
	Where results make sense

Topic: Theory of analytic functions

Related Topics

functional analysis

functional programming

trigonometric function

functional group

exponential function

probability density function

continuous functions

graph of a function

Riemann zeta function

function domain

holomorphic function

In the News (Wed 30 Dec 09)

Northern Trust

Marvin Harrison

Peter Chernin

Gary Locke

Match Play

Penelope Cruz

Mickey Rourke

AR Rahman

Danny Boyle

Hugh Jackman

	Several complex variables (Site not responding. Last check: 2007-11-02)
		From this point onwards there was a foundational theory, which could be applied to analytic geometry (a name adopted, confusingly, for the geometry of zeroes of analytic functions — this is not the analytic geometry learned at school), automorphic forms of several variables, and PDEs.
		The deformation theory of complex structures and complex manifolds was described in general terms by Kunihiko Kodaira and D.C. Spencer.
		C.L. Siegel was heard to complain that the new theory of functions of several complex variables had few functions in it — meaning that the special function side of the theory was subordinated to sheaves.
	www.sciencedaily.com /encyclopedia/several_complex_variables (741 words)

	m165a (Site not responding. Last check: 2007-11-02)
		This is the first course in a two quarter introduction to the theory of analytic functions of a complex variable.
		Functions of a complex variable, interpretation as mappings, limits over the complex numbers, the point at infinity, continuity, derivatives and differentiation formulas, the Cauchy-Riemann equations, polar coordinates, analytic and harmonic functions.
		The complex exponential function and its properties, trigonometric functions, hyperbolic functions, the logarithmic function and its branches, complex exponents, inverse trigonometric and hyperbolic functions.
	math.ucr.edu /home/UndergradInfo/pages/m165a (196 words)

	[No title]
		Assuming that the size of a generation of a population depends solely on the size of the previous generation and may thus be expressed as a function of it, questions concerning the further development of the population reduce to iteration of this function.
		From this point of view, the iteration theory of entire functions and of meromorphic functions with one pole which is an omitted value is quite different from that of general meromorphic functions, which have at least two poles or only one pole which is not omitted.
		For transcendental functions, $\Si(f^{-1})$ may of course be infinite (and simple examples like $f(z)=e^z+z+1$ or $f(z)=e^z+z+2$ show that there may, in fact, be infinitely many cycles of immediate attractive basins and Leau domains), but for a rational function $f$ of degree $d$ there are at most $2d-2$ singularities of $f^{-1}$.
	www.ams.org /journals/bull/pre-1996-data/199329-2/Bergweiler (9297 words)

	[No title]
		The theory of integrable systems represents a real breakthrough in the development of mathematics at the end of the 20th century.
		The theory of integrable systems found various applications in physics, ranging from hydrodynamics and non-linear optics to astrophysics and elementary particle theory.
		According to the their taste, students may pursue the study of the more abstract aspects of the theory, or choose its more practical applications, for instance in non-linear optics, condensed matter physics or astrophysics.
	math.u-bourgogne.fr /mma2/eproint.html (1034 words)

	Lecture (Site not responding. Last check: 2007-11-02)
		The theory of analytic functions studies the consequences of the assumption that a function of a complex variable is differentiable in the sense of complex analysis.
		The basic notion is the one of a holomorphic function and the main result is the Cauchy integral theorem.
		The theory of analytic functions is employed by many branches of mathematics.
	www.wmid.amu.edu.pl /lecture.do?lang=en&id=39 (89 words)

	National Academy of Sciences - Members (Site not responding. Last check: 2007-11-02)
		This led to work on the theory of analytic functions of several complex variables, which has important applications to the theory of linear PDE with constant coefficients and conversely can be developed with methods from that theory.
		In [1] the theory of linear PDE with variable coefficients was restricted to elliptic or hyperbolic equations apart from a beginning of a general theory of PDE with simple real characteristics.
		The development of this topic since the mid 1960's has led to the new and powerful methods in the study of linear PDE often referred to as microlocal analysis, consisting roughly speaking in a detailed study of the singularities of solutions in phase space, taking both location and harmonic decomposition into account.
	www4.nationalacademies.org /nas/naspub.nsf/(urllinks)/NAS-58N2P7?opendocument (347 words)

	Weierstrass (Site not responding. Last check: 2007-11-02)
		Weierstrass attended Gudermann's lectures on elliptic functions, some of the first lectures on this topic to be given, and Gudermann strongly encouraged Weierstrass in his mathematical studies.
		The concepts on which Weierstrass based his theory of functions of a complex variable in later years after 1857 are found explicitly in his unpublished works written in Münster from 1841 through 1842, while still under the influence of Gudermann.
		The transformation of his conception of an analytic function from a differentiable function to a function expansible into a convergent power series was made during this early period of Weierstrass's mathematical activity.
	www-groups.mcs.st-and.ac.uk /~history/Mathematicians/Weierstrass.html (2473 words)

	Research at our department (Site not responding. Last check: 2007-11-02)
		The research is focused upon pluripotential theory and certain aspects of analytic function theory.
		In the theory of analytic functions we are mainly interested in ideals and spectrum for algebras of analytic functions, in particular Gleason's problem.
		For example, potential theory in spaces of homogeneous type, the connection between a function and different types of its maximal functions, characteristic properties of relevant weight-classes for weighted spaces, the dimension of fractal curves, and questions about harmonic majorization of subharmonic functions, are all subjects being investigated.
	abel.math.umu.se /Forskning/survey.html (441 words)

	Critical_Foundations (Site not responding. Last check: 2007-11-02)
		In his 1863/64 course on The general theory of analytic functions Weierstrass began to formulate his theory of the real numbers.
		Its contents were: numbers, the function concept with Weierstrass's power series approach, continuity and differentiability, analytic continuation, points of singularity, analytic functions of several variables, in particular Weierstrass's "preparation theorem", and contour integrals.
		As well as his analysis of the nature of number, his work on mathematical induction, including the definition of finite and infinite sets, and his work in number theory, particularly in algebraic number fields, is of major importance.
	www.humboldt.edu /~mef2/book/Critical_Foundations.htm (318 words)

	Analytic Number Theory course/college Analytische Getaltheorie, Leiden
		The Mathematical atlas, in particular the subcategories Number Theory (MSC2000 11), Zeta and L-functions: analytic theory (MSC2000 11M) and Multiplicative number theory (MSC2000 11N).
		Riemann viewed ζ(s) as a function in the complex variable s, and showed that it has an analytic continuation to C\{1}, with a simple pole with residue 1 at s=1.
		Dirichlet may be viewed as the founder of analytic number theory.
	www.math.leidenuniv.nl /~evertse/ant.shtml (1203 words)

	Elementary Theory of Analytic Functions of One or Several Complex Variables
		Noted mathematician offers basic treatment of theory of analytic functions of a complex variable, touching on analytic functions of several real or complex variables as well as the existence theorem for solutions of differential systems where data is analytic.
		Also included is a systematic, though elementary, exposition of theory of abstract complex manifolds of one complex dimension.
		Step-by-step coverage of fundamentals of analytic function theory—plus lucid exposition of 5 important applications: potential theory, ordinary differential equations, Fourier transforms, Laplace transforms, asymptotic expansions.
	www.doverdirect.com /0486685438.html (187 words)

	Generalized Analytic Functions in Fractional Spaces
		This book studies the foundations of the general theory of generalized analytic functions in fractional spaces.
		The employment of fractional spaces and embedding theorems support applications of the theory of generalized analytic functions.
		The results obtained are applicable to the theory of singular integral equations, boundary value problems for elliptic differential equations, functions of a complex variable, as well as the theory of plates and shells.
	www.ramex.com /ch/ch-3001.html (131 words)

	GENERALIZED ANALYTIC FUNCTIONS IN FRACTIONAL SPACES (Site not responding. Last check: 2007-11-02)
		I.N. Vekua's generalized analytic functions which possess the basic classified properties of analytic functions of a complex variable have found many real of application.
		The theory of these functions which is closely linked to many branches of analysis has found organic links to many diversions of analysis, geometry and mechanics (quasi-conformal mappings, surface theory, shell theory, gas dynamics, etc) [106].
		With the aid analytic apparatus of this theory, I.N. Vekua has studied very important problems of geometry and mechanics which arise during the study of the infinitesimal bending of surfaces of positive curvature and states of momentless stress equilib-rium of convex shells [103], [102], [100].
	www.math.kz /MNBLIEV.HTM (246 words)

	John Conway's home page
		I like to look at problems in operator theory that are susceptible to an application of complex function theory, and I have specialized in those operators where this naturally occurs.
		Fix a bounded open set $G$ in the complex plane and let $H$ be the Hilbert space of all analytic functions on $G$ that are square integrable with respect to area measure on $G$.
		Additions and Changes for "Functions of One Complex Variable." Some of these are comments on the exercises and some are references to the literature.
	www.math.utk.edu /~conway (860 words)

	Analytic Functions
		Representations of analytic functions in terms of local values by means of the Riemann mapping function.
		On the continuability of multivalued analytic functions to an analytic subset.
		The Analyticity of the Roots of a Polynomial as Functions of the Coefficients
	mathews.ecs.fullerton.edu /c2003/AnalyticFunBib/Links/AnalyticFunBib_lnk_3.html (763 words)

	Jules Henri Poincaré (Site not responding. Last check: 2007-11-02)
		Before the age of 30 he developed the concept of automorphic functions which he used to solve second order linear differential equations with algebraic coefficients.
		Poincaré can be said to have been the originator of algebraic topology and of the theory of analytic functions of several complex variables.
		In applied mathematics he studied optics, electricity, telegraphy, capillarity, elasticity, thermodynamics, potential theory, quantum theory, theory of relativity and cosmology.
	www.stetson.edu /~efriedma/periodictable/html/Pr.html (360 words)

	1997
		The main subject of the Conference was the theory of function spaces and related topics (geometry of Banach spaces, interpolation theory, approximation theory, analytic functions and convex analysis).
		We construct $p$-adic analytic families of $p$-ordinary cohomology classes in the cohomology of arithmetic subgroups of $GL(n)$ with coefficients in a family of representation spaces for $GL(n)$.
		It is proved that, in contrast to the case of $H^p$-spaces, the space $\Smir$ is not isomorphic to the Smirnov class of holomorphic functions on the unit disc.
	www.imub.ub.es /collect/1997.html (5244 words)

	Karl Theodor Wilhelm Weierstrass
		There he attended Gudermann's lectures on elliptic functions, some of the first lectures on this topic to be given, and Gudermann strongly encouraged Weierstrass in his mathematical studies.
		The topics of his lectures included the application of Fourier series and integrals to mathematical physics, an introduction to the theory of analytic functions, the theory of elliptic functions, applications to problems in geometry and mechanics, the foundations of analysis, and the integral calculus.
		The courses were "Introduction to the theory of analytic functions", "Elliptic functions","Abelian functions", and "Calculus of variations or applications of elliptic functions".
	helmet.stetson.edu /~efriedma/periodictable/html/W.html (835 words)

	Dense Analytic Subspaces In Fractal L²-Spaces (ResearchIndex)
		We show that for certain self-similar measures ¯ with support in the interval 0 x 1, the analytic functions \Phi e i2ßnx : n = 0; 1; 2; : : : \Psi contain an orthonormal basis in L 2 (¯).
		Moreover, we identify subsets P ae N0 = f0; 1; 2; : : : g such that the functions fen : n 2 Pg form an orthonormal basis for L 2 (¯).
		6 Elementary Theory of Analytic Functions of One or Several Co..
	citeseer.ist.psu.edu /117685.html (333 words)

	Complex Variables
		Using the formulas for the trig functions of the sum and difference of two angles, it is easy to prove that (cos θ + i sin θ)(cos θ' + i sin θ') = cos(θ + θ') = i sin(θ + θ').
		Laurent extended Taylor's theorem to the case where the domain of analyticity was the region between two concentric circles, a large one and a small one surrounding a point (the origin, for purposes of argument) which could be a singularity.
		A function that is analytic in the finite plane except for a certain number of poles is called meromorphic.
	www.du.edu /~jcalvert/math/complex.htm (4434 words)

	a directory of all known zeta functions
		Ruelle explains that Artin-Mazur zeta functions are Weil zeta functions in the case where we have a diffeomorphism on a compact manifold.
		Tamagawa, "On the zeta function of a division algebra", Annals of Mathematics 77 (1963) 387-405.
		Ruelle defines the Weil zeta function for an algebraic variety over a finite field in terms of the numbers of fixed points of all iterations of the Frobenius map on the extension of the algebraic variety to the algebraic closure of the finite field.
	www.maths.ex.ac.uk /~mwatkins/zeta/directoryofzetafunctions.htm (3251 words)

	math lessons - Lagrange inversion theorem
		If it can be formulated for functions of several variables, it can be extended to provide a ready formula for F(g(z)) for any analytic function F, and it can be generalized to the case f '(a) = 0, where the inverse g is a multivalued function.
		The theorem was proved by Lagrange and generalized by Bürmann, both in the late 18th century.
		There is a straightforward derivation using complex analysis and contour integration (the complex formal power series version is clearly a consequence of knowing the formula for polynomials, so the theory of analytic functions may be applied).
	www.mathdaily.com /lessons/Lagrange_inversion_theorem (303 words)

	Clyde Davenport's Hypercomplex Electromagnetic Theory Page (Site not responding. Last check: 2007-11-02)
		Similarly, analytic functions of a complex variable are automatically conformal in two dimensions.
		In fact, the theory of analytic functions of a complex variable is the same as the theory of 2-D electromagnetic fields in free space.
		Branch cuts, if any, of the analytic potential function represent equipotential surfaces (conducting objects), isolated singularities represent point or line charges, and any equipotential surface of the analytic function can represent an equipotential conducting object.
	home.usit.net /~cmdaven/electro.htm (975 words)

	COMPLEX ANALYSIS (Site not responding. Last check: 2007-11-02)
		Analytic functions: algebraic and geometric representation of complex numbers; elementary functions, including the exponential functions and its relatives (log, cos, sin, cosh, sinh,...); functions defined by power series; concept of holomorphic (analytic) function, complex derivative and the Cauchy-Riemann equations; harmonic functions.
		Calculus of residues: meromorphic functions, the Residue Theorem, calculation of definite integrals by the evaluation of residues, including improper integrals (principal values) and integrands with branch points; the argument principle (for counting zeroes and poles) and the Rouche Theorem.
		Conformal mapping: geometrical interpretation of an analytic function; explicit mappings defined by elementary functions; linear fractional (bilinear) transformations and their action on the Riemann sphere; the Riemann Mapping Theorem (statement); solution of specific problems in potential theory (boundary-value problems for harmonic functions) by the conformal mapping technique.
	www.math.umass.edu /Progs_Events/Grad_Program/axiom/node37.html (174 words)

	Perturbation Theory for Analytic Matrix Functions: The Semisimple Case
		Perturbation Theory for Analytic Matrix Functions: The Semisimple Case: SIAM Journal on Matrix Analysis and Applications Vol.
		The eigenvalue problem for non-self-adjoint, analytic matrix functions of two variables, $L(\lambda,\alpha)$, is examined with emphasis on the case when, at a fixed $\al_0$, $L(\lambda,\alpha_0)$ has a multiple, semisimple eigenvalue $\lambda_0$.
		New sufficient conditions for analytic dependence of eigenvalue functions, $\lambda(\alpha)$, on $\alpha$ in a neighborhood of $\alpha_0$ are obtained.
	epubs.siam.org /sam-bin/dbq/article/42379 (160 words)

Try your search on: Qwika (all wikis)