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Topic: Holomorphic function

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	Table of contents for Library of Congress control number 2001046415
		Holomorphic Functions, the Cauchy-Riemann Equations, and Harmonic Functions 14 §1.5.
		The Elliptic Modular Function and Picard's Theorem 314 §10.6.
		Schlicht Functions and the Bieberbach Conjecture 384 §13.2.
	www.loc.gov /catdir/toc/fy031/2001046415.html (548 words)

	Meromorphic function - Wikipedia, the free encyclopedia
		In complex analysis, a meromorphic function on an open subset D of the complex plane is a function that is holomorphic on all D except a set of isolated points, which are poles for the function.
		Every meromorphic function on D can be expressed as the ratio between two holomorphic functions (with the denominator not constant 0) defined on D: the poles then occur at the zeroes of the denominator.
		In the language of Riemann surfaces, a meromorphic function is the same as a holomorphic function that maps to the Riemann sphere and which is not constant ∞.
	en.wikipedia.org /wiki/Meromorphic_function (384 words)

	Holomorphic function - Wikipedia, the free encyclopedia
		Holomorphic functions are the central object of study of complex analysis; they are functions defined on an open subset of the complex number plane C with values in C that are complex-differentiable at every point.
		A function that is holomorphic on the whole complex plane is called an entire function.
		A complex analytic function of several complex variables is defined to be analytic and holomorphic at a point if it is locally expandable (within a polydisk, a cartesian product of disks, centered at that point) as a convergent power series in the variables.
	en.wikipedia.org /wiki/Holomorphic_function (727 words)

	Complex analysis - Wikipedia, the free encyclopedia (Site not responding. Last check: 2007-10-07)
		The integral around a closed path of a function which is holomorphic everywhere inside the area bounded by the closed path is always zero; this is the Cauchy integral theorem.
		The remarkable behavior of holomorphic functions near essential singularities is described by the Weierstrass-Casorati theorem.
		Sometimes, as in the case of the natural logarithm, it is impossible to analytically continue a holomorphic function to a non-simply connected domain in the complex plane but it is possible to extend it to a holomorphic function on a closely related surface known as a Riemann surface.
	www.eastcleveland.us /project/wikipedia/index.php/Complex_analysis (697 words)

	Smooth function article - Smooth function mathematics derivatives continuous function exponential function - ... (Site not responding. Last check: 2007-10-07)
		For example, the exponential function is evidently smooth because the derivative of the exponential function is the exponential function itself.
		Smooth functions with given closed support are used in the construction of smooth partitions of unity (see topology glossary for partition of unity); these are essential in the study of smooth manifolds, for example to show that Riemannian metrics can be defined globally starting from their local existence.
		From what has just been said, partitions of unity don't apply to holomorphic functions; their different behaviour relative to existence and analytic continuation is one of the roots of sheaf theory.
	www.what-means.com /encyclopedia/Smooth (548 words)

	[No title] (Site not responding. Last check: 2007-10-07)
		Holomorphic functions are the central object of study of complex analysis; they are functions defined on an open subset of the complex number plane C with values in C which are complex differentiable at every point.
		A function that is holomorphic on the whole complex plane is called entire.
		The inverse trigonometric functions likewise have seams and are holomorphic everywhere except the seams.
	www.informationgenius.com /encyclopedia/h/ho/holomorphic_function.html (509 words)

	PlanetMath: entire function (Site not responding. Last check: 2007-10-07)
		For example, a polynomial is holomorphic everywhere, as is the exponential function.
		is not holomorphic at zero, so it is not entire; it is meromorphic.
		This is version 6 of entire function, born on 2001-12-28, modified 2004-11-28.
	planetmath.org /encyclopedia/Entire.html (80 words)

	[No title] (Site not responding. Last check: 2007-10-07)
		Sept 27: line integrals in the complex plane, existence of a holomorphic primitive of a holomorphic function on an open rectangle, Morera's theorem, Cauchy's integral formula for a disk, integral representation of the derivatives of a holomorphic function.
		Oct 18: punctured neighborhood of infinity and related notions, a function meromorphic on the complex plane and infinity is a rational function.
		Nov 22: biholomorphic maps, $n$-th root and logarithm as a holomorphic function, geometry of a holomorphic function, Schwarz lemma.
	www.math.uu.nl /people/looijeng/cfies.html (481 words)

	sciforums.com - Analytic Functions
		A holomorphic function; is that analytic or not?
		From what I understand, holomorphic is just another term for analytic (as is regular), the only difference being that people whose field of study is not Mathematics use tend to use analytic, while Mathematicians are more partial to holomorphic.
		Well, you could either redefine your set to make the function analytic, or simply say that the function has singular points, (Note: for a point to be singular the function must be analytic in the neighborhood of the singularity) e.g., f(z) = 1/z is analytic everywhere except at the singularity z = 0 + 0i.
	www.sciforums.com /showthread.php?t=27572 (2147 words)

	PlanetMath: harmonic function (Site not responding. Last check: 2007-10-07)
		Indeed, a holomorphic function is harmonic, and a real harmonic function
		This is version 6 of harmonic function, born on 2002-06-04, modified 2005-03-25.
		A couple of the entries attached to Harmonic Functions should be moved over to the separate entry on harmonic functions on graphs.
	planetmath.org /encyclopedia/HarmonicFunction.html (161 words)

	Exponential function - Wikipedia, the free encyclopedia (Site not responding. Last check: 2007-10-07)
		The exponential function is one of the most important functions in mathematics.
		The importance of exponential functions in mathematics and the sciences stems mainly from properties of their derivatives.
		The exponential function maps any line in the complex plane to a logarithmic spiral in the complex plane with the center at the origin.
	xahlee.org /_p/wiki/Exponential_function.html (900 words)

	Fundamental theorem of calculus - Wikipedia, the free encyclopedia (Site not responding. Last check: 2007-10-07)
		This means that if a continuous function is first integrated and then differentiated, the original function is retrieved.
		An important consequence of this, sometimes called the second fundamental theorem of calculus, allows one to compute integrals by using an antiderivative of the function to be integrated.
		The derivative of this function is equal to the infinitesimal change in x per infinitesimal change in time (of course, the derivative itself is dependent on time).
	xahlee.org /_p/wiki/Fundamental_theorem_of_calculus.html (519 words)

	[No title] (Site not responding. Last check: 2007-10-07)
		A function f is holomorphic on an open set of the complex plane when f maps to the complex plane also, and for each z in that open set, the limit as w
		A Riemann surface is a topological space with a collection of "charts": homeomorphisms phi_i of open subsets of X to open subsets of the complex plane, such that the charts cover X, and wherever two charts overlap, the functions phi_i composed with {phi_j}^{-1} is a holomorphic function.
		Rational functions are holomorphic maps from the sphere to itself; the points where the denominator goes to zero but the numerator doesn't get mapped to infinity, and infinity gets mapped to the limit of the function as you approach infinity.
	www.math.niu.edu /~rusin/papers/known-math/95/modularity (2492 words)

	sciforums.com - Cauchy's Integral Theorem
		Now if we have a function which is entire in its domain (is analytic everywhere) say exp(z) or cos(z) or even sec(z), then the contour integral is zero.
		a meromorphic function (a holomorphic function everywhere but for a point where it has a simple pole) can be considered a holomorphic funcition on the region with the point of its pole removed.
		So for the contour integral of f(z) which is holomorphic (analytic) at each point in and on the contour C, the value of the integral is independent of path along the contour between the two endpoints.
	www.sciforums.com /showthread.php?t=33287 (1584 words)

	[No title]
		The second shows that not every domain in several variables is the natural domain of existence of a holomorphic function.
		Besides using ideas from function theory, several complex variables routinely employs techniques from differential geometry, partial differential equations, harmonic analysis, real analysis, operator theory, geometric measure theory, commutative algebra, Lie group theory, and even mathematical logic.
		The purpose of this holomorphic mappings conference was to bring together 35 recognized experts to share recent progress in subject areas that find their roots in Poincar\'{e}'s theorem described above.
	www.math.lsa.umich.edu /~fornaess/sk (1170 words)

	Pick functions related to the Gamma function (ResearchIndex) (Site not responding. Last check: 2007-10-07)
		Abstract: had to establish that the only zeros of the holomorphic branch of log # in A are those at z = 1 and z = 2.
		a holomorphic function in the upper half-plane with a non-negative imaginary part, so the result in [2] implies that f is a Pick function.
		We now show directly that f is a Pick function, which has two advantages: it easily implies that f # is completely monotone and also that log #(z) is zero-free in C \ R.
	citeseer.ist.psu.edu /423280.html (244 words)

	Holomorphic and Meromorphic Functions (Site not responding. Last check: 2007-10-07)
		A holomorphic function on the plane (or other complex manifold) is analytic.
		A meromorphic function is the quotient of two holomorphic functions f/g.
		Technically, every holomorphic function is meromorphic, since you can always set g to 1.
	www.mathreference.com /cx-pow,holom.html (149 words)

	Zeta function - Metaweb (Site not responding. Last check: 2007-10-07)
		(In that expression, Re means the real part of a number.) Bernhard Riemann realized that the zeta function can be extended by analytic continuation in a unique way to a holomorphic function ζ(s) defined for all complex numbers s with s ≠ 1.
		It is this function that is the object of the Riemann hypothesis.
		The zeros of ζ(s) are important because certain path integrals involving the function ln(1/ζ(s)) can be used to approximate the prime counting function π(x) (see prime number theorem).
	www.metaweb.com /wiki/wiki.phtml?title=Zeta_function (504 words)

	No Title
		The function w(x,y) = u(x,y)v(x,y) is harmonic in D.
		Determine, by integration, the conjugate harmonic function and the corresponding analytic function (up to a constant).
		are simultanously holomorphic (this is a special case of the Reflection Principle).
	www.math.umass.edu /~markman/m621_spring00_html/hw2/hw2.html (284 words)

	AoPS Math Forum :: View topic - Holomorphic function sum (Site not responding. Last check: 2007-10-07)
		Prove that there exists an entire function h such that p, q, r are constant multiples of h.
		Then f and g are holomorphic functions non-vanishing, and f+g=1.
		The statement that if a holomorphic function skips 2 points from it's image, then f is constant I think is due to Picard (NOT Patrick Stewart).
	www.artofproblemsolving.com /Forum/topic-1277.html (296 words)

	Math 428: Topics in Complex Analysis, Fall 98
		Weierstrass's work on the analytic continuation of holomorphic functions led Riemann to develop the notion of the Riemann surface as a natural domain for an algebraic function.
		It is well-known that Riemann surface theory is not only a point of departure but even at the historical roots of a large part of mathematics, in particular, topology, differential geometry, algebraic geometry, algebraic number theory, and partial differential equations.
		Cohn: sec.8.3#5,#6, sec.2.2#1,#2, and Prove: Any holomorphic function on a compact Riemann surface is constant.
	math.rice.edu /~hardt/428 (1647 words)

	Introduction (Site not responding. Last check: 2007-10-07)
		The objective of the first sections is the relationship between the factorisation of scalar functions and the theory of singular integral equations as far as they are relevant to boundary value problems for holomorphic functions.
		Since the kernel has uncountably many singularities, the corresponding operator fails to be compact, but its boundedness can be secured on certain classes of functions and the boundary values are described by the formulas of Sokhotski/Plemelj.
		After these preliminaries, these problems will be generalized to matrix valued functions and the matrix Riemann as well as the Riemann-Hilbert problems will be dealt with followed by applications in the theory of differential equations.
	www.gang.umass.edu /~kilian/mathesis/node2.html (208 words)

	Introduction
		The process invests old theorems with new meanings, and bestows upon functional analysis an intriguing class of concrete linear operators.
		The setting is the simplest one consistent with serious ``function-theoretic operator theory:'' the unit disc U of the complex plane, and the Hilbert space H^2 of functions holomorphic on U with square summable power series coefficients.
		Here our story comes full circle: the crucial inequality on the Counting Function is due to Littlewood, and when reinterpreted it becomes a striking generalization of the Schwarz Lemma.
	www.mth.msu.edu /~shapiro/Pubvit/Book/intro.html (963 words)

	A BIRKHOFF THEOREM FOR RIEMANN SURFACES (Site not responding. Last check: 2007-10-07)
		is dense in the space of entire functions.
		of holomorphic self-mappings of R such that there exists a holomorphic function f on R such that
		is dense in the space of the holomorphic functions on R.
	math.la.asu.edu /~rmmc/rmj/Vol28-2/MON (113 words)

	No Title
		Let f be a holomorphic function defined and having a continuous derivative f' in an open set U containing a rectangle R.
		a sequence of holomorphic functions which converges, uniformly on compact subsets of U, to a function f.
		Ahlfors, page 130 Problem 2: Show that a function which is analytic in the whole plane and has a non-essential singularity at
	www.math.umass.edu /~markman/m621_spring00_html/hw4/hw4.html (278 words)

	No Title (Site not responding. Last check: 2007-10-07)
		Moreover, F is unique as a function defined on the range of x(t).
		In fact it will be shown that the other derivatives of x in the arguments of F in equation (2) are not required to determine F.
		which is clearly an analytic function defined on the range of x, namely (c,d).
	math.uc.edu /ode/2000sols/2000sols.html (210 words)

	The Riemann-problem (Site not responding. Last check: 2007-10-07)
		If a function f is to be composed with any branch of the logarithm, its argument mustn't vary by more than
		and in the above proof, the function g was factorized, split into three factors, of which the right-hand side one was holomorphic in the interior, the left one holomorphic in the exterior of the contour, both invertible in their domains of holomorphicity and the middle factor was a monomial.
		the individual factors should be invertible in their domains of holomorphicity.
	www.gang.umass.edu /~kilian/mathesis/node5.html (426 words)

	Banach Algebras where the Singular Elements are Removable Singularities (ResearchIndex) (Site not responding. Last check: 2007-10-07)
		C has a holomorphic extension to A 0.
		In this definition we consider the identity component of A inv rather than A inv since if A inv is not connected the holomorphic function on A inv which is 1 on one of the components and 0 on the others has no holomorphic extension to A 0.
		It follows from Theorem 3 given in the next section (or see [12, Theorem 3]) that our definition is no more stringent when the range of f is allowed to be any Banach space.
	citeseer.ist.psu.edu /283218.html (368 words)

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