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Topic: Entire function

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	Entire function - Wikipedia, the free encyclopedia
		In complex analysis, an entire function is a function that is holomorphic everywhere (ie complex-differentiable at every point) on the whole complex plane.
		Neither the natural logarithm nor the square root function is entire.
		Note that an entire function may have a singularity or even an essential singularity at the complex point at infinity.
	en.wikipedia.org /wiki/Entire_function (206 words)

	Holomorphic function - Wikipedia, the free encyclopedia (Site not responding. Last check: 2007-10-21)
		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.
	www.sciencedaily.com /encyclopedia/holomorphic_function (876 words)

	PlanetMath: error function
		The name “error function” comes from the role that these functions play in the theory of the normal random variable.
		It is also worth noting that the error function is a special case of the confluent hypergeometric functions and of the Mittag-Leffler function.
		This is version 3 of error function, born on 2004-10-28, modified 2004-11-05.
	planetmath.org /encyclopedia/ErrorFunction.html (134 words)

	PlanetMath: entire function
		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)

	Entire function: Facts and details from Encyclopedia Topic (Site not responding. Last check: 2007-10-21)
		Complex analysis is the branch of mathematics investigating holomorphic functions, i.e....
		In mathematics, a function is a relation, such that each element (mathematics)element of a set (the domain) is associated with a unique element of...
		Typical examples of entire functions are the polynomials[Follow this hyperlink for a summary of this subject], EHandler: no quick summary.
	www.absoluteastronomy.com /encyclopedia/e/en/entire_function.htm (707 words)

	Content
		As with any other function, the graphs of trigonometric functions can be shifted, shrunk, stretch and reflected about one of the axis through very simple mathematical operations.
		The mathematical operation that leads to this type of graph modifications is the multiplication of the entire functional expression by a constant.
		When we multiply the actual x-variable in the function's expression by a constant number, the graph transformation we observe is either a shrinking or a stretching of the original graph but on the horizontal dimension.
	web1.shastacollege.edu /cberisso/m2/m2_50301.htm (607 words)

	[No title]
		Suppose we have an entire function with a poles at 1 and 2i.
		Suppose you have a holomorphic function in a strip, continuous and bounded in absolute value by 1 on the boundary, and bounded everywhere.
		Elliptic functions ------------------ Talk about doubly periodic functions on C. Prove that the sum of the residues of such a function in a period parallelogram is 0.
	www.princeton.edu /~missouri/Generals/generals/complex.txt (1516 words)

	Liouville's theorem (complex analysis) - Wikipedia, the free encyclopedia
		≤ M for all z in C) entire function (a holomorphic function f(z) defined on the whole complex plane C) must be constant.
		The theorem is considerably improved by Picard's little theorem, which says that every entire function whose image omits at least two complex numbers must be constant.
		In the language of Riemann surfaces, the theorem can be generalized as follows: if M is a parabolic Riemann surface (such as the complex plane C) and N is a hyperbolic one (such as an open disk), then every holomorphic function f : M → N must be constant.
	en.wikipedia.org /wiki/Liouville%27s_theorem_(complex_analysis) (268 words)

	Creating Functions in JavaScript (Site not responding. Last check: 2007-10-21)
		Generally, a function is designed to accomplish one particular task and then return a value to the main body of the script.
		Declaring a function is when you tell the browser, "I'm going to have this function, and this is what it's going to do." When the browser loads a page, it will make note of the different functions that you've declared so that it knows to return to them when they're called.
		The entire function call is replaced with the value that is passed back from the function declaration.
	www.quepublishing.com /content/images/0789728958/webresources/0789728958_web01/web01_que_0789728958_02.html (1210 words)

	Wolfram Research, Inc.
		is the sine function or the cosine function?
		But the overall function is not periodic because the pieces are not identical: as the input increases, the size of the pieces (as measured by the size of the output values) is decreasing.
		The phase shift is the horizontal shift away from the standard graph of the sine function or of the cosine function, whichever you have.
	documents.wolfram.com /teachersedition/Teacher/ATM.html (1719 words)

	Nat' Academies Press, Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics (2003)
		Functions whose domain is all complex numbers and which are decently well behaved (there is a precise mathematical definition of that!) are called entire functions.
		Those rational functions I showed in Chapter 17.ii, however, are not entire functions, since their denominators can be zero.
		The log function is not an entire function, either: it has no value at argument zero.
	www.nap.edu /books/0309085497/html/332.html (624 words)

	[No title] (Site not responding. Last check: 2007-10-21)
		The phrase "function scope" is a technical term in C and C++ which in its most narrow sense applies only to statement labels.
		The scope of 'i' at #2 is not really the entire function, since there is a "hole in the scope" at #3 where a new 'i' is declared.
		The label l1 has function scope; it extends into inner scopes and cannot be redefined.
	cpptips.hyperformix.com /cpptips/func_scope (261 words)

	Afton H. Cayford (Site not responding. Last check: 2007-10-21)
		The set of values has been enlarged, the point at which the function values are taken on has been increased to be a given set or region in the plane, the function has been replaced by a set of fu nctions, related by some condition like membership in a differential ring, and so on.
		The metric properties are enti rely different but the concept of relating growth to the characteristics of the function on certain sets or to the maximal term in a power series expansion seems to be convertible.
		Meromorphic functions have many of the properties of entire functions an d I have recently considered growth rate in the neighbourhood of a singularity, instead of at infinity, in this variable.
	www.math.ubc.ca /~cayf/Cayford.html (490 words)

	The Universe as a Mathematical Function (Site not responding. Last check: 2007-10-21)
		In the case of the universe function, this is (I hope) even clearer: the outcome is caused by the internal workings of the function; the consistency of that outcome is a consequence of the requirement that the function be well-behaved.
		The universe, viewed over its entire lifetime, is a static function with a fixed single output at each point, and thus quite deterministic; yet the function remains (in part, at least) intrinsically unpredictable for observers within the universe.
		The functional model of the universe is also compatible with genuine moral agency or volition: each such agent is represented by a nondeterministic (but "wilful" or "self-shaping", rather than "random") component of the total "universe function".
	www.nutters.org /docs/universe-function (3255 words)

	Lab 7: Scope, Lifetime and More Functions (Site not responding. Last check: 2007-10-21)
		The parameters in the function header and function prototype.
		The value that you pass to that function is called an argument or the actual parameters.
		This is a bad function design because the user is expecting the function to return a value, and may not be aware that it increments n.
	www.towson.edu /~mgillich/cosc236/labs/lab7.html (1796 words)

	Ksh Functions
		Functions are the key to writing just about ANY program that is longer than a page or so of text.
		Once you have done that, you then know the entire function is correct, for that particular set of input(s), and you can look for errors elsewhere.
		This makes the function be called in a separate ksh process, which cannot dynamically share variables back to the parent shell.
	www.bolthole.com /solaris/ksh-functions.html (940 words)

	University of Wisconsin-Department of Biostatistics&Medical Informatics
		Functional data analysis, semiparametric models, nonlinear random effects models, free knot spline regression, analysis of clustered data, statistical computing.
		My research concentrates on methods for functional data analysis, that is, the analysis of data where the ideal observation for each experimental unit is a function.
		Since it is not possible to observe the entire function, the data for each experimental unit consist of a number of noisy observations of the function at various points in its domain.
	www.biostat.wisc.edu /faculty/lindstrom.html (1032 words)

	[No title]
		Roughly, an entire function f is a product f(z) = Prod(z - c_i) * exp(g(z)) for some other entire function g; then c_i are the zeros of f.
		Then the function h(z) = Prod E(p_n, z/c_n) converges to an entire function with roots precisely at the c_n, where E(0,z)=1-z and E(k,z)=(1-z)*exp(z+z^2/2+...+z^k/k) for k>0.
		Then (if f has a root of multiplicity n at zero) f/(z^n h) is an entire function with no zeros, so g(z)=log(f(z)/(z^n h(z))) is a well-defined entire function and we have a factorization f(z) = z^n * Prod E(p_n, z/c_n) * exp(g(z)) This is the Weierstrass Factorization Theorem.
	www.math.niu.edu /~rusin/known-math/95/hilbert_irred (815 words)

	polynomial (Site not responding. Last check: 2007-10-21)
		As said, here the polynomial in x, each term is an entire power of x, and the function is also called an entire (rational) function.
		An entire function of 2nd degree is called a quadratic function; an entire function of 4th degree is called an biquadratic function.
		A broken function is a function that is the quotient of two polynomials.Together with the polynomials do they form the group of rational functions, also named the rational polynomial functions.
	www.2dcurves.com /polynomial/polynomial.html (178 words)

	Springer Online Reference Works
		are transcendental functions which do not satisfy any linear differential equation with rational coefficients (Hölder's theorem).
		The exceptional importance of the gamma-function in mathematical analysis is due to the fact that it can be used to express a large number of definite integrals, infinite products and sums of series (see, for example, Beta-function).
		In addition, it is widely used in the theory of special functions (the hypergeometric function, of which the gamma-function is a limit case, cylinder functions, etc.), in analytic number theory, etc.
	eom.springer.de /G/g043310.htm (373 words)

	HR Magazine: Carve out training? Outsourcing the entire training function is a huge change that—when handled ...
		But as the pressure builds to cut costs further and to focus solely on core competencies, companies are turning to the growing trend of business process outsourcing (BPO), or total outsourcing of an entire function.
		In HR, the BPO industry is expected to grow from $2.5 billion in 2003 to $15 billion next year, according to estimates.
		Training can be a perfect candidate for BPO because, as a function, it's not seen as a revenue-producer and much of it is administrative.
	www.findarticles.com /p/articles/mi_m3495/is_2_49/ai_n6040865 (451 words)

	Operation and Use, Volume 2 (Site not responding. Last check: 2007-10-21)
		If there are multiple functions in your program that include the string you enter in their names, the filter applies to each one.
		There may be times when you may want to see the function boxes for both the parent and child functions of the function in which you are interested, without erasing the rest of the function call tree.
		The function boxes and call arcs of the shared libraries remain within their cluster boxes, which now appear smaller to make room for the unclustered executable function boxes.
	www.gwdg.de /service/rechenanlagen/parallelrechner/sp_documentation/ppe_3.1/d3d31mst30.html (4992 words)

	Russell's Research
		Much of the theory built around the iteration of entire functions was developed by Fatou, extending results obtained for rational functions by Fatou and Julia independently.
		The major difference between iteration theory for rational and entire functions is in the proof that, for a function f, the Julia set is the closure of the repelling periodic points.
		There are two basic examples in the literature of entire functions f for which J(f) (or some subset thereof) is given as the attractor of an iterated function system.
	www.cfm.brown.edu /people/jackson/research.html (1560 words)

	[No title]
		The function exp(-t)/(1+zt) is indeed analytic for all complex z except real and negative values.
		The function is not even continuous near any point of the negative real axis.
		This is closely akin to the exponential integral function Ei(z) = -integral_{-z}^infinity exp(-t)dt / t (1) where we take the principal value of this integral to deal with the singularity at t = 0.
	www.math.niu.edu /~rusin/known-math/00_incoming/divseq (882 words)

	STEPP - Function (Site not responding. Last check: 2007-10-21)
		During this unit, students will focus on the sense of vision and the human eye by studying some of the parts of the eye and the functions of these parts.
		An improved knowledge of sensory function may give children a better awareness of the world around them.
		Moreover, they will be given the opportunity to recognize how important the interaction between the senses and the brain is for all of their daily activities.
	www.shriverstepp.org /function (490 words)

	Bajpai: On the mean values of an entire function represented by Dirichlet series
		Bajpai, S. On the mean values of an entire function represented by Dirichlet series.
		For entire functions defined by absolutely convergent Dirichlet series, a theorem on their mean values is established which include the results of Kamthan, Juneja and Awasthi.
		JUNEJA, On the mean values of an entire function and its derivatives represented by Dirichlet series, Ann.
	www.numdam.org /numdam-bin/item?id=AIF_1971__21_2_31_0 (176 words)

	Analytic functions.
		A polynomial function being the sum of a finite number of such functions, again by Prop.
		2.4, together with the previous alinea, as a rational function is the quotient of two polynomial functions.
		Example 3.5.5 A polynomial function is an entire function, as a consequence of Prop.
	ndp.jct.ac.il /tutorials/complex/node21.html (177 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 the congruence subgroup of the modular group, its fundamental domain, and the construction of the modular function that is invariant under the action of the congruence subgroup.
		We discussed an integral representation for the ζ function, the functional equation for the ζ function, the trivial zeroes of the ζ function, and the values of the ζ function at the positive even integers.
	www.math.tamu.edu /~harold.boas/courses/618-2004a/daily.html (1723 words)

	Everest automates your entire payroll function - Everest Software, Inc.
		The fully-integrated payroll function in Everest-US Payroll-allows complete automation of payroll including supplementary payroll, employee deductions, company contributions, statutory forms, and much more, from within one cohesive system, saving countless man-hours and simplifying the processing of your company's payroll.
		The Everest payroll functionality is seamlessly built-in with the high-coverage 'all-in-one' capabilities of Everest, so there is no requirement for additional staff or hardware to manage this specialized area of your business.
		When your payroll is managed internally via Everest's US Payroll function, all employee data is always current and available to the system, round-the-clock, so you never have to compile data to ship to a vendor or worry about validity of information.
	www.everestsoftwareinc.com /Everest/payroll.asp (673 words)

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