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| {{Infobox scientist
| | == 「あなたが戻って == |
| | name = Karl Weierstrass
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| | image = Karl Weierstrass.jpg|300px
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| | caption = Karl Theodor Wilhelm Weierstrass (Weierstraß)
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| | birth_date = {{birth date|1815|10|31|df=y}}
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| | birth_place = [[Ennigerloh|Ostenfelde]], [[Province of Westphalia]], [[Kingdom of Prussia]]
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| | death_date = {{death date and age|1897|2|19|1815|10|31|df=y}}
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| | death_place = [[Berlin]], [[Province of Brandenburg]], [[Kingdom of Prussia]]
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| | residence = [[Germany]]
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| | nationality = [[Germans|German]]
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| | field = [[Mathematics]]
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| | work_institution = [[Technical University of Berlin|Gewerbeinstitut]]
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| | alma_mater = [[University of Bonn]]<br>[[University of Münster|Münster Academy]]
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| | doctoral_advisor = [[Christoph Gudermann]]
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| | 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]]
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| | known_for = [[Weierstrass function]]
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| | prizes =
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| | religion =
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| | footnotes =
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| }}
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| '''Karl Theodor Wilhelm Weierstrass''' ({{lang-de|Weierstraß}}; 31 October 1815 – 19 February 1897) was a [[Germany|German]] [[mathematics|mathematician]] who is often cited as the "father of modern [[mathematical analysis|analysis]]".
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| == Biography ==
| | 表、あごはシャオヤンは、「陰陽」「陰陽」の道を笑っ直面し、指が交差し立ち往生。<br><br>は「触れ」「触る」これはシャオヤンが少し気持ちで元の外観を、隠さ久しい笑った [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-4.html casio 腕時計 g-shock]。<br><br>「あなたが戻って？に行くxiaojia行く、クラウドのLAN物事を成し遂げる？ '雅 [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-7.html カシオ 腕時計 gps] - [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-13.html casio 腕時計 edifice] フェイは微笑んで尋ねた。<br><br>「訪問する戻りますが、後に、カナンの病院に行きます。 '<br><br>「カナンの学校は何」聞いた雅飛モーメントが、すぐにささやいたか、何を考えているようだ。「薫の子供たちにそれを見つけることです？が。 [http://alleganycountyfair.org/sitemap.xml http://alleganycountyfair.org/sitemap.xml] '<br>「これらの理由で。」<br>シャオヤンは、彼にお茶の弓の一口を取った雅 - [http://nrcil.net/sitemap.xml http://nrcil.net/sitemap.xml] 飛橋リアンつかの間の失望を見ていなかった、笑った。<br><br>「今日私は残っている場合は、この愛、xiaojiaの世話を助けるために、トラブルあなたにしたい、、私が思うに、マイテルファミリーの男の手のひらに本当の力ですが、私は将来的に返されます。」シャオヤン茶を保持 |
| Weierstrass was born in Ostenfelde, part of [[Ennigerloh]], [[Province of Westphalia]].
| | 相关的主题文章： |
| | <ul> |
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| | <li>[http://gacha.heteml.jp/tobata/yybbs/yybbs.cgi http://gacha.heteml.jp/tobata/yybbs/yybbs.cgi]</li> |
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| | <li>[http://hb145.com/forum.php?mod=viewthread&tid=1951226 http://hb145.com/forum.php?mod=viewthread&tid=1951226]</li> |
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| | <li>[http://www.baolong1.com/plus/view.php?aid=16437 http://www.baolong1.com/plus/view.php?aid=16437]</li> |
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| | </ul> |
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| 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.
| | == 「それはそう、今回は彼らシャオヤン == |
| 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.
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| 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>
| | 突然陵シャドウでグリップが直面している日に、ハイペリオン東雲山の手のひらを飲み込む飛行経路のpythonをブロックした後、突然、風のロープの本質から風によってまとめられ、圧倒的な、最終的なように、お互いに巻きつけ、風のロープの外に現れたボイドとシャトル内条ハイドラ一般、時間の瞬間、ハイペリオン東、及びリンの影が激しいです、それが既に縛ら縛られたときに二人の男は、実際に知らない自分を見つけた瞬間体執念深い気になるチョン、残念ながら、すべての彼らはいくつかの風の後にロープをZhenduanが、それは空気から轟音以上であれば、より多くの二人はより強固に接続されています [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-5.html カシオ 腕時計 スタンダード]。<br><br>「コールは確かに好調に推移しました、攻撃は明らかに、使用する色を計算しない彼の手の中に、2つの強い皇室闘争かもしれないが、それが戻って戦うために多くのない強さはないと判明しており、この二つの間にどのようなギャップで戦っている [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-12.html カシオ 時計 電波]？」というただ1ラウンドの空を見て、助けることが叫んだことができなかったハイペリオン東雲山2体格、刑罰の日に、縛られています [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-12.html 腕時計 メンズ casio]。<br><br>「それはそう、今回は彼らシャオヤン |
| | 相关的主题文章： |
| | <ul> |
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| | <li>[http://www.dolooks.com/plus/feedback.php?aid=212 http://www.dolooks.com/plus/feedback.php?aid=212]</li> |
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| | <li>[http://www.k8k9.com/plus/feedback.php?aid=10669 http://www.k8k9.com/plus/feedback.php?aid=10669]</li> |
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| | <li>[http://www.sylviasmother.com/cgi-bin/activeguestbook/guestbook.cgi http://www.sylviasmother.com/cgi-bin/activeguestbook/guestbook.cgi]</li> |
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| | </ul> |
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| 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]].
| | == は、ここに、薫の子供たちは、小さな医療セントに見て言った == |
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| == Mathematical contributions ==
| | 古代部族ようになりまし位置家族インテリジェンス機能に加え、燻製子どもたちは、限り唇が動くように、でも、彼女の口なしで、私はそれが多くの人が彼女の知性の前面にカラースケールと気になりますですが怖いです [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-1.html カシオ スタンダード 腕時計]。<br>「これはどのように？私はまだ彼の前に激怒し、その後彼はカラースケールを殺しましょうアムすることができますか？ [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-7.html カシオ電波ソーラー腕時計レディース] '薫の子供の口の少しすするhuanhangrn、曲がった浅い弧を動かし、口、笑顔にしたいようだが、最終的にはまたは、ため息をため息つぶやいた：でもまた、彼を憎むが、私は、これは不可能であることを知っていました、その場合、これのために [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-4.html カシオ 腕時計 バンド] '実際には、そのようなことは、私は頭痛を持って、他の人は私が去ることが望ましいだろうやって満足していない見つけるために自分自身を与えるか？ '<br><br>は、ここに、薫の子供たちは、小さな医療セントに見て言った。、その時間を「彼は率先してするためにこの種のことは、私が待っていると私は言って、私は彼を得るために、残酷なことを装って彼を教えることができた理解し、マッチョ男が、 [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-2.html 腕時計 casio] '結果'良いがありません！ [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-11.html カシオ ソーラー電波腕時計] ' |
| === Soundness of calculus ===
| | 相关的主题文章： |
| 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,
| | <ul> |
| and many had only vague definitions of [[Limit of a function|limits]] and [[Continuous function|continuity]] of functions.
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| | | <li>[http://ncndx.com/home.php?mod=space&uid=5366 http://ncndx.com/home.php?mod=space&uid=5366]</li> |
| [[(ε, δ)-definition of limit|Delta-epsilon]] proofs are first found in the works of [[Augustin Louis Cauchy|Cauchy]] in the 1820s.<ref>{{citation
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| |title=Who Gave You the Epsilon? Cauchy and the Origins of Rigorous Calculus
| | <li>[http://bbs.zg-gly.com/forum.php?mod=viewthread&tid=441719&fromuid=41242 http://bbs.zg-gly.com/forum.php?mod=viewthread&tid=441719&fromuid=41242]</li> |
| |first=Judith V.
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| |last=Grabiner
| | <li>[http://www.we366.com/forum.php?mod=viewthread&tid=3007 http://www.we366.com/forum.php?mod=viewthread&tid=3007]</li> |
| |journal=The American Mathematical Monthly
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| |date=March 1983
| | </ul> |
| |volume=90
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| |pages=185–194
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| |url=http://www.maa.org/sites/default/files/pdf/upload_library/22/Ford/Grabiner185-194.pdf
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| |doi=10.2307/2975545
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| |issue=3
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| |jstor=2975545
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| }}</ref><ref>{{citation
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| |first=A.-L.
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| |last=Cauchy
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| |author-link=Augustin Louis Cauchy
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| |title=Résumé des leçons données à l’école royale polytechnique sur le calcul infinitésimal
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| |place=Paris
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| |year=1823
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| |url=http://math-doc.ujf-grenoble.fr/cgi-bin/oeitem?id=OE_CAUCHY_2_4_9_0
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| |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
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| |chapterurl=http://gallica.bnf.fr/ark:/12148/bpt6k90196z/f45n5.capture
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| |postscript=, [http://gallica.bnf.fr/ark:/12148/bpt6k90196z.image.f47 p. 44].
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| }}</ref>
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| 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).
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| 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.
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| The formal definition of continuity of a function, as formulated by Weierstrass, is as follows:
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| <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>, <math> \displaystyle \ |x-x_0| < \delta \Rightarrow |f(x) - f(x_0)| < \varepsilon.</math>
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| Using this definition and the concept of uniform convergence,
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| 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]].
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| === Calculus of variations ===
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| 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.
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| === Other analytical theorems ===
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| :: ''See also'' [[List of things named after Karl Weierstrass]].
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| * [[Stone–Weierstrass theorem]]
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| * [[Weierstrass–Casorati theorem]]
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| * [[Weierstrass's elliptic functions]]
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| * [[Weierstrass function]]
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| * [[Weierstrass M-test]]
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| * [[Weierstrass preparation theorem]]
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| * [[Lindemann–Weierstrass theorem]]
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| * [[Weierstrass factorization theorem]]
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| * [[Enneper–Weierstrass parameterization]]
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| * [[Sokhatsky–Weierstrass theorem]]
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| == Selected works ==
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| *''Zur Theorie der Abelschen Funktionen'' (1854)
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| *''Theorie der Abelschen Funktionen'' (1856)
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| * ''[http://name.umdl.umich.edu/AAN8481.0001.001 Abhandlungen-1]''// Math. Werke. Bd. 1. Berlin, 1894
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| * ''[http://name.umdl.umich.edu/AAN8481.0002.001 Abhandlungen-2]''// Math. Werke. Bd. 2. Berlin, 1895
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| * ''[http://name.umdl.umich.edu/AAN8481.0003.001 Abhandlungen-3]''// Math. Werke. Bd. 3. Berlin, 1903
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| * ''[http://name.umdl.umich.edu/AAN8481.0004.001 Vorl. ueber die Theorie der Abelschen Transcendenten]''// Math. Werke. Bd. 4. Berlin, 1902
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| * ''[http://name.umdl.umich.edu/AAN8481.0007.001 Vorl. ueber Variationsrechnung]''// Math. Werke. Bd. 7. Leipzig, 1927
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| ==Students of Karl Weierstrass== | |
| * [[Edmund Husserl]]
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| * [[Sofia Kovalevskaya]]
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| * [[Gösta Mittag-Leffler]]
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| * [[Hermann Schwarz]]
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| * [[Carl Johannes Thomae]]
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| ==Honours and awards==
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| The lunar [[Impact crater|crater]] [[Weierstrass (crater)|Weierstrass]] is named after him.
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| == See also ==
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| * [[List of things named after Karl Weierstrass]]
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| ==References==
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| {{Reflist}}
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| ==External links==
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| {{commons|Karl Weierstrass}}
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| {{Wikiquote}}
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| *{{MacTutor Biography|id=Weierstrass}}
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| *{{MathGenealogy |id=7486}}
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| * [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]''.
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| *{{gutenberg author | id=Karl_Weierstrass | name=Karl Weierstrass }}
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| {{Copley Medallists 1851-1900}}
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| {{Authority control|PND=11876618X|LCCN=n/84/806249|VIAF=36999173|SELIBR=238824}}
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| {{Persondata<!-- Metadata: see [[Wikipedia:Persondata]] -->
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| |NAME= Weierstrass, Karl
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| |ALTERNATIVE NAMES=
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| |SHORT DESCRIPTION= [[Mathematician]]
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| |DATE OF BIRTH= 31 October 1815
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| |PLACE OF BIRTH= [[Ostenfelde]], [[Westphalia]]
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| |DATE OF DEATH= 19 February 1897
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| |PLACE OF DEATH= [[Berlin]], [[Germany]]
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| }}
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| {{DEFAULTSORT:Weierstrass, Karl}}
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| [[Category:1815 births]]
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| [[Category:1897 deaths]]
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| [[Category:19th-century German mathematicians]]
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| [[Category:Mathematical analysts]]
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| [[Category:People from the Province of Westphalia]]
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| [[Category:People from Braniewo]]
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| [[Category:Recipients of the Copley Medal]]
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| [[Category:University of Bonn alumni]]
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| [[Category:University of Königsberg alumni]]
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| [[Category:University of Münster alumni]]
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| [[Category:Humboldt University of Berlin faculty]]
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| [[Category:Berlin Institute of Technology faculty]]
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| [[Category:Foreign Members of the Royal Society]]
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| [[Category:Corresponding Members of the St Petersburg Academy of Sciences]]
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| [[Category:Honorary Members of the St Petersburg Academy of Sciences]]
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| [[Category:German Roman Catholics]]
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