bprim.org/arsm.pdf

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Exercise on Runge's theorem

Apr 3, 2006 ... standing of Runge's Theorem: If K is a compact sub- set of C, then every function
holomorphic in a neigh- borhood of K can be approximated ...

www.math.tamu.edu/~boas/courses/618-2006a/apr03.pdf

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MATH 215 SOLUTION TWO 2.6-15. Suppose f is a non-vanishing ...

D. 2.7-4. Prove the converse to Runge's Theorem: that if K is a compact set whose
complement is not connected, then there exists a function f holomorphic in a.

math.stanford.edu/~jli/Math215A/Math215.SolutionTwo%20copy.pdf

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Runge's Theorem in science

Carl David Tolmé Runge - Wikipedia, the free encyclopedia

[PDF]A quantitative version of Runge's theorem on diophantine equations

improves on the special case of Runge's Theorem proved in Theorem 3.31 of ...... [
21] R. J. W a l k e r, Algebraic Curves, Princeton University Press, Princeton, ...

[PDF]A quantitative version of Runge's theorem on diophantine equations

Runge's Theorem asserts that if. F satisfies ... In [3] it was shown more precisely
that under the hypotheses of Runge's Theorem, all ... UNIVERSITY OF OTTAWA.

MAT9800 - Complex Analysis - University of Oslo - University of Oslo

Applications of the residue theorem, Montel's theorem, Cauchy-estimates,
solutions to d-bar, Runge's Theorem, Cousin I and II, Ahlfors-Schwarz-Pick
Lemma, ...

MAT4800 - Complex Analysis - University of Oslo - University of Oslo

Applications of the residue theorem, Montel's theorem, Cauchy-estimates,
solutions to d-bar, Runge's Theorem, Cousin I and II, Ahlfors-Schwarz-Pick
Lemma, ...

Runge's Theorem - Springer

Runge's Theorem. D. H. Luecking,; L. A. Rubel ... Author Affiliations. 3.
Department of Mathematics, University of Arkansas, Fayetteville, AR, 72701, USA
; 4.

Mathematics (MATH) - Calendars - Carleton University

Complex differentiation and integration, harmonic functions, maximum modulus
principle, Runge's Theorem, conformal mapping, entire and meromorphic ...

Runge's theorem in hypercomplex function theory - ResearchGate

Runge's Theorem in hypercomplex function theory. R DELANGHE. Seminar of
Mathematical Analysis, State University of Ghent, Gent B-9000, Belgium.

Sabancı University Faculty of Engineering and Natural Sciences ...

One of the exam subjects may be from another University Program; in this case
the ... Zeros and Poles of Analytic Functions: Runge's Theorem, Meromorphic ...

Books on the term Runge's Theorem

Function Theory of One Complex Variable

Chapter. 12. Rational. Approximation. Theory. 12.1. Runge's. Theorem. A rational
function is, by definition, a quotient of polynomials. A function from C to C is
rational if and only if it is meromorphic on all of C (Theorem 4.7.7). A rational
function ...

Fundamentals of Time-Dependent Density Functional Theory (Lecture Notes in Physics, Vol. 837)

There have been many significant advances in time-dependent density functional theory over recent years, both in enlightening the fundamental theoretical basis of the theory, as well as in computational algorithms and applications. This book, as successor to the highly successful volume Time-Dependent Density Functional Theory (Lect. Notes Phys. 70...

A Course in Functional Analysis

(Though it is difficult to see at this moment, this fact is related to the proof of (c) in
Theorem 7. ... Runge's. Theorem. The symbol (Cx denotes the extended complex
plane. 8.1. Runge's Theorem. Let K be a compact subset of

Complex Analysis (Princeton Lectures in Analysis, No. 2)

With this second volume, we enter the intriguing world of complex analysis. From the first theorems on, the elegance and sweep of the results is evident. The starting point is the simple idea of extending a function initially given for real values of the argument to one that is defined when the argument is complex. From there, one proceeds to the m...

Complex function theory

Runge's Theorem ON POLYNOMIAL APPROXIMATION We return to the Runge theorem concerning approximation by rational functions (Theorem 8.1, Ch
. V) and apply it to obtain the following theorem of Runge on the possibility of ...

Introductory Differential Equations, Third Edition: with Boundary Value Problems

This text is for courses that are typically called (Introductory) Differential Equations, (Introductory) Partial Differential Equations, Applied Mathematics, Fourier Series and Boundary Value Problems. The text is appropriate for two semester courses: the first typically emphasizes ordinary differential equations and their applications while the se...

Complex Analysis: In the Spirit of Lipman Bers

However, truncation of the Laurent series expansion for f on Ω shows that f is
indeed uniformly approximated on K by rational functions whose poles lie outside
Ω. This fact is generalized to arbitrary open sets Ω by Runge's Theorem.

Complex Analysis (Undergraduate Texts in Mathematics)

An introduction to complex analysis for students with some knowledge of complex numbers from high school. It contains sixteen chapters, the first eleven of which are aimed at an upper division undergraduate audience. The remaining five chapters are designed to complete the coverage of all background necessary for passing PhD qualifying exams in com...

Complex Function Theory

Sharpened Form of Runge's Theorem Let K be a compact subset of C and let S
be a subset of C\K that contains at least one point in each connected component
of C\K . Then any function holomorphic in an open set containing K can be ...

A Mathematician's Survival Guide: Graduate School and Early Career Development

Graduate school marks the first step toward a career in mathematics for young mathematicians. During this period, they make important decisions which will affect the rest of their careers. Here now is a detailed guide to help students navigate graduate school and the years that follow. In his inimitable and forthright style, Steven Krantz addresses...

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Online sources for the term

Runge's Theorem

Runge's Theorem

Hazewinkel, Michiel, ed. (2001), "Runge theorem", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

www.encyclopediaofmath.org/index.php?title=p/r082830
Blog posts on the term

Runge's Theorem

Runge's Theorem

complex analysis - Runge's theorem and polynomially convex hull - Mathematics Stack Exchange
math.stackexchange.com/questions/571904/runges-theorem-and-polynomially-convex-hull

Some operation systems might schedule more ti - Infinite Positive

Heit: Well, as i said, it sounds like a normal distribution.
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Lecture Notes on Diophantine Analysis

These lecture notes originate from a course delivered at the Scuola Normale in Pisa in 2006. Generally speaking, the prerequisites do not go beyond basic mathematical material and are accessible to many undergraduates. The contents ...

www.springer.com/birkhauser/mathematics/scuola+normale+superiore/book/978-88-7642-341-3
Phys. Rev. Lett. 111, 023001 (2013): Fragment-Based Time-Dependent Density Functional Theory
prl.aps.org/abstract/PRL/v111/i2/e023001

pehvoffdx - Download ebook Classical topics in complex function theory pdf
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[1107.4810] Numerical Stability of Explicit Runge-Kutta Finite-Difference Schemes
for the Nonlinear Schr\"odinger Equation
arxiv.org/abs/1107.4810

Halmos on writing and education | Annoying Precision

John Ewing wrote up a nice collection of quotes from Paul Halmos for the Notices of the AMS; let's meditate on his words. For example: The best notation is no notation; whenever possible to avoid the use of a complicated alphabetic apparatus, avoid it. A good attitude to the preparation of written mathematical exposition is…

qchu.wordpress.com/2009/08/05/halmos-on-writing-and-education/
zdmarioy » Blog Archive » Download ebook Classical Topics in Complex Function Theory (Graduate pdf

Classical Topics in Complex Function Theory (Graduate Texts in Mathematics 172) download pdf book.
Download Classical Topics in Complex Function Theory (Graduate Texts in Mathematics 172) here.

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fsmarcus » Blog Archive » Download ebook Classical topics in complex function theory pdf

Classical topics in complex function theory download pdf book by Remmert R. .

fsmarcus.blogdetik.com/2013/08/26/download-ebook-classical-topics-in-complex-function-theory-pdf/
lagrangian formalism - A kind of Noether's theorem for the Hamiltonian - Physics Stack Exchange
physics.stackexchange.com/questions/69271/a-kind-of-noethers-theorem-for-the-hamiltonian

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