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{{Infobox scientist
| name = Karl Weierstrass
| image = Karl Weierstrass.jpg|300px
| caption = Karl Theodor Wilhelm Weierstrass (Weierstraß)
| birth_date = {{birth date|1815|10|31|df=y}}
| birth_place = [[Ennigerloh|Ostenfelde]], [[Province of Westphalia]], [[Kingdom of Prussia]]
| death_date = {{death date and age|1897|2|19|1815|10|31|df=y}}
| death_place = [[Berlin]], [[Province of Brandenburg]], [[Kingdom of Prussia]]
| residence = [[Germany]]
| nationality = [[Germans|German]]
| field = [[Mathematics]]
| work_institution = [[Technical University of Berlin|Gewerbeinstitut]]
| alma_mater = [[University of Bonn]]<br>[[University of Münster|Münster Academy]]
| doctoral_advisor = [[Christoph Gudermann]]
| doctoral_students = [[Nikolai Bugaev]]<br>[[Georg Cantor]]<br>[[Georg Frobenius]]<br>[[Lazarus Fuchs]]<br>[[Wilhelm Killing]]<br>[[Leo Königsberger]]<br>[[Sofia Kovalevskaya]]<br>[[Mathias Lerch]]<br>[[Hans von Mangoldt]]<br>[[Eugen Netto]]<br>[[Adolf Piltz]]<br>[[Carl Runge]]<br>[[Arthur Schoenflies]]<br>[[Friedrich Schottky]]<br>[[Hermann Schwarz]]<br>[[Ludwig Stickelberger]]
| known_for = [[Weierstrass function]]
| prizes =
| religion =
| footnotes =
}}
'''Karl Theodor Wilhelm Weierstrass''' ({{lang-de|Weierstraß}}; 31 October 1815 &ndash; 19 February 1897) was a [[Germany|German]] [[mathematics|mathematician]] who 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 he was a ''[[Gymnasium (school)|Gymnasium]]'' student at [[:de:Gymnasium Theodorianum|Theodorianum]] in [[Paderborn]].  He was sent to the [[University of Bonn]] upon graduation to prepare for a government position. Because his studies were to be in the fields of [[law]], economics, and finance, he was immediately in conflict with his hopes to study mathematics. He resolved the conflict by paying little heed to his planned course of study, but continued private study in mathematics. The outcome was to leave the university without a degree. After that he studied mathematics at the [[University of Münster]] (which was even at this time very famous for mathematics) and his father was able to obtain a place for him in a teacher training school in [[Münster]].  Later he 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 function]]s.
In 1843 he taught in Deutsch-Krone in Westprussia and since 1848 he taught at the [[Lyceum Hosianum]] in [[Braunsberg]]. Besides mathematics he also taught physics, botanics and gymnastics.
 
Weierstrass may have had an illegitimate child named Franz with the widow of his friend Borchardt.<ref>See [http://www.ams.org/mathscinet-getitem?mr=1388786 here]</ref>
 
After 1850 Weierstrass suffered from a long period of illness, but was able to publish papers that brought him fame and distinction. He took a chair at the [[Technical University of Berlin]], then known as the Gewerbeinstitut. He was immobile for the last three years of his life, and died in Berlin from [[pneumonia]].
 
== Mathematical contributions ==
=== Soundness of calculus ===
Weierstrass was interested in the [[soundness]] of calculus. At the time, there were somewhat ambiguous definitions regarding the foundations of calculus, and hence important theorems could not be proven with sufficient rigour.  While [[Bernard Bolzano|Bolzano]] had developed a reasonably rigorous definition of a [[Limit of a function|limit]] as early as 1817 (and possibly even earlier) his work remained unknown to most of the mathematical community until years later,
and many had only vague definitions of [[Limit of a function|limits]] and [[Continuous function|continuity]] of functions.
 
[[(ε, δ)-definition of limit|Delta-epsilon]] proofs are first found in the works of [[Augustin Louis Cauchy|Cauchy]] in the 1820s.<ref>{{citation
|title=Who Gave You the Epsilon? Cauchy and the Origins of Rigorous Calculus
|first=Judith V.
|last=Grabiner
|journal=The American Mathematical Monthly
|date=March 1983
|volume=90
|pages=185–194
|url=http://www.maa.org/sites/default/files/pdf/upload_library/22/Ford/Grabiner185-194.pdf
|doi=10.2307/2975545
|issue=3
|jstor=2975545
}}</ref><ref>{{citation
|first=A.-L.
|last=Cauchy
|author-link=Augustin Louis Cauchy
|title=Résumé des leçons données à l’école royale polytechnique sur le calcul infinitésimal
|place=Paris
|year=1823
|url=http://math-doc.ujf-grenoble.fr/cgi-bin/oeitem?id=OE_CAUCHY_2_4_9_0
|chapter=Septième Leçon – Valeurs de quelques expressions qui se présentent sous les formes indéterminées <math>\frac{\infty}\infty, \infty^0, \ldots</math> Relation qui existe entre le rapport aux différences finies et la fonction dérivée
|chapterurl=http://gallica.bnf.fr/ark:/12148/bpt6k90196z/f45n5.capture
|postscript=, [http://gallica.bnf.fr/ark:/12148/bpt6k90196z.image.f47 p. 44].
}}</ref>
Cauchy did not clearly distinguish between continuity and uniform continuity on an interval. Notably, in his 1821 ''Cours d'analyse,'' Cauchy argued that the (pointwise) limit of (pointwise) continuous functions was itself (pointwise) continuous, a statement interpreted as being incorrect by many scholars. The correct statement is rather that the [[uniform limit|''uniform'' limit]] of continuous functions is continuous (also, the uniform limit of uniformly continuous functions is uniformly continuous).
This required the concept of [[uniform convergence]], which was first observed by Weierstrass's advisor, [[Christoph Gudermann]], in an 1838 paper, where Gudermann noted the phenomenon but did not define it or elaborate on it. Weierstrass saw the importance of the concept, and both formalized it and applied it widely throughout the foundations of calculus.
 
The formal definition of continuity of a function, as formulated by Weierstrass, is as follows:
 
<math>\displaystyle f(x)</math> is continuous at <math>\displaystyle x = x_0</math> if <math> \displaystyle \forall \ \varepsilon > 0\ \exists\ \delta > 0</math> such that for every <math>x</math> in the domain of <math>f</math>, &nbsp; <math> \displaystyle \ |x-x_0| < \delta \Rightarrow |f(x) - f(x_0)| < \varepsilon.</math>
 
Using this definition and the concept of uniform convergence,
Weierstrass was able to write proofs of several then-unproven theorems such as the [[intermediate value theorem]] (for which [[Bernard Bolzano|Bolzano]] had already given a rigorous proof), the [[Bolzano–Weierstrass theorem]], and [[Heine–Borel theorem]].
 
=== Calculus of variations ===
 
Weierstrass also made significant advancements in the field of [[calculus of variations]]. Using the apparatus of analysis that he helped to develop, Weierstrass was able to give a complete reformulation of the theory which paved the way for the modern study of the calculus of variations. Among the several significant axioms, Weierstrass established a necessary condition for the existence of [[strong extrema]] of variational problems. He also helped devise the [[Weierstrass–Erdmann condition]], which gives sufficient conditions for an extremal to have a corner along a given extrema, and allows one to find a minimizing curve for a given integral.
 
=== Other analytical theorems ===
 
:: ''See also'' [[List of things named after Karl Weierstrass]].
 
* [[Stone–Weierstrass theorem]]
* [[Weierstrass–Casorati theorem]]
* [[Weierstrass's elliptic functions]]
* [[Weierstrass function]]
* [[Weierstrass M-test]]
* [[Weierstrass preparation theorem]]
* [[Lindemann–Weierstrass theorem]]
* [[Weierstrass factorization theorem]]
* [[Enneper–Weierstrass parameterization]]
* [[Sokhatsky–Weierstrass theorem]]
 
== Selected works ==
*''Zur Theorie der Abelschen Funktionen'' (1854)
*''Theorie der Abelschen Funktionen'' (1856)
* ''[http://name.umdl.umich.edu/AAN8481.0001.001 Abhandlungen-1]''// Math. Werke. Bd. 1. Berlin, 1894
* ''[http://name.umdl.umich.edu/AAN8481.0002.001 Abhandlungen-2]''// Math. Werke. Bd. 2. Berlin, 1895
* ''[http://name.umdl.umich.edu/AAN8481.0003.001 Abhandlungen-3]''// Math. Werke. Bd. 3. Berlin, 1903
* ''[http://name.umdl.umich.edu/AAN8481.0004.001 Vorl. ueber die Theorie der Abelschen Transcendenten]''// Math. Werke. Bd. 4. Berlin, 1902
* ''[http://name.umdl.umich.edu/AAN8481.0007.001 Vorl. ueber Variationsrechnung]''// Math. Werke. Bd. 7. Leipzig, 1927
 
==Students of Karl Weierstrass==
* [[Edmund Husserl]]
* [[Sofia Kovalevskaya]]
* [[Gösta Mittag-Leffler]]
* [[Hermann Schwarz]]
* [[Carl Johannes Thomae]]
 
==Honours and awards==
The lunar [[Impact crater|crater]] [[Weierstrass (crater)|Weierstrass]] is named after him.
 
== See also ==
 
* [[List of things named after Karl Weierstrass]]
 
==References==
{{Reflist}}
==External links==
{{commons|Karl Weierstrass}}
{{Wikiquote}}
*{{MacTutor Biography|id=Weierstrass}}
*{{MathGenealogy |id=7486}}
* [http://bibliothek.bbaw.de/bibliothek-digital/digitalequellen/schriften/autoren/weierstr/ Digitalized versions of Weierstrass's original publications] are freely available online from the library of the ''[http://bibliothek.bbaw.de/bibliothek-digital Berlin Brandenburgische Akademie der Wissenschaften]''.
*{{gutenberg author | id=Karl_Weierstrass | name=Karl Weierstrass }}
 
{{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 NAMES=
|SHORT DESCRIPTION= [[Mathematician]]
|DATE OF BIRTH= 31 October 1815
|PLACE OF BIRTH= [[Ostenfelde]], [[Westphalia]]
|DATE OF DEATH= 19 February 1897
|PLACE OF DEATH= [[Berlin]], [[Germany]]
 
}}
{{DEFAULTSORT:Weierstrass, Karl}}
[[Category:1815 births]]
[[Category:1897 deaths]]
[[Category:19th-century German mathematicians]]
[[Category:Mathematical analysts]]
[[Category:People from the Province of Westphalia]]
[[Category:People from Braniewo]]
[[Category:Recipients of the Copley Medal]]
[[Category:University of Bonn alumni]]
[[Category:University of Königsberg alumni]]
[[Category:University of Münster alumni]]
[[Category:Humboldt University of Berlin faculty]]
[[Category:Berlin Institute of Technology faculty]]
[[Category:Foreign Members of the Royal Society]]
[[Category:Corresponding Members of the St Petersburg Academy of Sciences]]
[[Category:Honorary Members of the St Petersburg Academy of Sciences]]
[[Category:German Roman Catholics]]

Revision as of 22:11, 23 January 2014

Template:Infobox scientist Karl Theodor Wilhelm Weierstrass (Template:Lang-de; 31 October 1815 – 19 February 1897) was a German mathematician who is often cited as the "father of modern 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 he was a Gymnasium student at Theodorianum in Paderborn. He was sent to the University of Bonn upon graduation to prepare for a government position. Because his studies were to be in the fields of law, economics, and finance, he was immediately in conflict with his hopes to study mathematics. He resolved the conflict by paying little heed to his planned course of study, but continued private study in mathematics. The outcome was to leave the university without a degree. After that he studied mathematics at the University of Münster (which was even at this time very famous for mathematics) and his father was able to obtain a place for him in a teacher training school in Münster. Later he 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. In 1843 he taught in Deutsch-Krone in Westprussia and since 1848 he taught at the Lyceum Hosianum in Braunsberg. Besides mathematics he also taught physics, botanics and gymnastics.

Weierstrass may have had an illegitimate child named Franz with the widow of his friend Borchardt.[1]

After 1850 Weierstrass suffered from a long period of illness, but was able to publish papers that brought him fame and distinction. He took a chair at the Technical University of Berlin, then known as the Gewerbeinstitut. He was immobile for the last three years of his life, and died in Berlin from pneumonia.

Mathematical contributions

Soundness of calculus

Weierstrass was interested in the soundness of calculus. At the time, there were somewhat ambiguous definitions regarding the foundations of calculus, and hence important theorems could not be proven with sufficient rigour. While Bolzano had developed a reasonably rigorous definition of a limit as early as 1817 (and possibly even earlier) his work remained unknown to most of the mathematical community until years later, and many had only vague definitions of limits and continuity of functions.

Delta-epsilon proofs are first found in the works of Cauchy in the 1820s.[2][3] Cauchy did not clearly distinguish between continuity and uniform continuity on an interval. Notably, in his 1821 Cours d'analyse, Cauchy argued that the (pointwise) limit of (pointwise) continuous functions was itself (pointwise) continuous, a statement interpreted as being incorrect by many scholars. The correct statement is rather that the uniform limit of continuous functions is continuous (also, the uniform limit of uniformly continuous functions is uniformly continuous). This required the concept of uniform convergence, which was first observed by Weierstrass's advisor, Christoph Gudermann, in an 1838 paper, where Gudermann noted the phenomenon but did not define it or elaborate on it. Weierstrass saw the importance of the concept, and both formalized it and applied it widely throughout the foundations of calculus.

The formal definition of continuity of a function, as formulated by Weierstrass, is as follows:

f(x) is continuous at x=x0 if ∀ε>0∃δ>0 such that for every x in the domain of f,   |x−x0|<δ⇒|f(x)−f(x0)|<ε.

Using this definition and the concept of uniform convergence, Weierstrass was able to write proofs of several then-unproven theorems such as the intermediate value theorem (for which Bolzano had already given a rigorous proof), the Bolzano–Weierstrass theorem, and Heine–Borel theorem.

Calculus of variations

Weierstrass also made significant advancements in the field of calculus of variations. Using the apparatus of analysis that he helped to develop, Weierstrass was able to give a complete reformulation of the theory which paved the way for the modern study of the calculus of variations. Among the several significant axioms, Weierstrass established a necessary condition for the existence of strong extrema of variational problems. He also helped devise the Weierstrass–Erdmann condition, which gives sufficient conditions for an extremal to have a corner along a given extrema, and allows one to find a minimizing curve for a given integral.

Other analytical theorems

See also List of things named after Karl Weierstrass.

Selected works

Students of Karl Weierstrass

Honours and awards

The lunar crater Weierstrass is named after him.

See also

References

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On February 19 we had an appointment with the proprietor and his agent (A and H!) at the condominium to hand over the keys. They went by means of all the things with a wonderful tooth comb. An important lesson we learned over all this is that you simply MUST ENGAGE YOUR PERSONAL AGENT and never rely on the homeowners agent as his priority is to the proprietor not you. Nevertheless, last night time my own agent called me and informed me suddenly that ECG instructed them a buyer goes handy them a check within the morning, so we higher act fast or we may lose the property. Stamp responsibility is to be paid inside 14 days from the date of acceptance of the OTP or Sale and buy a house in singapore (click hyperlink) (S&P) Settlement. For more information, please go to www.iras.gov.sg - Gown Up Your House Woodlands East Industrial & Industrial Affiliation

There may also be a Code of Ethics and a Skilled Conduct Commonplace, as well as the introduction of disciplinary motion in opposition to errant brokers/businesses and dispute decision mechanisms. Preparations shall be made to manage the transition of existing agencies and agents to these new standards, which have but to be finalized. The Proposed Enchancment in High quality for Actual Property Businesses Wheelock Properties put up 95 items of The Panorama in Ang Mo Kio for balloting. With a reduction of 12 p.c, they claimed to promote 80 to eighty five units. Whereas developers are clearing their existing stock, every month there are new projects acquiring their HIGH and new sites released by the government to construct more private housing. The due date of each rental payment; or

To know who pays actual property commissions - whether or not it's sellers or buyers or both or if it is Landlord's or Tenant's or both Divisions vary. All Brokers work on a commission scheme that is determined by the experience, efficiency and various other elements equivalent to recruitment and many others. New brokers can receive from a range of 60%-70% of the full fee received by them from the closure of a deal. High producing brokers would possibly obtain 100% and pay the company (broker) a desk fee. Everybody else falls somewhere in between. Kindly discuss with the FAQ part of the CEA web site-www.cea.gov.sg Co-Broking / sharing of fee There isn't a set formulation. This is based on the demand and supply circumstances in the market. present agents have tertiary education.

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  1. ↑ See here
  2. ↑ Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
  3. ↑ Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
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