Infobox scientist name KarlWeierstrass image Karl Weierstrass.jpg 300px caption Karl Theodor Wilhelm Weierstrass Weierstra birth date birth date 1815 10 31 mf y birth place Ennigerloh Ostenfelde , Province ... for Weierstrass function prizes religion footnotes Karl Theodor Wilhelm Weierstrass lang de Weierstra .... Other analytical theorems See also List of topics named after KarlWeierstrass . Stone Weierstrass ... Vorl. ueber Variationsrechnung Math. Werke. Bd. 6. Berlin, 1927 Students of KarlWeierstrass ... also List of topics named after KarlWeierstrass References Reflist External links commons KarlWeierstrass ... digital Berlin Brandenburgische Akademie der Wissenschaften . gutenberg author id KarlWeierstrass name KarlWeierstrass Copley Medallists 1851 1900 Authority control PND 11876618X LCCN n 84 806249 VIAF 36999173 SELIBR 238824 Persondata Metadata see Wikipedia Persondata NAME Weierstrass, Karl ALTERNATIVE ... ca KarlWeierstrass cs Karl Theodor Wilhelm Weierstrass da KarlWeierstrass de Karl Weierstra es KarlWeierstrass eo KarlWeierstrass eu KarlWeierstrass fa fr KarlWeierstrass ko hy id KarlWeierstrass is KarlWeierstrass it KarlWeierstrass he ht KarlWeierstrass la Carolus Weierstra lv K rlis Veier tr ss hu KarlWeierstrass nl KarlWeierstrass ja no KarlWeierstrass nn KarlWeierstrass pms KarlWeierstrass pl KarlWeierstrass pt KarlWeierstrass ro KarlWeierstrass ru , simple Karl Weierstra sk KarlWeierstrass sl KarlWeierstrass sr fi KarlWeierstrass sv KarlWeierstrass ta th tr KarlWeierstrass uk vi KarlWeierstrass zh ... is often cited as the father of modern mathematical analysis analysis . Biography Weierstrass was born in Ostenfelde, part of Ennigerloh , Province of Westphalia . Weierstrass was the son of Wilhelm Weierstrass, a government official, and Theodora Vonderforst. His interest in mathematics began while ... more details
This is a list of things named after the German mathematician KarlWeierstrass 1815 1897 . Mathematical concepts, theorems, and the like Named after Weierstrass and other persons Bolzano Weierstrass theorem Casorati Weierstrass theorem Durand Kerner method , also called the method of Weierstrass Enneper Weierstrass parameterization Lindemann Weierstrass theorem Sokhatsky Weierstrass theorem Stone Weierstrass theorem Weierstrass Casorati theorem see Casorati Weierstrass theorem Weierstrass Enneper parameterization Weierstrass Erdmann condition Named after Weierstrass alone Weierstrass approximation theorem Weierstrass s elliptic functions Weierstrass factorization theorem Weierstrass function Weierstrass M test Weierstrass point Weierstrass preparation theorem Weierstrass product inequality Weierstrass ring Weierstrass substitution Weierstrass theorem any of several theorems Weierstrass transform Typography Weierstrass p , a form of the letter p used to denote the Weierstrass elliptic function Celestial bodies or features of them Weierstrass crater 14100 Weierstrass Category Lists of things named after mathematicians Weierstrass, Karl ... more details
Several theorems are named after KarlWeierstrass . These include The Weierstrass approximation theorem , of which one well known generalization is the Stone Weierstrass theorem The Bolzano Weierstrass theorem , which ensures compactness of closed and bounded sets in R sup n sup The Weierstrass extreme value theorem , which states that a continuous function on a closed and bounded set obtains its extreme values The Weierstrass Casorati theorem describes the behavior of holomorphic functions near essential singularities The Weierstrass preparation theorem describes the behavior of analytic functions near a specified point The Lindemann Weierstrass theorem concerning the transcendental numbers The Weierstrass factorization theorem asserts that entire functions can be represented by a product involving their zeroes The Sokhatsky Weierstrass theorem which helps evaluate certain Cauchy type integrals See also List of topics named after KarlWeierstrass mathdab fr Th or me de Weierstrass ko sv Weierstrass sats ... more details
In mathematics, a Weierstrass ring , named by harvtxt Nagata 1962 loc section 45 after KarlWeierstrass , is a commutative ring commutative local ring that is Henselian , pseudo geometric , and such that any quotient ring by a prime ideal is a finite extension of a regular local ring . Examples The Weierstrass preparation theorem can be used to show that the ring of convergent power series over the complex numbers in a finite number of variables is a Wierestrass ring. The same is true if the complex numbers are replaced by a perfect field with a valuation. Every ring that is a finite module over a Weierstrass ring is also a Weierstrass ring. References springer id W w097500 first V. I. last Danilov M. Nagata, Local rings , Interscience 1962 algebra stub Category commutative algebra ... more details
In mathematics , the Weierstrass functions are special function s of a complex variable that are auxiliary to the Weierstrass elliptic function . They are named for KarlWeierstrass . Weierstrass sigma function The Weierstrass sigma function associated to a two dimensional fundamental pair of periods lattice math Lambda subset Complex math is defined to be the product math sigma z Lambda z prod w in Lambda left 1 frac z w right e z w frac 1 2 z w 2 math where math Lambda math denotes math Lambda 0 math . Weierstrass zeta function The Weierstrass zeta function is defined by the sum math zeta z Lambda frac sigma z Lambda sigma z Lambda frac 1 z sum w in Lambda left frac 1 z w frac 1 w frac z w 2 right . math Note that the Weierstrass zeta function is basically the logarithmic derivative of the sigma function. The zeta function can be rewritten as math zeta z Lambda frac 1 z sum k 1 infty mathcal G 2k 2 Lambda z 2k 1 math where math mathcal G 2k 2 math is the Eisenstein series of weight math 2k 2 math . Also note that the derivative of the zeta function is math wp z math , where math wp z math is the Weierstrass elliptic function The Weierstrass zeta function should not be confused with the Riemann zeta function in number theory. Weierstrass eta function The Weierstrass eta function is defined to be math eta w Lambda zeta z w Lambda zeta z Lambda , mbox for any z in Complex math It can ... on w . The Weierstrass eta function should not be confused with the Dedekind eta function Dedekind eta function . Weierstrass p function The Weierstrass p function is defined to be math wp z Lambda zeta z Lambda , mbox for any z in Complex math The Weierstrass p function is an even elliptic function of order N 2 with a double pole at each lattice and no others. See also Weierstrass function PlanetMath attribution id 4650 title Weierstrass sigma function Category Elliptic functions Category Analytic functions es Funciones de Weierstrass eo Sigmo funkcio de Weierstrass ku Fonksiyon n Weierstrass ... more details
Notability Astro date February 2012 Infobox planet minorplanet yes width 25em bgcolour FFFFC0 apsis name Weierstrass symbol image caption discovery yes discovery ref discoverer P. G. Comba discovery site Prescott Observatory Prescott discovered September 8, 1997 designations yes mp name 14100 alt names 1997 RQ5 named after Karl Weierstrass mp category orbit ref epoch May 14, 2008 aphelion 3.4264329 perihelion 2.8235023 semimajor eccentricity 0.0964699 period 2017.7469815 avg speed inclination 0.57836 asc node 190.38695 mean anomaly 117.51123 arg peri 37.11961 satellites physical characteristics yes dimensions mass density surface grav escape velocity sidereal day axial tilt pole ecliptic lat pole ecliptic lon albedo temperatures temp name1 mean temp 1 max temp 1 temp name2 max temp 2 spectral type abs magnitude 13.3 14100 Weierstrass 1997 RQ5 is a Asteroid belt main belt asteroid discovered on September 8, 1997 by P. G. Comba at Prescott Observatory Prescott . References Reflist External links http ssd.jpl.nasa.gov sbdb.cgi?sstr 14100 Weierstrass JPL Small Body Database Browser on 14100 Weierstrass MinorPlanets Navigator 14099 1997 RQ3 14101 1997 SD1 MinorPlanets Footer DEFAULTSORT Weierstrass Category Main Belt asteroids Category Asteroids named for people Category Discoveries by Paul G. Comba Category Astronomical objects discovered in 1997 Beltasteroid stub fa it 14100 Weierstrass pl 14100 Weierstrass pt 14100 Weierstrass uk 14100 vi 14100 Weierstrass yo 14100 Weierstrass ... more details
lunar crater data image Image 1025 med crop.png 200px Jenkins crater and surroundings caption The craters Weierstrass lower center , Van Vleck lower right , Nobili upper left and Jenkins upper right from Lunar Orbiter 1 . NASA L&PI image . latitude 1.3 N or S S longitude 77.2 E or W E diameter 33 km depth 2.4 km colong 283 eponym Karl Weierstrass Weierstrass is a small Moon lunar Impact crater crater that is attached to the northern rim of the walled plain Gilbert lunar crater Gilbert , in the eastern part of the Moon . It also lies very near the crater Van Vleck crater Van Vleck , a similar formation just to the southeast that is almost attached to the outer rim. Due to its location, the crater appears foreshortened as seen from the Earth . The crater has an oval shaped outer rim that is longer along an east west axis. There are some slumped shelves along the inner walls to the north and south. The interior floor is nearly featureless, with only a few tiny impacts. Neither the rim nor the interior are marked by impact craters of significance. Prior to being named by the International Astronomical Union IAU , this crater was designated Gilbert N. References Lunar crater references External links http www.lpi.usra.edu resources mapcatalog LTO lto81a2 1 LTO 81A2 Gilbert &mdash L&PI topographic map Category Impact craters on the Moon Moon crater stub de Weierstrass Mondkrater fa it Weierstrass cratere ... more details
, there is a brief mention that on July 18th, Hr. Weierstrass las ber stetige Funktionen ohne bestimmte Differentialquotienten Mr. Weierstrass read a paper about continuous functions without definite i.e., well defined derivatives to members of the Academy . However, Weierstrass s paper was not published in the Monatsberichte . ref ref KarlWeierstrass, http books.google.com books?id 1FhtAAAAMAAJ ... Akademie der Wissenschaften, Mathematische Werke von KarlWeierstrass Berlin, Germany Mayer & Mueller, 1895 , vol. 2, pages 71 74. ref ref See also KarlWeierstrass, Abhandlungen aus der Functionenlehre ... June 1964 . KarlWeierstrass, http books.google.com books?id 1FhtAAAAMAAJ&pg PA71 v onepage&q ..., Mathematische Werke von KarlWeierstrass Berlin, Germany Mayer & Mueller, 1895 , vol. 2, pages ...dablink Weierstrass function may also refer to the Weierstrass elliptic function math wp math or the Weierstrass sigma function Weierstrass sigma, zeta, or eta functions . Image WeierstrassFunction.svg 300px right thumb Plot of Weierstrass Function over the interval &minus 2, 2 . Like fractal s, the function ... , the Weierstrass function is a pathological mathematics pathological example of a real valued ... function continuous everywhere but differentiable nowhere. It is named after its discoverer KarlWeierstrass . Historically, the Weierstrass function is important because it was the first published ... before Weierstrass, but their findings were not published in their lifetimes. Around 1831, Bernard ... differentiable function that closely resembles Weierstrass s function. Cell rier s discovery was, however .... 14, pages 142 160. ref Construction In Weierstrass original paper, the function was defined by math ..., was first given by Weierstrass in a paper presented to the Prussian Academy of Sciences ... of the terms is uniform by the Weierstrass M test with math M n a n math . Since each partial ... should be small in some sense. According to Weierstrass in his paper, earlier mathematicians ... more details
, named after KarlWeierstrass , is the function F defined by math F x frac 1 sqrt 2 pi int infty infty ...In mathematics , the Weierstrass transform ref Ahmed I. Zayed, Handbook of Function and Generalized Function ... thumb right 320px The graph of a function f x gray and its generalized Weierstrass transforms for t 0.2 red , t 1 green and t 3 blue . The standard Weierstrass .... The Weierstrass transform F can be viewed as a smoothed version of f the value F x is obtained by averaging ... are not changed by the Weierstrass transform. The Weierstrass transform is intimately related ... define the generalized Weierstrass transform of f . The generalized Weierstrass transform provides ... Weierstrass used this transform in his original proof of the Weierstrass approximation theorem . It is also known as the Gauss transform or Gauss Weierstrass transform after Carl Friedrich Gauss and as the Hille ... above, every constant function is its own Weierstrass transform. The Weierstrass transform of any polynomial ... physicist s Hermite polynomial of degree n , then the Weierstrass transform of H sub n sub x 2 is simply ... polynomials is closely related to the Gaussian kernel used in the definition of the Weierstrass transform. The Weierstrass transform of the function e sup ax sup where a is an arbitrary constant is e sup a sup 2 sup sup e sup ax sup . The function e sup ax sup is thus an eigenvector for the Weierstrass ... a bi where i is the imaginary unit , and using Euler s identity , we see that the Weierstrass transform of the function cos bx is e sup &minus b sup 2 sup sup cos bx and the Weierstrass transform of the function sin bx is e sup &minus b sup 2 sup sup sin bx . The Weierstrass transform ... 1 2 and undefined if a 1 2. In particular, by choosing a negative, we see that the Weierstrass ... The Weierstrass transform assigns to each function f a new function F this assignment is linear ... Lp space L sup 1 sup R , then so is its Weierstrass transform F , and if furthermore f x ... more details
div class thumb tright style width 131px Image Weierstrass p.svg 100px Weierstrass p div class thumbcaption Weierstrass p div div In mathematics , the Weierstrass p , or math wp math at a larger font size, or in bold form, or in italicized form, a stylized letter p , also called pe , is used for the Weierstrass s elliptic functions Weierstrass s elliptic function . It is occasionally used for the power set , ref Citation last de Haan first Lex author link Lex de Haan last2 Koppelaars first2 Toon title Applied Mathematics for Database Professionals publisher Apress year 2007 chapter Chapter 2 &mdash Set Theory Introduction page 35 isbn 978 1 59059 745 3 ref although for that purpose a cursive capital, rather than lower case, p is more widespread. It is named after the German mathematician KarlWeierstrass . Its Unicode code point is unichar 2118 script capital p html . The naming is incorrect, because it is always a lowercase letter, but this error is not corrected in later versions in order to keep the Unicode standard stable. ref cite web url http unicode.org notes tn27 title Unicode Technical Note 27 Known Anomalies in Unicode Character Names publisher Unicode Consortium date 2006 05 08 accessdate 2011 06 11 ref The TeX code for this character is code wp code . Starting with Unicode 3.0.1, a separate, capital symbol is available for power set, namely unichar 1D4AB MATHEMATICAL SCRIPT CAPITAL P html , which is available as code & Pscr code in HTML5 . ref cite web url http www.w3.org TR html5 named character references.html title 8.5 Named character references HTML 5 publisher W3.org date accessdate 2010 03 15 ref ref cite web url http www.fileformat.info info unicode char 1d4ab index.htm title Unicode Character MATHEMATICAL SCRIPT CAPITAL P U 1D4AB publisher Fileformat.info date accessdate 2010 03 15 ref As of 2009, only a few Category Mathematical OpenType typefaces ... Category Typographical symbols de Weierstra p es P de Weierstrass ... more details
File WeierstrassSubstitution.svg thumb right 300px The Weierstrass substitution, here illustrated as stereographic projection of the circle. In integral calculus , the Weierstrass substitution , named after KarlWeierstrass , is used for finding antiderivative s, and hence definite integrals, of rational function s of trigonometric function s. Without loss of generality No generality is lost by taking these to be rational function s of the sine and cosine. The substitution involves tangent half angle formula tangents of half angles . Michael Spivak wrote that The world s sneakiest substitution is undoubtedly this technique. ref Michael Spivak, Calculus , Cambridge University Press , 2006, pages 382 383. ref The substitution One starts with the problem of finding an antiderivative of a rational function of the sine and cosine, and replaces sin x , cos x , and the Differential mathematics differential d x with rational functions of a variable t and the product of a rational function of t with the differential d t , as follows ref James Stewart, Calculus Early Transcendentals , Brooks Cole, 1991, page 439 ref math begin align sin x & frac 2t 1 t 2 8 pt cos x & frac 1 t 2 1 t 2 8 pt mathrm d x & frac 2 , mathrm d t 1 t 2 . end align math Derivation Let math t tan frac x 2 . math By the double angle formula for the sine function, math begin align sin x& 2 sin frac x 2 cos frac x 2 & 2t cos 2 frac x 2 & frac 2t sec 2 frac x 2 & frac 2t 1 t 2 . end align math ... 400px thumb right The Weierstrass substitution parametrizes the unit circle centered ... 350px thumb right How the Weierstrass substitution is related to the stereographic projection ... in the second quadrant from 0, 1 to &minus 1, 0 . File Weierstrass substitution.png ... Weierstrass substitution formulas at PlanetMath Category Integral calculus de Generalsubstitution nl Weierstrass substitutie ru ... more details
In mathematics , a Weierstrass point P on a nonsingular algebraic curve C defined over the complex numbers is a point such that there are extra functions on C , with their pole mathematics poles restricted to P only, than would be predicted by looking at the Riemann Roch theorem . That is, looking at the vector spaces L 0 , L P , L 2 P , L 3 P , ..., where L k P is the space of meromorphic functions on C whose order at P is at least k and with no other poles. We know three things the dimension is at least 1, because of the constant functions on C , it is non decreasing, and from the Riemann Roch theorem the dimension eventually increments by exactly 1 as we move to the right. In fact if g is the genus mathematics genus of C , the dimension from the k th term is known to be l kP k &minus g 1, for k &ge 2 g &minus 1. Our knowledge of the sequence is therefore 1, ?, ?, ..., ?, g , g 1, g 2, ... . What we know about the ? entries is that they can increment by at most 1 each time this is a simple ..., so the cases g 0 or 1 need no further discussion and do not give rise to Weierstrass points. Assume .... A non Weierstrass point of C occurs whenever the increments are all as far to the right as possible ... case is a Weierstrass point . A Weierstrass gap for P is a value of k such that no function on C has exactly a k fold pole at P only. The gap sequence is 1, 2, ..., g for a non Weierstrass point. For a Weierstrass point it contains at least one higher number. The Weierstrass gap theorem or L ckensatz ... , ... the weight of the Weierstrass point is a &minus 1 b &minus 2 c &minus 3 ... . This is introduced because of a counting theorem on a Riemann surface the sum of the weights of the Weierstrass points is g g sup 2 sup &minus 1 . For example a hyperelliptic Weierstrass point, as above, has weight ... this exhausts all the Weierstrass points on C . Further information on the gaps comes from applying ... 16. A definition of Weierstrass point for a nonsingular curve over a field of positive characteristic ... more details
In mathematics , the Weierstrass product inequality states that given real numbers 0 a , b , c , d 1, it follows that math 1 a 1 b 1 c 1 d a b c d geq 1. , math The inequality is named after the Germany German mathematician KarlWeierstrass . Category Inequalities mathanalysis stub ... more details
Weierstrass and Heine Borel theorems are essentially the same. History The Bolzano Weierstrass theorem is named after mathematicians Bernard Bolzano and KarlWeierstrass . It was actually first proved ...In real analysis , the Bolzano Weierstrass theorem is a fundamental result about convergence in a finite dimensional Euclidean space R sup n sup . The theorem states that each bounded sequence in R sup n sup has a limit of a sequence convergent subsequence . An equivalent formulation is that a subset of R sup n sup is Sequentially compact space sequentially compact if and only if it is closed set closed and bounded set bounded . Proof First we prove the theorem when n 1, in which case the ordering on R can be put to good use. Indeed we have the following result. Lemma Every sequence nowrap ... by Weierstrass. It has since become an essential theorem of Real analysis analysis . Application to economics ... of the existence of which often require variations of the Bolzano Weierstrass theorem. One example ... matrix must be rankable by a preference relation . The Bolzano Weierstrass theorem allows ... Weierstrass theorem http planetmath.org ?op getobj&from objects&id 2129 PlanetMath proof of Bolzano Weierstrass Theorem http www.youtube.com watch?v dfO18klwKHg A proof of the Bolzano Weierstrass ... Weierstrass theorem Category Theorems in real analysis Category Compactness theorems az Boltsano ... Weierstrass cy Theorem Bolzano Weierstrass de Satz von Bolzano Weierstra et Bolzano Weierstrassi teoreem es Teorema de Bolzano Weierstrass fr Th or me de Bolzano Weierstrass ko is Bolzano Weierstrass setningin it Teorema di Bolzano Weierstrass he hu Bolzano Weierstrass t tel nl Stelling van Bolzano Weierstrass ja pl Twierdzenie Bolzano Weierstrassa pt Teorema de Bolzano Weierstrass ru fi Bolzanon Weierstassin lause sv Bolzano Weierstrass sats tr Bolzano Weierstrass teoremi uk ... more details
was established by KarlWeierstrass in 1885 using the Weierstrass transform . Marshall H. Stone considerably ...In mathematical analysis , the Weierstrass approximation theorem states that every continuous function ... is known as the Stone Weierstrass theorem . The Stone Weierstrass theorem generalizes the Weierstrass .... The Stone Weierstrass theorem is a vital result in the study of the algebra of continuous functions on a compact Hausdorff space . Further, there is a generalization of the Stone Weierstrass theorem ... Weierstrass theorem and described below. A different generalization of Weierstrass original theorem ... plane. Weierstrass approximation theorem The statement of the approximation theorem as originally discovered by Weierstrass is as follows Suppose is a continuous complex number complex valued function ... s is outlined on that page. Applications As a consequence of the Weierstrass approximation theorem ... . Stone Weierstrass theorem, real version The set C a , b of continuous real valued functions on a , b ... under multiplication of functions , and the content of the Weierstrass approximation theorem ... Weierstrass is Suppose X is a compact Hausdorff space and A is a subalgebra of C X , R which .... This implies Weierstrass original statement since the polynomials on a , b form a subalgebra of C ... Weierstrass theorem is also true when X is only locally compact . Let nowrap C sub 0 sub X ,&thinsp ... R C X ,&thinsp R . There are also more general versions of the Stone Weierstrass that weaken the assumption ... math 0402150 Variations on a theme of Gelfand and Naimark . ref Applications The Stone Weierstrass theorem can be used to prove the following two statements which go beyond Weierstrass s result. If f ... s on 0,1 . Stone Weierstrass theorem, complex version Slightly more general is the following theorem ... version of the Stone Weierstrass theorem states harv Hewitt Stromberg 1965 loc Theorem 7.29 Suppose ... max f , g and min f , g also belong to L . The lattice version of the Stone Weierstrass theorem ... more details
In mathematics , the Weierstrass factorization theorem in complex analysis , named after KarlWeierstrass , asserts that entire function s can be represented by a product involving their zero complex analysis zeroes . In addition, every sequence tending to infinity has an associated entire function with zeroes at precisely the points of that sequence. A second form extended to meromorphic function s allows one to consider a given meromorphic function as a product of three factors the function s poles, zeroes, and an associated non zero holomorphic function. Motivation The consequences of the fundamental theorem of algebra are twofold. ref name knopp citation last Knopp first K. contribution Weierstrass s Factor Theorem title Theory of Functions, Part II location New York publisher Dover pages 1 7 year 1996 . ref Firstly, any finite sequence math c n math in the complex plane has an associated polynomial that has zeroes precisely at the points of that sequence , math , prod n z c n . math Secondly, any polynomial function math p z math in the complex plane has a factorization math ,p z a prod n z c n , math where a is a non zero constant and c sub n sub are the zeroes of p . The two forms of the Weierstrass factorization theorem can be thought of as extensions of the above to entire ... those prescribed. Enter the genius of Weierstrass elementary factors . These factors serve the same ... with specified zeroes Sometimes called the Weierstrass theorem . ref name mw wst MathWorld urlname WeierstrasssTheorem title Weierstrass s Theorem ref Let math a n math be a sequence of non zero complex ... . The Weierstrass factorization theorem Sometimes called the Weierstrass product factor theorem. ref name mw wpt MathWorld urlname WeierstrassProductTheorem title Weierstrass Product Theorem ref Let ... Produktsatz es Teorema de factorizaci n de Weierstrass fr Th or me de factorisation de Weierstrass ko nl Factorisatiestelling van Weierstrass ru ... more details
In mathematics , Weierstrass s elliptic functions are elliptic function s that take a particularly simple form they are named for KarlWeierstrass . This class of functions are also referred to as P functions and generally written using the symbol & 8472 or math wp math , and known as Weierstrass P . div class thumb tright div style width 131px Image Weierstrass p.svg 100px Symbol for Weierstrass P function div class thumbcaption Symbol for Weierstrass P function div div div Definitions div class ... appearance. The letter has the appearance of being created with a brush 200px Weierstrass P function div class thumbcaption Weierstrass P function defined over a subset of the complex plane using a standard ... lattice of poles, and two interleaving lattices of zeros. div div div The Weierstrass elliptic function ..., for fixed z the Weierstrass functions become modular function s of &tau . In terms of the two periods, Weierstrass s elliptic function is an elliptic function with periods &omega sub 1 sub and &omega ... omega 1, omega 2 math for any pair of generators of the lattice defines the Weierstrass function as a function ... m n tau 2 right . math The above sum is homogeneous of degree minus two, from which we may define the Weierstrass ... functions shows that the condition on Weierstrass s function is determined up to addition of a constant ... the relation above. Integral equation The Weierstrass elliptic function can be given as the inverse ... periods sub 1 sub 2 and sub 2 sub 2 of Weierstrass elliptic function are related to the roots ... 1 omega 2 math . Since the square of the derivative of Weierstrass s elliptic function equals the above ... int e 3 infty frac dz sqrt 4z 3 g 2 z g 3 . math Addition theorems The Weierstrass elliptic functions ... , so that the imaginary part of is positive, we define the Weierstrass & 8472 function by math wp ... is any non zero complex number, math wp cz c tau wp z tau c 2 math from which we may define the Weierstrass ... Bbb C wp, wp , math so that all such functions are rational function s in the Weierstrass function and its ... more details
Karl also Carl is a variant of the given name Charles see there . People Royalty See Carl name Notable people or Charles Royalty for royalty with similar names. Karl of Austria last Austrian Emperor Charlemagne Karl der Gro e commonly known in English as Charlemagne Karl as first name Karl mythology , the ancestor of the peasants according to Norse mythology, see R g Norse god Karl Auerbach , computer scientist Karl Bartos , German musician Karl Becker general Karl Becker , German engineer and officer Karl Becker painter Karl Becker , German painter Karl Becker philologist Karl Becker , German philologist Karl Benz , German engineer Karl Blossfeldt , German artist Karl Briullov , Russian Painter Karl Davies , English actor Karl D nitz 1891 1980 , German admiral Karl Amadeus Hartmann , German composer Karl Haushofer , German geopolitics geopolitician Karl Hess , American comedian Karl Jaspers , German philosopher Karl Otto Koch , German Nazi commandant of the Nazi concentration camps at Buchenwald and Sachsenhausen Karl Kraus , Austrian writer Karl Christian Friedrich Krause , German philosopher Karl Kruszelnicki , Australian television and radio personality Karl Lagerfeld , German fashion designer Karl Landsteiner , Austrian biologist and physician Karl Lucas .English Actor and Comedian Karl Malden 1912 2009 , American actor Karl Malone , American basketball player Karl Marx , Prussian philosopher, father of communism Karl Pilkington , English podcaster, author and former radio producer, star of The Ricky Gervais Show and An Idiot Abroad Karl Popper , Austrian British philosopher Karl Power , British prankster Karl Rapp , German engineer Karl Rove , American political advisor Karl Stromberg , fictional James Bond villain Karl Urban , New Zealand actor Karl Virtanen born 1971 , Sweden Finnish journalist Karl Edward Wagner , American writer, editor and publisher Karl Walken , fictional Black Cat mayor KarlWeierstrass , German mathematician Karl Wendlinger , Austrian race ... more details
In mathematics , the Weierstrass Enneper parameterization of minimal surface s is a classical piece of differential geometry . Alfred Enneper and KarlWeierstrass studied minimal surfaces as far back as 1863. Let &fnof and g be functions on either the entire complex plane or the unit disk, where g is meromorphic function meromorphic and &fnof is analytic function analytic , such that wherever g has a pole of order m , f has a zero of order 2 m or equivalently, such that the product &fnof g sup 2 sup is holomorphic , and let c sub 1 sub , c sub 2 sub , c sub 3 sub be constants. Then the surface with coordinates x sub 1 sub , x sub 2 sub , x sub 3 sub is minimal, where the x sub k sub are defined using the real part of a complex integral, as follows math begin align x k zeta & Re left int 0 zeta varphi k z , dz right c k , qquad k 1,2,3 varphi 1 & f 1 g 2 2 varphi 2 & bold i f 1 g 2 2 varphi 3 & fg end align math The converse is also true every nonplanar minimal surface defined over a simply connected domain can be given a parametrization of this type. ref name DHWK Dierkes, U., Hildebrandt, S., K ster, A., Wohlrab, O. Minimal surfaces , vol. I, p. 108. Springer 1992. ISBN 3540531696 ref For example, Enneper s surface has &fnof z 1, g z z . References reflist Category Differential geometry Category surfaces differential geometry stub ... more details
In complex analysis , a branch of mathematics, the Casorati Weierstrass theorem describes the behaviour of meromorphic function s near essential singularity essential singularities . It is named for Karl Theodor Wilhelm Weierstrass and Felice Casorati mathematician Felice Casorati . In Russian literature it is called Yulian Sokhotski Sokhotski s theorem. Formal statement of the theorem Start with some open set open subset U in the complex number complex plane containing the number z sub 0 sub , and a function f that is holomorphic function holomorphic on U z sub 0 sub , but has an essential singularity at z sub 0 sub . The Casorati Weierstrass theorem then states that if V is any Neighbourhood mathematics neighbourhood of z sub 0 sub contained in U , then f V z sub 0 sub is dense set dense in C . This can also be stated as follows for any &epsilon 0, &delta 0, and complex number w , there exists a complex number z in U with z &minus z sub 0 sub &delta and f z &minus w &epsilon . Or in still more descriptive terms f comes arbitrarily close to any complex value in every neighbourhood of z sub 0 sub . This form of the theorem also applies if f is only meromorphic . The theorem is considerably strengthened by Picard s great theorem , which states, in the notation above, that f assumes every complex value, with one possible exception, infinitely often ... It was published by Weierstrass in 1876 in German and by Sokhotski in 1873 in Russian . So it was called Sokhotski s theorem in the Russian literature and Weierstrass s theorem in the Western literature ... singularities de Satz von Weierstra Casorati es Teorema de Weierstrass Casorati fa fr Th or me de Weierstrass Casorati it Teorema di Casorati Weierstrass he nl Stelling van Weierstrass Casorati ja ru tr Weierstrass Casorati teoremi zh Sokhotsky Weierstrass ... more details
In mathematics , the Weierstrass preparation theorem is a tool for dealing with analytic function s of several complex variables , at a given point P. It states that such a function is, up to multiplication by a function not zero at P, a polynomial in one fixed variable z , which is monic polynomial monic , and whose coefficient s are analytic functions in the remaining variables and zero at P. There are also a number of variants of the theorem, that extend the idea of factorization in some ring mathematics ring R as u w , where u is a Unit ring theory unit and w is some sort of distinguished Weierstrass polynomial . C.L. Siegel has disputed the attribution of the theorem to Weierstrass , saying that it occurred under the current name in some of late nineteenth century Trait s d analyse without ... variables as z , z sub 2 sub , ..., z sub n sub . A Weierstrass polynomial W z is z sup k sup g sub ... sub , ..., z sub n sub with h analytic and h 0, ..., 0 not 0, and W a Weierstrass polynomial. This has ... f cannot have an isolated zero. A related result is the Weierstrass division theorem , which states that if f and g are analytic functions, and g is a Weierstrass polynomial of degree N , then there exists ... to the preparation theorem, since the Weierstrass factorization of f may be obtained ... the desired Weierstrass polynomial is then z sup N sup j h . For the other direction, we use the preparation theorem on g , and the normal polynomial remainder theorem on the resulting Weierstrass ... to as the Weierstrass preparation theorem, for power series rings over the ring of integers in a p ... Weierstrass year 1969 title Number Theory and Analysis Papers in Honor of Edmund Landau pages 297 ... pages 1 8 isbn 0 387 09374 5 springer id W w097510 title Weierstrass theorem first E.D. last Solomentsev ... Weierstrass title Mathematische Werke. II. Abhandlungen 2 pages 135 142 publisher Mayer & M ller ... de Weierstrass ... more details
Lindemann and KarlWeierstrass . Lindemann proved in 1882 that math e sup math sup is transcendental ...stack Pi box stack E mathematical constant In mathematics , the Lindemann Weierstrass theorem is a result that is very useful in establishing the transcendental number transcendence of numbers. It states that if math sub 1 sub , ..., math sub math n sub are algebraic number s which are linearly independent over the rational number s math Q , then math e sup math sub 1 sub sup , ..., math e sup math sub math n sub sup are algebraically independent over Q in other words the extension field math Q math e sup math sub 1 sub sup , ..., math e sup math sub math n sub sup has transcendence degree math n over math Q . An equivalent formulation Harv Baker 1975 loc Chapter 1, Theorem 1.4 , is the following If math sub 1 sub , ..., math sub math n sub are distinct algebraic numbers, then the exponentials math e sup sub 1 sub sup , ..., ... below . Weierstrass proved the above more general statement in 1885. The theorem, along with the Gelfond ... known variously as the Hermite Lindemann theorem and the Hermite Lindemann Weierstrass theorem ... Weierstrass obtained the full result, ref Zu Hrn. Lindemanns Abhandlung ber die Ludolph sche Zahl ... be algebraic as well, and then by the Lindemann Weierstrass theorem math e sup pi math i sup 1 see ... p adic Lindemann Weierstrass conjecture is that a math p adic analog of this statement is also ... n ne 0 math . ref fr icon http nombrejador.free.fr article lindemann weierstrass ttj.htm Proof s Lindemann Weierstrass HTML ref Lemma A If math K and math c are non zero integers, and math sub 1 ... Theorems in number theory de Satz von Lindemann Weierstra es Teorema de Lindemann Weierstrass fr Th or me de Lindemann Weierstrass it Teorema di Lindemann Weierstrass he nl Stelling van Lindemann Weierstrass ja pt Teorema de Lindemann Weierstrass ru ... more details
Wikify date February 2011 Unreferenced date March 2009 The Weierstrass&ndash Erdmann condition is a technical tool from the calculus of variations . This condition gives the sufficient conditions for an extremal to have a corner. Conditions The condition says that, along a piecewise smooth extremal x t i.e. an extremal which is smooth except at a finite number of corners for an integral math J int f t,x,y ,dt math , the partial derivative math partial f partial x math must be continuous at a corner T . That is, if one takes the limit of partials on both sides of the corner as one approaches the corner T , the result must be the same answer. Applications The condition allows one to prove that a corner exists along a given extremal. As a result, there are many applications to differential geometry. In calculations of the Weierstrass E Function, it is often helpful to find where corners exist along the curves. Similarly, the condition allows for one to find a minimizing curve for a given integral. DEFAULTSORT Weierstrass Erdmann Condition Category Calculus of variations ... more details
In mathematics , the Weierstrass M test is a criterion for determining whether an infinite series of function mathematics functions converges uniform convergence uniformly . It applies to series whose terms are functions with real number real or complex number complex values, and is analogous to the comparison test for determining the convergence of series of real or complex numbers. Statement Suppose that math f n math is a sequence of real or complex valued functions defined on a Set mathematics set math A math , and that there is a sequence of positive numbers math M n math satisfying math f n x leq M n math for all math n geq 1 math and all math x in A math . Suppose also that the series math sum n 1 infty M n math is convergent. Then the series math sum n 1 infty f n x math converges uniform convergence uniformly on math A math . Remark The result is often used in combination with the uniform limit theorem . Together they say that if, in addition to the above conditions, the set math A math is a topological space and the functions math f n math are continuous on math A math , then the series converges to a continuous function. Generalization A more general version of the Weierstrass M test holds if the codomain of the functions math f n math is any Banach space , in which case the statement math f n leq M n math may be replaced by math f n leq M n math , where math cdot math is the norm on the Banach space. For an example of the use of this test on a Banach space, see the article Fr chet derivative . Proof Suppose that the conditions stated above hold. We must show that math ... of Weierstrass M test References cite book last Rudin first Walter title Functional Analysis year 1991 ... M de Weierstrass eo M provo de Weierstrass ko M it Criterio di Weierstrass he M ja M pl Kryterium Weierstrassa pt Teste M de Weierstrass fi Weierstrassin M testi sv Weierstrass majorantsats tr Weierstrass M testi zh ... more details
Other uses Karl disambiguation Karl Infobox Radio station name KARL image Image KARL FM.jpg Station logo city Tracy, Minnesota area Marshall, Minnesota slogan Today s New Hit Country branding 105.1 KARL frequency 105.1 FM band FM Megahertz MHz airdate format Commercial radio Commercial Country music Country erp 45,000 watt s haat 153 meters class C2 facility id 35129 coordinates coord 44 19 32 N 95 52 19 W region US MN type landmark display inline,title callsign meaning pronounced as carl former callsigns owner Linder Radio Group licensee KMHL Broadcasting Company sister stations KARL, KARZ FM KARZ , KKCK , KMHL , KNSG webcast website http www.1051karl.com affiliations CD Country Jones CD Country KARL 105.1 FM broadcasting FM is a radio station broadcasting a country music format. City of license Licensed to Tracy, Minnesota , the station serves the Marshall, Minnesota area. The station is currently owned by Linder Radio Group . ref cite web url http www.fcc.gov fcc bin fmq?call KARL title KARL Facility Record work United States Federal Communications Commission , audio division ref References reflist External links http www.1051karl.com 105.1 KARL official website FM station data KARL Worthington Marshall Radio Country Radio Stations in Minnesota Category Radio stations in Minnesota Category Country radio stations in the United States Minnesota radio station stub ... more details
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