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Topic: Weierstrass factorization theorem

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	Karl Weierstrass
		Karl Theodor Wilhelm Weierstrass (WeierstraÃ) (October 31, 1815 â“ February 19, 1897) was a German mathematician who is often cited as the "father of modern analysis".
		After that he studied mathematics at the University of MÃ¼nster which was even to this time very famous for mathematics and his father was able to obtain a place for him in a teacher training school in MÃ¼nster, and he later was certified as a teacher in that city.
		During this period of study, Weierstrass attended the lectures of Christoph Gudermann and became interested in elliptic functions.
	www.anime.co.za /wiki/Karl_Weierstrass (423 words)

	PlanetMath: fundamental theorems in complex analysis (Site not responding. Last check: 2007-10-21)
		The following is a list of fundamental theorems in the subject of complex analysis (single complex variable).
		If a theorem does not yet appear in the encyclopedia, please consider adding it -- Planet Math is a work in progress and even some basic results have not yet been entered.
		This is version 19 of fundamental theorems in complex analysis, born on 2005-01-22, modified 2006-07-19.
	planetmath.org /encyclopedia/FundamentalTheoremsInComplexAnalysis.html (142 words)

	Karl Weierstrass - Wikipedia, the free encyclopedia
		He took a chair at the Technical University of Berlin, then known as the Gewerbeinstitut.
		Weierstrass was interested in the soundness of calculus.
		Weierstrass also formulated similar definitions of limit and derivative still taught today.
	en.wikipedia.org /wiki/Karl_Weierstrass (445 words)

	Springer Online Reference Works
		The Jackson theorem is a strengthening of this theorem.
		Weierstrass in 1860 as a preparation lemma, used in the proofs of the existence and analytic nature of the implicit function of a complex variable defined by an equation
		The analogue of the Weierstrass preparation theorem for differentiable functions is variously known as the differentiable preparation theorem, the Malgrange preparation theorem or the Malgrange–Mather preparation theorem.
	eom.springer.de /w/w097510.htm (1150 words)

	Weierstrass factorization theorem - Wikipedia, the free encyclopedia (via CobWeb/3.1 planetlab2.netlab.uky.edu) (Site not responding. Last check: 2007-10-21)
		The consequences of the fundamental theorem of algebra are twofold
		The theorem generalizes to: sequences in open subsets (and hence regions) of the Riemann sphere have associated functions that are holomorphic in those subsets and have zeroes at the points of the sequence.
		The theorem may be generalized to the space of meromorphic functions, in which case, the factorization is unique.
	en.wikipedia.org.cob-web.org:8888 /wiki/Weierstrass_factorization_theorem (539 words)

	PlanetMath: Weierstrass factorization theorem
		There are several different statements of this theorem, but in essence this theorem will allow us to prescribe zeros and their orders of a holomorphic function.
		It also allows us to factor any holomorphic function into a product of zeros and a non-zero holomorphic function.
		This is version 4 of Weierstrass factorization theorem, born on 2004-04-22, modified 2006-09-17.
	planetmath.org /encyclopedia/WeierstrassProductTheorem.html (336 words)

	[No title] (Site not responding. Last check: 2007-10-21)
		The theorem prover Weierstrass was built by combining the symbolic computation facilities of Mathpert with an inference engine, earlier built for the theorem-prover Gentzen.
		Therefore, Weierstrass needed the extra apparatus of a theorem-prover, based on a backwards construction of a proof in Gentzen sequent calculus using unification to instantiate meta-variables for terms.
		To make a long story short, Weierstrass was able to generate several proofs involving epsilon-delta arguments (in analysis), and in number theory, it was able to prove the irrationality of e.
	www.mathcs.sjsu.edu /faculty/beeson/Papers/weier.html (873 words)

	Karl Weierstrass - Biocrawler (Site not responding. Last check: 2007-10-21)
		Karl Weierstrass was the son of Wilhem Weierstrass, a government official, and Theodora Vonderforst.
		His father was able to obtain a place for him in a teacher training school in Muenster, and he later was certified as a teacher in that city.
		fr:Karl Weierstrass it:Karl Weierstrass he:קארל ויירשטראס nl:Karl Weierstrass ja:カール・ワイエルシュトラス pl:Karl Weierstrass pt:Karl Weierstrass ru:Вейерштрасс, Карл zh:卡尔·魏尔施特拉斯
	www.biocrawler.com /encyclopedia/Karl_Weierstrass (367 words)

	Record of daily activities and homework, Math 618, Theory of Functions of a Complex Variable II, Spring 2004, Texas ...
		We discussed Hadamard's factorization theorem for entire functions: the genus μ and the order λ are related by the double inequality μ ≤ λ ≤ μ+1.
		We discussed proofs of the area theorem and of the estimate for the second coefficient of a normalized schlicht function in the unit disc.
		We discussed the statement of the prime number theorem and some numerical evidence for it; the Skewes number; and the three functions ζ, Φ, and ϑ that appear in the proof of the prime number theorem.
	www.math.tamu.edu /~harold.boas/courses/618-2004a/daily.html (1723 words)

	IUPUI Course Bulletin: School of Science-Math
		Continuation of differentiation, the mean value theorem and applications, the inverse and implicit function theorems, the Riemann integral, the fundamental theorem of calculus, point-wise and uniform convergence, convergence of infinite series, series of functions.
		Curvature and torsion, Frenet-Serret apparatus and theorem, fundamental theorem of curves.
		Compactness and convergence in the space of analytic functions, Riemann mapping theorem, Weierstrass factorization theorem, Runge’s theorem, Mittag-Leffler theorem, analytic continuation and Reimann surfaces, Picard theorems.
	www.indiana.edu /~bltindy/science/math/mathcourses.html (3059 words)

	[No title] (Site not responding. Last check: 2007-10-21)
		It played a key role in Euler's solution of the "Basel problem", which was to compute the sum of the reciprocals of the squares of the positive integers.
		It appears that Euler viewed the infinite product expansion of sin(x) as the factorization of (the infinite degree polynomial!) sin(x) into monomials, as indicated by its roots at x=0, and x=(+/-)k*pi for k=1,2,....
		Nowadays one may view the expansion as an instance of Weierstrass's factorization theorem which (in the present situation) says that () sin(z) = zexp(g(z))Pi_{n=1}^{infty} [1-(z^2)/(pin)^2] where g is an entire function.
	www.math.niu.edu /~rusin/known-math/00_incoming/weier_fact (258 words)

	Graduate Courses
		Serre duality, the existence of meromorphic functions on Riemann surfaces, the equality of the topological and analytic genera, the equivalence of algebraic curves and compact Riemann surfaces, the Riemann-Roch theorem.
		The Weierstrass preparation and division theorems, properties of the local ring of germs of holomorphic functions, complex analytic varieties, the Ruckert Nullstellensatz.
		Sheaves and cohomology, coherent analytic sheaves, Oka's coherence theorem, Dolbeault cohomology.
	mathnt.mat.jhu.edu /new/grad/grad-courses.htm (614 words)

	Mathematics (MATH)
		Analytic functions as mappings, Cauchy theorems, Laurent series, maximum modulus theorems and ramifications, normal families, Riemann mapping theorem, Weierstrass factorization theorem, Mittag-Leffler theory, analytic continuation, and harmonic functions.
		Hilbert and Banach space theory, linear operator theory, the closed graph theorem, the open mapping theorem, the principle of uniform boundedness, linear functionals, dual spaces and weak topologies, distribution theory, topological vector spaces, spectral theory of compact and unbounded self-adjoint and unitary operators, and semigroup theory.
		Measure and integration, axiomatic foundations of probability theory, random variables, distributions and their characteristic functions, stable and infinitely divisible laws, limit theorems for sums of independent random variables, conditioning, Martingales.
	www.depts.ttu.edu /officialpublications/courses/MATH.html (1727 words)

	PlanetMath:
		Weierstrass division theorem (=Weierstrass preparation theorem) owned by jirka
		Weierstrass extreme value theorem (=extreme value theorem) owned by classicleft
		Weierstrass' theorem on addition formulas owned by rspuzio
	planetmath.org /encyclopedia/W (855 words)

	IUPUI Course Bulletin (Site not responding. Last check: 2007-10-21)
		Divergence theorem, Stokes' Theorem, complex variables, contour integration, calculus of residues and applications, conformal mapping, and potential theory.
		Compactness and convergence in the space of analytic functions, Riemann mapping theorem, Weierstrass factorization theorem, Runge's theorem, Mittag-Leffler theorem, analytic continuation and Riemann surfaces, Picard theorems.
		General probability rules, conditional probability, Bayes theorem, discrete and continuous random variables, moments and moment generating functions, continuous distributions and their properties, law of large numbers, and central limit theorem.
	bulletin.iupui.edu /2004-html/science/dept_math_courses.html (4777 words)

	Weierstrass Approximation Theorem. Bernstein's Polynomials. (Site not responding. Last check: 2007-10-21)
		There are generalizations of this theorem and non-constructive proofs.
		Factorization of a polynomial, which defines values of sine function (angles n*pi/17).
		Factorization of polynomials with roots cos(2kpi/n), where n is Fermat number.
	www.mathandcomp.com /mathcountry/approximation.htm (387 words)

	Texas A&M University Graduate Catalog 2004-2005 Edition (Site not responding. Last check: 2007-10-21)
		Among the factors considered in admission decisions are: GRE General Test, undergraduate and graduate GPR, undergraduate academic background and achievement, letters of recommendation, GRE Subject Test in Mathematics (encouraged but not required).
		Banach spaces, theorems of Hahn-Banach and Banach-Steinhaus, the closed graph and open mapping theorems, Hilbert spaces, topological vector spaces and weak topologies.
		Instability mechanisms; instability of interfacial and free surface flows; thermal instability, centrifugal instability, instability of inviscid and viscous parallel shear flows; fundamental concepts and applications of nonlinear instability; the onset of turbulence; various transitions to turbulence.
	www.tamu.edu /admissions/GCatalog2004-5/Course_Descriptions/MATH.htm (1780 words)

	Mathematics Department - Wayne State University
		Core material: divisibility, prime numbers, greatest common divisors, the Euclidean algorithm, linear Diophantine equations, congruences, mathematical induction, the Fundamental Theorem of Arithmetic (unique factorization theorem), number and sum of divisors of an integer, linear congruences, the Chinese remainder theorem, Fermat's little theorem, Euler's theorem, Wilson's theorem, quadratic reciprocity.
		This includes the Central Limit Theorem, the convergence of sequences and sums of random variables, infinitely divisible distributions, and the Strong Law of Large Numbers.
		The remaining six chapters deal with various topics, including the Riemann mapping theorem, the Weierstrass factorization theorem, the gamma function and the Riemann zeta function, Runge's theorem, the Mittag-Leffler theorem, the Schwarz reflection principle, the monodromy theorem, Riemann surfaces, harmonic functions, Jensen's formula, the Hadamard factorization theorem, Bloch's theorem, the Picard theorems, and Schottky's theorem.
	www.math.wayne.edu /pinkbook.html (3379 words)

	[No title] (Site not responding. Last check: 2007-10-21)
		Weierstrass factorization for entire function Find all positive-valued harmonic functions on the complex plane.
		Explicit equation of p and p' Divisors of a curve.
		Abel's theorem Definition of arithematic genus and geometric genus.
	www.math.princeton.edu /graduate/generals/wu_zhongtao (194 words)

	Math Course Descriptions
		Planar graphs and the theorems of Euler and Kuratowski.
		Theorems of Lie, Engel, and Weyl; Cartan's Criterion; the classification of root systems; and abstract theory of weights.
		Reducible and irreducible representations, Maschke's theorem, characters, Schur's lemmas, orthogonality theorems, the group algebra, induced representations and Frobenius reciprocity, Young tableaux and representations of the symmetric group, applications in chemistry and physics.
	www.sci.csuhayward.edu /mathcs/coursesMath (3192 words)

	Math 542. Complex Variables I
		Basic definitions and properties; the local Cauchy theory, the Cauchy integral theorem and integral formula for a disk; integrals of Cauchy type; consequences.
		The residue theorem, evaluation of certain improper real integrals; argument principle, Rouche's theorem, the local mapping theorem.
		Ascoli-Arzela theorem, normal families, theorems of Montel and Hurwitz, the Riemann mapping theorem.
	www.math.uiuc.edu /Bourbaki/Syllabi/syl542.html (165 words)

	Record of daily activities, Math 618, Spring 2006, Texas A&M University
		Also we discussed part 9 of the exercise on simple connectivity and observed that the case of the square would be solved if we knew that the Riemann map from the square to the disc can be approximated uniformly on compact sets by polynomials.
		We finished the second part of the exercise on Picard's theorems and recapitulated the key idea in the proof of the homology version of Cauchy's integral formula.
		We discussed an improvement of the version of Mittag-Leffler's theorem stated in the book (Theorem 8.3.8): namely, in addition to prescribing finite chunks of the Laurent series of a meromorphic function at prescribed points, one can guarantee that the function has no extraneous zeroes.
	www.math.tamu.edu /~harold.boas/courses/618-2006a/daily.xml (2040 words)

	[No title]
		Note that the resulting factorization is _not_ unique; for example, for any _finite_ collection of the roots you may replace the 'E' factors above by (z-c_n).
		Naturally, there are extensions of this theorem, such as to other open domains in C. There are also applications, such as factorizations of the trigonometric functions, the gamma function, and so on.
		Oh, you can factor the zeta function too, but as an exercise you'll have to find its zeros first.
	www.math.niu.edu /~rusin/known-math/95/hilbert_irred (815 words)

(via CobWeb/3.1 planetlab2.netlab.uky.edu)MAA 6407 - Complex Analysis II (via CobWeb/3.1 ...</u> <i>(Site not responding. Last check: 2007-10-21)</i></td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> One of these is the Little Picard <b>Theorem</b>, which asserts that the range of a nonconstant <a href="/topics/Entire-function" title="Entire function" class=fl>entire function</a> must be the entire <a href="/topics/Complex-plane" title="Complex plane" class=fl>complex plane</a>, excepting possibly at most one point. </td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> A more local result is the Great Picard <b>Theorem</b>, which demonstrates that there are singularities of analytic functions in each neighborhood of which the function takes every complex value, with one possible exception, an infinite number of times. </td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> The student will be expected to (continue to) develop the ability to reason through and coherently and correctly write proofs of <b>theorems</b>, as well as to develop relevant computational skills.</td></tr> <tr><td></td><td colspan=2><font color=gray>www.math.ufl.edu.cob-web.org:8888 /~sjs/MAA6407.html</font> (600 words)</td></tr> </table> </td> </tr> </table><body face="Arial"> <br> <table cellpadding=0> <tr> <td> </td> <td> <table > <tr><td> </td><td colspan=2><u>PlanetMath 2005-06-08 Snapshot: W</u> <i>(Site not responding. Last check: 2007-10-21)</i></td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> weak ergodic <b>theorem</b>, a (=fundamental <b>theorem</b> of demography) </td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> <b>Weierstrass</b> eta function (defined in <b>Weierstrass</b> sigma function) </td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> <b>Weierstrass</b> sigma function (defined in <b>Weierstrass</b> sigma function)</td></tr> <tr><td></td><td colspan=2><font color=gray>202.41.85.103 /manuals/planetmath/W.html</font> (122 words)</td></tr> </table> </td> </tr> </table><body face="Arial"> <br> <table cellpadding=0> <tr> <td> </td> <td> <table > <tr><td> </td><td colspan=2><a href="http://planetmath.org/encyclopedia/E">PlanetMath:</a></td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> Zorn's lemma and the well-ordering <b>theorem</b> equivalence of Hausdorff's maximum principle owned by mathcam </td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> Euler's <b>theorem</b> on homogeneous functions owned by igor </td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> extended mean value <b>theorem</b> (=extended mean-value <b>theorem</b>) owned by mathwizard</td></tr> <tr><td></td><td colspan=2><font color=gray>planetmath.org /encyclopedia/E</font> (2301 words)</td></tr> </table> </td> </tr> </table><body face="Arial"> <br> <table cellpadding=0> <tr> <td> </td> <td> <table > <tr><td> </td><td colspan=2><a href="http://www.math.okstate.edu/grad/gradbk.html">OSU Department of Mathematics Graduate Student Handbook</a></td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> Number theory is famed not just for the beauty of its <b>theorems</b>, but for the enormous wealth and variety of techniques involved in discovering and proving these <b>theorems</b>. </td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> Current research projects are the extension of Hecke's converse <b>theorem</b> to GL(n) and applications of this converse <b>theorem</b> to liftings of automorphic forms and indirectly to class field theory. </td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> First work proved <b>theorems</b> on the distribution of cubic extensions of number fields using the theory of equivalence of binary cubic forms.</td></tr> <tr><td></td><td colspan=2><font color=gray>www.math.okstate.edu /grad/gradbk.html</font> (14669 words)</td></tr> </table> </td> </tr> </table><body face="Arial"> <br> <table cellpadding=0> <tr> <td> </td> <td> <table > <tr><td> </td><td colspan=2><a href="http://www.ma.utexas.edu/dev/math/Graduate/Prelims/Exam_Syllabi/Analysis.html">Analysis Syllabus</a></td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> Integration over paths, the local and global forms of Cauchy's <b>Theorem</b>, winding number and residue <b>theorem</b>, harmonic functions, Schwarz's Lemma and the Maximum Modulus <b>theorem</b>, isolated singularites, entire and meromorphic functions, Laurent series, <a href="/topics/Infinite-product" title="Infinite product" class=fl>infinite products</a>, <b>Weierstrass</b> <b>factorization</b>, conformal mapping, Riemann mapping <b>theorem</b>, analytic continuation, "little" Picard <b>theorem</b>. </td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> Students should be familiar with Monotone and Dominated Convergence <b>theorems</b>, Fatou's lemma, Egorov's <b>theorem</b>, Lusin's <b>theorem</b>, Radon-Nikodym <b>theorem</b>, Fubini-Tonelli <b>theorems</b> about product measures and integration on product spaces, Cauchy's <b>theorem</b> and integral formulas, Maximum Modulus <b>theorem</b>, Rouche's <b>theorem</b>, Residue <b>theorem</b>, and Fundamental <b>Theorem</b> of Calculus for Lebesgue Integrals. </td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> Differentiation on the line: The definition and geometric significance of the derivative of a real-valued function of a real variable; the Mean Value <b>Theorem</b> and its consequences; Taylor's <b>theorem;</b> L'Hospital's rules.</td></tr> <tr><td></td><td colspan=2><font color=gray>www.ma.utexas.edu /dev/math/Graduate/Prelims/Exam_Syllabi/Analysis.html</font> (562 words)</td></tr> </table> </td> </tr> </table><br> <p style="margin-left:30px;font-size:13px;"><b>Try your search on: <a href="http://www.qwika.com/find/Weierstrass factorization theorem">Qwika</a> (all wikis)</b></p> <form action=http://www.factbites.com/search.php><table width="100%" cellspacing=0 cellpadding=0 border=0><tr><td background="/images/f1.gif"><table cellspacing=0 cellpadding=0 border=0 background="/images/b.gif"><tr><td><img src="/images/f2.gif" width=38 height=37 alt=" "/></td><td><table cellspacing=0 cellpadding=0 border=0><tr><td><a href="/"><img src="/images/f3.gif" width=95 height=37 alt="Factbites" border=0 /></a><img src="/images/b.gif" width=15 height=1 alt=" "/></td><td valign=bottom><input type=text size=30 name=kp><img src="/images/b.gif" width=2 height=1 alt=" " /><input type=submit value=" Find » " class=b2></td></tr></table></td></tr><tr><td> </td><td><span class=f> <a href="http://www.factbites.com/about_us.php">About us</a> | <a href="http://www.factbites.com/why_use_us.php">Why use us?</a> | <a href="http://www.factbites.com/reviews.php">Reviews</a> | <a href="http://www.factbites.com/press.php">Press</a> | <a href="http://www.factbites.com/contact_us.php">Contact us</a> <br />Copyright © 2005-2007 www.factbites.com Usage implies agreement with <a href=http://www.factbites.com/terms_and_conditions.php>terms</a>.</span></td></tr></table><img src="/images/b.gif" width=450 height=1 alt=" " /></td></tr></table></form> <script src="http://www.google-analytics.com/urchin.js" type="text/javascript"> </script> <script type="text/javascript"> _uacct = "UA-317061-4"; 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Topic: Weierstrass factorization theorem

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