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 (r_n)_{n\in\N} of rational functions which converges uniformly to f on K and such that all the poles of the functions (r_n)_{n\in\N} are in A.
Note that not every complex number in A needs to be a pole of every rational function of the sequence (r_n)_{n\in\N}. We merely know that for all members of (r_n)_{n\in\N} 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 (p_n) 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.
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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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Carl David Tolmé Runge - Wikipedia, the free encyclopedia
Physics. Institutions, University of Hanover (1886–1904) Georg-August University of Göttingen (1904–1925) ... Laplace–Runge–Lenz vector · Runge's Theorem ...
[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 ...
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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 ...
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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...
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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 ...
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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...
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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.
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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...
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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
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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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