Lihat sumber Pengguna:Dedhert.Jr/Uji halaman 01/2

[[Berkas:Witch_of_Agnesi_curves.svg|ka|jmpl|Kurva penyihir Agnesi yang diwarnai hijau, serta lingkaran yang dikonstruksi dengan parameter jari-jari <math>a=1</math>, <math>a=2</math>, <math>a=4</math>, dan <math>a=8</math>.]]
Dalam [[matematika]], '''penyihir Agnesi''' adalah sebuah [[kurva bidang kubik]] yang terdefinisi dari dua titik lingkaran yang berlawanan. Nama kurva ini diperoleh dari seorang matematikawan berkebangsaan Italia bernama [[Maria Gaetana Agnesi]], dan juga diperoleh dari kata Italia untuk [[Sheet (sailing)|sailing sheet]] yang salah diterjemahkan. Kurva ini sudah dipelajari oleh [[Pierre de Fermat|Fermat]], [[Luigi Guido Grandi|Grandi]] dan [[Isaac Newton|Newton]] sebelum Agnesi.

[[Grafik]] turunan dari fungsi [[Fungsi invers trigonometri|arctangen]] membentuk sebuah contoh dari penyihir Agnesi. Kurva penyihir Agnesi mempunyai penerapan dalam [[teori probabilitas]] (peluang), yakni sebagai [[fungsi kerapatan probabilitas]] dari [[distribusi Cauchy]]. Kurva ini juga menimbulkan [[fenomena Runge]] dalam aproksimasi fungsi dengan menggunakan polinomial, yang dipakai untuk mengaproksimasi distribusi energi dari [[garis spektral]] serta menggambarkan bentuk bukit.

Kurva penyihir Agnesi bersinggungan dengan lingkaran yang didefinisikan di salah satu dari dua titik yang didefinisikan, serta kurva ini [[asimtotik]] dengan [[Garis singgung dengan lingkaran|garis yang singgung dengan lingkaran]] di titik yang lain. Kurva ini mempunyai satu buah [[Verteks (kurva)|verteks]] (sebuah titik dari kelengkungan ekstrem) di titik singgung dengan lingkaran yang terdefinisi darinya, yang juga berarti bahwa [[Lingkaran osilasi|lingkaran itu terosilasi]] di titik tersebut. Kurva ini mempunyai dua buah [[titik belok]] terhingga dan satu buah titik belok tak terhingga. Luas di antara kurva penyihir Agnesi dengan garis asimtotik sama dengan empat kali dari luas lingkaran yang terdefinisikan, serta volume kurva yang diputar di sekitar garis tersebut sama dengan dua kalinya volume [[torus]] dari putaran lingkaran yang terdefinisikan.

== Konstruksi ==
[[Berkas:Witch_of_Agnesi,_construction.svg|jmpl|300x300px|Kurva penyihir Agnesi ''MP'' dengan titik yang berlabel]]
[[Berkas:Agnesi.gif|ka|jmpl|Animasi yang memperlihatkan konstruksi dari kurva penyihir Agnesi]]
Cara mengonstruksi kurva ini dimulai dengan sebarang dua titik ''O'' dan ''M'', dan kemudian gambar sebuah lingkaran dengan ''OM'' sebagai diameternya. Untuk sebarang titik lain ''A'' di lingkaran, misalkan ''N'' adalah titik perpotongan dari sebuah [[garis sekan]] ''OA'' dan sebuah garis menyinggung di titik ''M''. Misalkan ''P'' adalah titik perpotongan dari sebuah garis yang tegak lurus dengan ''OM'' melalui ''A'', dan sebuah garis sejajar dengan ''OM'' melalui ''N''. Maka ''P'' terletak di kurva penyihir Agnesi. Kurva ini terdiri dari semua titik ''P'' yang dapat dikonstruksi dengan cara tadi dengan memilih sebarang titik ''O'' dan ''M''.{{r|eagles}} It includes, as a limiting case, the point ''M'' itself.

== Persamaan ==
Misalkan ''O'' adalah [[Titik asal (geometri)|titik asal]] dan misalkan ''M'' adalah sebuah titik yang terletak di sumbu-<math>y</math> positif, sehingga membentuk diameter lingkaran ''OM'' dan mempunyai {{nowrap|jari-jari <math>a</math>.}} Maka, kurva yang dikonstruksi dari ''O'' {{nowrap|dan ''M''}} mempunyai persamaan Cartesius{{r|lawrence|yates}}<math display="block">y = \frac{8a^3}{x^2+4a^2}.</math>Persamaan di atas dapat disederhanakan dengan memilih {{nowrap|<math>a=\tfrac12</math>,}} sehingga bentuknya menjadi <math display="block">y = \frac{1}{x^2+1}.</math>atau sebagai [[persamaan aljabar]] kubik dengan menghabiskan penyebut pada kedua ruas.<math display="block">(x^2+1)y=1.</math>Kurva dengan bentuk yang telah disederhanakan di atas menggambarkan [[Grafik fungsi|grafik]] [[turunan]] dari fungsi [[Fungsi invers trigonometri|arctangen]].{{r|calc}}

Kurva penyihir Agnesi juga digambarkan dengan menggunakan [[persamaan parametrik]], dengan parameter {{mvar|θ}} adalah sudut di antara ''OM'' dan ''OA'', yang diukur searah jarum jam:{{r|lawrence|yates}}<math display="block">x = 2a \tan \theta,</math> <math display="block">y = 2a \cos ^2 \theta.</math>

== Sifat-sifat ==
Sifat utama dari kurva ini dapat diperoleh menggunakan [[kalkulus integral]]. Luas di antara kurva penyihir Agnesi dengan garis asimtotik sama dengan empat kalinya luas lingkaran, {{nowrap|<math>4\pi a^2</math>.{{r|lawrence|yates|larsen}}}} Volume kurva penyihir Agnesi [[Benda putar|yang diputar]] terhadap garis asimtot {{nowrap|sama dengan <math>4\pi^2a^3</math>,{{r|lawrence}}}} dua kalinya volume torys yang dibentuk dengan memutar lingkaran yang terdefinisikan dari kurva penyihir Agnesi terhadap garis yang sama.{{r|larsen}}

Kurva penyihir Agnesi mempunyai satu buah [[Verteks (kurva)|verteks]] di titik singgung dengan lingkaran yang terdefinisi darinya. Ini mengartikan bahwa titik tersebut adalah satu-satunya titik, dengan [[kurvatur]]<nowiki/>nya mencapai minimum lokal atau maksimum lokal.{{r|egdc}} Lingkaran yang terdefinisi dari kurva penyihir Agnesi juga merupakan [[Lingkaran oskulasi|lingkaran yang oskulasi]] di verteks,{{r|haftendorn}} sebuah lingkaran yang "menyinggung" kurva di titik tersebut dengan membagikan orientasi dan kurvatur yang sama.{{r|kiss}} Karena teroskulasi di verteks kurva, lingkaran tersebut mempunyai [[Kontak (matematika)|kontak orde-tiga]] dengan kurva.{{r|omnibus}}

Kurva penyihir Agnesi mempunyai dua [[titik belok]], yaitu titik<math display="block">\left( \pm\frac{2a}{\sqrt{3}}, \frac{3a}{2}\right)</math>yang korespondensi dengan {{nowrap|sudut <math>\theta=\pm\pi/6</math>.{{r|lawrence|yates}}}} Saat memandangnya sebagai sebuah kurva dalam [[bidang proyektif]], akan ada pula titik belok tak terhingga ketiga, yang terletak di sebuah titik ketika [[garis di takhingga]] dilalui garis asimtotik. Karena salah satu dari titik belok tersebut tak terhingga, kurva penyihir Agnesi mempunyai jumlah minimum yang mungkin dari titik belok real terhingga dari sebarang kurva kubik tak-singular.

Luas terbesar [[persegi panjang]] yang berada di antara kurva penyihir Agnesi dengan garis asimtot sama {{nowrap|dengan <math>4a^2</math>,}} dengan panjang dari persegi panjang adalah jari-jari lingkaran yang terdefinisikan dan lebarnya sama dengan dua kalinya diameter {{nowrap|lingkaran.{{r|larsen}}}}

== Sejarah ==

=== Kajian awal ===
[[Berkas:Witch_of_Agnesi_(Agnesi,_1748).jpg|jmpl|Sebuah ilustrasi dari kurva beserta konstruksinya dalam karya milik Agnesi  pada tahun 1748.{{r|agnesi}}]]
Kurva ini dipelajari [[Pierre de Fermat]] dalam risalah tentang [[Kuadratur (matematika)|kuadratur]] miliknya tahun 1659. Dalam risalah tersebut, Fermat menghitung luas di bawah kurva dan mengatakan bahwa metode yang sama dapat memperluas juga ke [[cissoid dari Diocles]]. Fermat menulis bahwa kurva tersebut ditujukan kepadanya "''ab erudito geometra''" [oleh seorang ahli geometri terpelajar].{{r|fermat}} {{harvtxt|Paradís|Pla|Viader|2008}} menduga ahli geometri yang menunjukkan kurva tersebut kepada Fermat adalah [[Antoine de Laloubère]].{{r|fmq}}

Konstruksi untuk kurva tersebut seperti yang dijelaskan sebelumnya ditemukan oleh {{harvtxt|Grandi|1718}}. Konstruksi ini sudah ditemukan sebelumnya oleh [[Isaac Newton]], tetapi konstruksinya diterbitkan setelah kematiannya, yakni pada tahun 1779.{{r|stigler}} {{harvtxt|Grandi|1718}} juga memberi julukan nama ''versiera'' dalam bahasa Italia, atau ''versoria'' dalam bahasa Latin, untuk kurva.<ref>In his notes to Galileo's "Trattato del moto naturalmente accelerato," Grandi had referred to "quella curva che io descrivo nel mio libro delle quadrature [1703], alla prop. IV, nata da' seni versi, che da me suole chiamarsi ''Versiera'', in latino però ''Versoria''." See Galilei, ''Opere'', 3: 393. One finds the new term in Lorenzo Lorenzini, ''Exercitatio geometrica'', xxxi: "sit pro exemplo curva illa, quam Doctissimus magnusque geometra Guido Grandus versoria nominat."</ref> Dalam bahasa Latin, kata tersebut juga berarti [[Sheet (sailing)|sheet]], the rope which turns the sail, but Grandi may have instead intended merely to refer to the [[versine]] function that appeared in his construction.{{r|larsen|stigler|truesdell|grandi}}

Pada tahun 1748, [[Maria Gaetana Agnesi]] menerbitkan buku cetak pertamanya tentang [[kalkulus]] yang berjudul ''Instituzioni analitiche ad uso della gioventù italiana''.{{r|agnesi}} Di dalam buku tersebut, Agnesi menulis kajian tentang kurva tersebut setelah mengulas dua kurva lainnya. Agnesi mendefinisikan kurva secara geometris sebagai lokus dari titk yang memenuhi proporsi, menentukan persamaan aljabar, serta mencari titik pojok, garis asimtotik, dan titik belok dari kurva tersebut.{{r|struik}}

=== Etymology ===
[[Maria Gaetana Agnesi]] named the curve according to Grandi, ''versiera''.{{r|truesdell|struik}} Coincidentally, at that time in Italy it was common to speak of the [[Devil]], the adversary of God, through other words like ''aversiero'' or ''versiero'', derived from Latin ''adversarius''. ''Versiera'', in particular, was used to indicate the wife of the devil, or "witch".{{r|fanfani}} Because of this, Cambridge professor [[John Colson]] mistranslated the name of the curve as "witch".{{r|mulcrone}} Different modern works about Agnesi and about the curve suggest slightly different guesses how exactly this mistranslation happened.{{r|singh|darling}} [[Dirk Jan Struik|Struik]] mentions that:{{r|struik}}
{{blockquote|The word [''versiera''] is derived from Latin ''vertere'', to turn, but is also an abbreviation of Italian ''avversiera'', female devil. Some wit in England once translated it 'witch', and the silly pun is still lovingly preserved in most of our textbooks in English language. ... The curve had already appeared in the writings of [[Pierre de Fermat|Fermat]] (''Oeuvres'', I, 279–280; III, 233–234) and of others; the name ''versiera'' is from Guido Grandi (''Quadratura circuli et hyperbolae'', Pisa, 1703). The curve is type 63 in [[Isaac Newton|Newton]]'s classification. ... The first to use the term 'witch' in this sense may have been B. Williamson, ''Integral calculus'', 7 (1875), 173;{{r|oed}} see ''Oxford English Dictionary''.}}
On the other hand, [[Stephen Stigler]] suggests that  Grandi himself "may have been indulging in a play on words", a double pun connecting the devil to the versine and the sine function to the shape of the female breast (both of which can be written as "seno" in Italian).{{r|stigler}}

== Penerapan ==
A scaled version of the curve is the [[probability density function]] of the [[Cauchy distribution]]. This is the probability distribution on the [[random variable]] <math>x</math> determined by the following [[Experiment (probability theory)|random experiment]]: for a fixed point <math>p</math> above the {{nowrap|<math>x</math>-axis,}} choose uniformly at random a line {{nowrap|through <math>p</math>,}} and let <math>x</math> be the coordinate of the point where this random line crosses the axis. The Cauchy distribution has a peaked distribution visually resembling the [[normal distribution]], but its [[Heavy-tailed distribution|heavy tails]] prevent it from having an [[expected value]] by the usual definitions, despite its symmetry. In terms of the witch itself, this means that the {{nowrap|<math>x</math>-coordinate}} of the [[centroid]] of the region between the curve and its asymptotic line is not well-defined, despite this region's symmetry and finite area.{{r|stigler|alexander}}

In [[numerical analysis]], when approximating functions using [[polynomial interpolation]] with equally spaced interpolation points, it may be the case for some functions that using more points creates worse approximations, so that the interpolation diverges from the function it is trying to approximate rather than converging to it. This paradoxical behavior is called [[Runge's phenomenon]]. It was first discovered by [[Carl David Tolmé Runge]] for Runge's function {{nowrap|<math>y=1/(1+25x^2)</math>,}} another scaled version of the witch of Agnesi, when interpolating this function over the {{nowrap|interval <math>[-1,1]</math>.}} The same phenomenon occurs for the witch <math>y=1/(1+x^2)</math> itself over the wider {{nowrap|interval <math>[-5,5]</math>.{{r|runge}}}}

The witch of Agnesi approximates the [[spectral energy distribution]] of [[spectral lines]], particularly [[X-ray]] lines.{{r|spencer}}

The cross-section of a smooth [[hill]] has a similar shape to the witch.{{r|coppin}} Curves with this shape have been used as the generic topographic obstacle in a flow in mathematical modeling.{{r|snyder|lamb}} [[Soliton|Solitary waves]] in deep water can also take this shape.{{r|benjamin|noonan}}

A version of this curve was used by [[Gottfried Wilhelm Leibniz]] to derive the [[Leibniz formula for π|Leibniz formula for {{pi}}]]. This formula, the [[infinite series]]<math display="block">\frac{\pi}{4} = 1 \,-\, \frac{1}{3} \,+\, \frac{1}{5} \,-\, \frac{1}{7} \,+\, \frac{1}{9} \,-\, \cdots,</math>can be derived by equating the area under the curve with the integral of the {{nowrap|function <math>1/(1+x^2)</math>,}} using the [[Taylor series]] expansion of this function as the infinite [[geometric series]] {{nowrap|<math>1-x^2+x^4-x^6+\cdots</math>,}} and integrating term-by-term.{{r|yates}}

== Catatan ==
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<ref name=fanfani>[[Pietro Fanfani]], ''Vocabolario dell'uso toscano'', [https://books.google.com/books?id=SOczmy2F2y0C&pg=PA334 p. 334]</ref>

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 | doi = 10.1364/josa.30.000415
 | issue = 9
 | journal = Journal of the Optical Society of America
 | page = 415
 | title = Properties of the Witch of Agnesi—Application to Fitting the Shapes of Spectral Lines
 | volume = 30| bibcode = 1940JOSA...30..415S
 }}</ref>

<ref name=stigler>{{citation
 | last = Stigler | first = Stephen M. | author-link = Stephen Stigler
 | date = August 1974
 | doi = 10.1093/biomet/61.2.375
 | issue = 2
 | journal = [[Biometrika]]
 | jstor = 2334368
 | mr = 0370838
 | pages = 375–380
 | title = Studies in the History of Probability and Statistics. XXXIII. Cauchy and the Witch of Agnesi: An Historical Note on the Cauchy Distribution
 | volume = 61}}</ref>

<ref name=struik>A translation of Agnesi's work on this curve can be found in: {{citation|title=A Source Book in Mathematics, 1200–1800|last=Struik|first=Dirk J.|author-link=Dirk Jan Struik|year=1969|publisher=Harvard University Press|location=Cambridge, Massachusetts|pages=178–180|url=https://archive.org/stream/B-001-001-112#page/n199/mode/2up}}</ref>

<ref name=truesdell>{{citation |first=C. |last=Truesdell |author-link=Clifford Truesdell|title=Correction and Additions for "Maria Gaetana Agnesi" |journal=Archive for History of Exact Sciences |volume=43 |issue=4 |year=1991 |pages=385–386 |doi=10.1007/BF00374764 |quote=[…] nata da' seni versi, che da me suole chiamarsi la ''Versiera'' in latino però ''Versoria'' […]|doi-access=free }}</ref>

<ref name=yates>{{citation|first=Robert C.|last=Yates|title=Curves and their Properties|publisher=National Council of Teachers of Mathematics|series=Classics in Mathematics Education|volume=4|year=1954|url=https://files.eric.ed.gov/fulltext/ED100648.pdf|contribution=Witch of Agnesi|pages=237–238}}</ref>}}