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Topic: Weierstrass function

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	PlanetMath: Weierstrass function
		The Weierstrass function is a continuous function that is nowhere differentiable, and hence is not an analytic function.
		The function is named after Karl Weierstrass who presented it in a lecture for the Berlin Academy in 1872 [1].
		This is version 15 of Weierstrass function, born on 2002-01-03, modified 2005-02-28.
	planetmath.org /encyclopedia/WeierstrassFunction.html (222 words)

	PlanetMath: Weierstrass sigma function
		Note that the Weierstrass zeta function is basically the derivative of the logarithm of the sigma function.
		The Weierstrass eta function must not be confused with the Dedekind eta function.
		This is version 1 of Weierstrass sigma function, born on 2003-08-25.
	planetmath.org /encyclopedia/SigmaFunction.html (117 words)

	Weierstrass biography
		Weierstrass had made a decision to become a mathematician but he was still supposed to be on a course studying public finance and administration.
		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Weierstrass based his theory of functions of a complex variable in later years after 1857 are found explicitly in his unpublished works written in Münster from 1841 through 1842, while still under the influence of Gudermann.
	www-groups.dcs.st-and.ac.uk /~history/Biographies/Weierstrass.html (2615 words)

	Weierstrass function - Wikipedia, the free encyclopedia
		Historically, the Weierstrass function is important because it was the first published example to challenge the notion that every continuous function was differentiable except on a set of isolated points.
		This construction along with the proof that it is nowhere differentiable was first given by Weierstrass in a paper presented to the 'Königliche Akademie der Wissenschaften' on the 18 of July 1872.
		The Weierstrass function could perhaps be described as one of the very first 'fractals', although this term was not used until much later.
	en.wikipedia.org /wiki/Weierstrass_function (753 words)

	Weierstrass' Nowhere Differentiable Function
		The function g is continuous in R, but not differentiable at any point in R.
		Incidentally, there are many other functions of this type, and they are best treated in a course on complex analysis.
		This function, incidentally, is the one which was used to generate the logo at the title pages.
	pirate.shu.edu /projects/reals/cont/fp_weier.html (153 words)

	Karl Weierstrass - Wikipedia, the free encyclopedia
		Karl Theodor Wilhelm Weierstrass (Weierstraß) (October 31, 1815 – 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 Münster 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 Münster, 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.
	en.wikipedia.org /wiki/Karl_Weierstrass (432 words)

	Karl Weierstrass Summary
		Weierstrass was directed by his father to attend the University of Bonn, where it was expected he would acquire an education in business and law.
		Weierstrass demonstrated that the integral of an infinite series is equal to the sum of the integrals of the separate terms when the series converges uniformly within a given region.
		Weierstrass finally gained recognition with the publication of a paper on the theory of Abelian functions in August Crelle's journal in 1854.
	www.bookrags.com /Karl_Weierstrass (2407 words)

	The Number Theory Room
		as a function of the square of the nome q=exp (i pi tau).
		The imaginary part of the Modular Discriminant (Dedekind eta to the 24'th), as a function of the square of the nome q=exp (i pi tau).
		The Maclaurin sum for the series is an analytic function, that is, a harmonic function, that is, a two-dimensional function with vanishing laplacian.
	www.linas.org /art-gallery/numberetic/numberetic.html (1008 words)

	function concept
		Ptolemy dealt with functions, but it is very unlikely that he had any understanding of the concept of a function.
		This again brings the concept of a function into the construction of a curve, for Descartes is thinking in terms of the magnitude of an algebraic expression taking an infinity of values as a magnitude from which the algebraic expression is composed takes an infinity of values.
		Thus began the long controversy about the nature of functions to be allowed in the initial conditions and in the integrals of partial differential equations, which continued to appear in an ever increasing number in the theory of elasticity, hydrodynamics, aerodynamics, and differential geometry.
	www-groups.dcs.st-and.ac.uk /~history/HistTopics/Functions.html (3049 words)

	Grace Chisholm Young Quarterly Journal Abstract
		The rest of the introduction defines the derivate of a function f(x), gives some examples and properties of derivates, and outlines the history of work about differentiable and non-differentiable functions.
		Young concludes the introduction with an imaginative discourse about an analogy between the study of conic sections and motion of the heavenly bodies, and that of the study of curves with no tangents and the movements of tiny atoms as observed by an ultramicroscope.
		This appendix is accessible to any student who has used calculus to analyze the graph of a function to determine where the graph is increasing or decreasing, and to find the locations of the maxima and minima.
	www.agnesscott.edu /lriddle/women/abstracts/young_abstract1.htm (714 words)

	PlanetMath: elliptic curve
		The extremely strange numbering of the coefficients is an artifact of the process by which the above equations are derived.
		is a meromorphic function with double poles at points in
		See Also: isogeny, complex multiplication, rank of an elliptic curve, height function, L-series of an elliptic curve, Birch and Swinnerton-Dyer conjecture, j-invariant, Mordell-Weil theorem, conductor of an elliptic curve
	planetmath.org /encyclopedia/MathfrakpFunction.html (558 words)

	Math Refresher: Bolzano-Weierstrass Theorem
		It was first proved by Bernhard Bolzano but it became well known with the proof by Karl Weierstrass who did not know about Bolzano's proof.
		A function f(x) is said to be strictly increasing if x is less than y implies f(x) is less than f(y).
		Remember, the set of natural numbers is a metric space, that is, they are a set combined with a distance function for any two elements from that set.
	mathrefresher.blogspot.com /2006/09/bolzano-weierstrass-theorem.html (1258 words)

	Elliptic and Modular Functions
		More information on elliptic functions can be found for example in Chandrasekharan [Cha85], and for modular functions and their use see Koblitz [Kob84].
		Given a lattice L = [a, b] in the complex plane, this function returns the value of the elliptic j-invariant of L. This is the j-invariant of tau where tau = a/b or tau = b / a, whichever is in the upper half complex plane.
		Given a pair L = [a,b] of complex numbers generating a lattice in C, return the q-series expansion of the discriminant Delta(q) evaluated at q = e^(2pi itau) where tau = a/b or tau = b / a, whichever is in the upper half complex plane.
	www.math.lsu.edu /magma/text582.htm (1211 words)

	mar04web
		Karl Weierstrass was born in Ostenfelde Westphalia on October 31, 1815.
		Weierstrass had many important results, including the first proof that the complex numbers were the only commutative algebraic extension of the real numbers.
		That function, along with much of the fundamental work on power series and convergence (much more significantly over the complex numbers), is due to Weierstrass.
	noether.uoregon.edu /~mathpeers/newsletter/mar04/page4.html (546 words)

	Earliest Uses of Function Symbols
		The function symbol f(x) was first used by Leonhard Euler (1707-1783) in 1734 in Commentarii Academiae Scientiarum Petropolitanae (Cajori, vol.
		I] of Karl Weierstrass "Mathematische Werke" [Berlin: Mayer and Mueller], saw the light.
		The use of ζ for this function was introduced by Bernhard Riemann (1826-1866) as early as 1857 (Cajori vol.
	members.aol.com /jeff570/functions.html (1076 words)

	The Mandelbrot Set as a Modular Form
		Modular forms are a particular kind of function on the complex upper half-plane studied in analytic number theory and the theory of elliptic curves.
		This mapping is curious because it is not infrequent in the literature, and because a periodic function on the upper half-plane takes the appearance of a self-similar function on the disk.
		Such a function is provided by not working with the modulus, but subtracting the divergence directly; this is explored in the next section.
	linas.org /math/dedekind/dedekind.html (3929 words)

	Elliptic Curves and Elliptic Functions
		For every algebraic function, it is possible to construct a specific surface such that the function is "single-valued" on the surface as a domain of definition.
		Classically, such doubly periodic functions were called elliptic functions, since they occurred in the elliptic integrals which represent the arc length of an ellipse.
		Especially so, because there are many other noteworthy analytic properties of elliptic functions that were discovered in the 19th century by Weierstrass and others and which we haven't even mentioned.
	cgd.best.vwh.net /home/flt/flt03.htm (3513 words)

	[No title]
		Lattices in (and the Field of Elliptic Functions EL A lattice L in the complex plane is the set of all integral linear combinations of two complex numbers (1 and (2, where (1 and (2 are linearly independent.
		For a given lattice L, a meromorphic function on (is an elliptic function iff it is doubly periodic.
		We define the Weierstrass function ((z) with respect to a normalized lattice L in the complex plane as follows: EMBED Equation.3 for l (L. It can be shown that: ((z) converges uniformly and absolutely on compact subsets of (/L; and ((z) (EL, and its only poles are double poles on its lattice points.
	www.ms.uky.edu /~uwenagel/ALG-GEOM-04/watson.doc (1428 words)

	Elliptic Integrals and Elliptic Functions
		Ordinary trigonometric functions are singly periodic, in the sense that
		The Weierstrass zeta and sigma functions are not strictly elliptic functions since they are not periodic.
		Modular elliptic functions are defined to be invariant under certain fractional linear transformations of their arguments.
	documents.wolfram.com /v4/MainBook/3.2.11.html (620 words)

	Series acceleration - Numerical Recipes Forum
		I mentioned in an earlier post that I was going to try to implement the Weierstrass P function in C. I came upon this series for the function.
		The Jacobi functions are generally easier to deal with, and the routine in NR is fine for them.
		N.B. the lambda is the modular lambda function; e1, e2, and e3 are the values of the Weierstrass P function at the so-called half-periods.) However, as you have said, they do fail at certain values; besides, I want to write something that has as little dependencies as possible.
	www.nr.com /forum/showthread.php?t=97 (536 words)

	Example of a Function... (Site not responding. Last check: 2007-11-04)
		Weierstrass was one of the mathematicians (including Dedekind, Cantor, Kronecker, and so forth) that heralded the 2nd age of rigor, which put on firm ground many concepts that were previously nebulous in definition and application.
		Assume it is not continuous at point C, then the limit of the slope approve to C from both sides should not be equal, thus derivative does not exist and nondifferentiable.
		There are functions that have the same derivative on both sides of discontinuities (like the greatest integer function), but the derivative at the discontinuity is undefined.
	www.physicsforums.com /showthread.php?t=99930 (671 words)

	Math Forum - Ask Dr. Math
		Weierstrass's example is quite complicated, and although searching the web with the phrase "nowhere differentiable" found a number of sites which state examples or give graphs or animations, I found no site which actually proves the statement.
		He says that using the same technique, one can prove the non-differentiability of the Weierstrass function.
		Let {x} denote the "distance to the nearest integer function": on [0,1), { x 0 <= x <=.5 {x} = { { 1 - x.5 < x < 1, and {x} is periodic with period 1 for other values of x.
	mathforum.org /library/drmath/view/64615.html (409 words)

	Safe Haven \| Prediction: The Future of the USA Stock Market
		Again, the continuous line is the fit and its extrapolation using the super-exponential power-law log-periodic function derived from the first order Landau expansion of the logarithm of the price, while the dashed line is the fit and its extrapolation by including in the function a second log-periodic harmonic.
		This 'zero-phase' Weierstrass-type function adds one additional ingredient: it attempts to capture the existence of 'critical' points within the anti-bubble, corresponding to accelerating waves of imitation within the large scale unraveling of the herding anti-bubble.
		The 'zero-phase' Weierstrass-type function, which up to May 18, 2003 selected a series of downward critical crashes, is now selecting as the dominant critical points the bullish accelerations.
	www.safehaven.com /showarticle.cfm?id=934 (3044 words)

	PlanetMath 2004-01-12 Snapshot: W
		Weierstrass eta function (defined in Weierstrass sigma function)
		Weierstrass sigma function (defined in Weierstrass sigma function)
		Weierstrass zeta function (defined in Weierstrass sigma function)
	simba.cs.uct.ac.za /~hussein/PlanetMath-snapshot_2004-01-12/W.html (66 words)

	Mathematical Background (Site not responding. Last check: 2007-11-04)
		It seems that such a function should be differentiable at all but some finite set of points.
		By beginning with a wave curve and superimposing smaller and smaller waves ad infinitum, Weierstrass was able to generate a curve that was never smooth.
		Mandelbrot cites the Weierstrass function as an example of a fractal curve, since its Hausdorff dimension is greater than one.
	home.comcast.net /~davebowser/fractals/math.html (1037 words)

	New_Chaos.html
		In both these cases the functions increase as n gets larger and in both cases they go toward a value of one.
		This may be one of the earliest of a sum type of fractal function like the Weierstrass functions.
		I used a Weierstrass level dimension in a Weierstrass to get a new kind of fractal in that scaled of of harmonic functions as well.
	victorian.fortunecity.com /carmelita/435/New_Chaos.html (760 words)

	Branching and Spatial Structure
		we obtain the Weierstrass function that is continuous but not differentiable, i.e.
		As we have seen in the case of the Wiertrauss function, fractal antennae naturally result in the generation of a spatial structure in the radiated power density.
		This interplay between the spatial structure and the increase in the peak radiated power are the essential ingredients of fractal antennae and why they are so important.
	macul.ciencias.uchile.cl /alejo/fractal_antenna/node5.html (545 words)

	Recreated Results
		Audio example 1 is an example of a Weierstrass function with
		Audio example 2 is an example of a Weierstrass function with
		Again, the sound is played at the original sampling rate (22050 Hz) and then played at twice the sampling rate (44100 Hz).
	www-crca.ucsd.edu /~syadegar/MasterThesis/node36.html (368 words)

	Graphics (Site not responding. Last check: 2007-11-04)
		Picture A. This visual description of the classical Weierstrass P-function, regarded as a branched two-fold cover of the Riemann sphere by the torus.
		Picture B. Here's another picture of the Weierstrass P-function, but this time it's regarded as a doubly periodic function on the complex plane (the universal cover of the torus in the previous example).
		The picture-rectangle is a region on the complex plane with the origin at its center.
	faculty.rmc.edu /rhammack/pictures.html (507 words)

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