Fungsi hipergeometris: Perbedaan antara revisi
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Baris 1:
{{Dalam perbaikan}}
Dalam [[matematika]], '''fungsi hipergeometris''' biasa atau Gaussia <sub>2</sub>''F''<sub>1</sub>(''a'',''b'';''c'';''z'') adalah sebuah [[fungsi istimewa]] yang diwakili oleh '''rangkaian hipergeometris''', yang meliputi sebagian besar fungsi istimewa lainnya sebagai [[kasus istimewa|kasus spesifik]] atau [[kasus pembatasan (matematika)|pembatasan]]. Fungsi tersebut adalah solusi dari [[persamaan diferensial biasa]] (ODE) [[fungsi linear|linear]] urutan kedua. Setiap ODE liberal urutan kedua dengan tiga [[titik tinggal reguler]] dapat bertransformasi menjadi persamaan tersebut.▼
{|class="wikitable" bgcolor="#ffffff" cellpadding="5" align="right" style="margin-left:10px" width="320"
!bgcolor=#e7dcc3 colspan=2|'''Fungsi hipergeometris'''
|-
|bgcolor=#e7dcc3|Fungsi hipergeometris biasa||<sub>2</sub>''F''<sub>1</sub>(''a'',''b'';''c'';''z'')
|-
|bgcolor=#e7dcc3|Deret hipergeometris||:<math>{}_2F_1(a,b;c;z) = \sum_{n=0}^\infty \frac{(a)_n (b)_n}{(c)_n} \frac{z^n}{n!}.</math>
|-
|bgcolor=#e7dcc3|Rumus Antiturunan||: <math>
\frac{d }{dz} \ {}_2F_1(a,b;c;z) = \frac{ab}{c} \ {}_2F_1(a+1,b+1;c+1;z)
</math>
dan lebih umum
: <math>
\frac{d^n }{dz^n} \ {}_2F_1(a,b;c;z) = \frac{(a)_n (b)_n}{(c)_n} \ {}_2F_1(a+n,b+n;c+n;z)
</math>
In the special case that <math>c = a + 1</math>, we have
: <math>
\frac{d }{dz} \ {}_2F_1(a,b;a+1;z) = \frac{d }{dz} \ {}_2F_1(b,a;a+1;z) = \frac{a((1-z)^{-b} - {}_2F_1(a,b;1+a;z))}{z}
</math>
|-
|bgcolor=#e7dcc3|Persamaan turunan Fungsi hipergeometris||:<math>z(1-z)\frac {d^2w}{dz^2} + \left[c-(a+b+1)z \right] \frac {dw}{dz} - ab\,w = 0.</math>
|-
|bgcolor=#e7dcc3|Pecahan berlanjut Gauus||:<math>\frac{{}_2F_1(a+1,b;c+1;z)}{{}_2F_1(a,b;c;z)} = \cfrac{1}{1 + \cfrac{\frac{(a-c)b}{c(c+1)} z}{1 + \cfrac{\frac{(b-c-1)(a+1)}{(c+1)(c+2)} z}{1 + \cfrac{\frac{(a-c-1)(b+1)}{(c+2)(c+3)} z}{1 + \cfrac{\frac{(b-c-2)(a+2)}{(c+3)(c+4)} z}{1 + {}\ddots}}}}}</math>
|}
▲Dalam [[matematika]], '''
== Sejarah ==
Baris 15 ⟶ 43:
* {{cite book | last1= Andrews | first1= George E. | authorlink= George Andrews (mathematician) | last2= Askey | first2= Richard | last3= Roy | first3= Ranjan | lastauthoramp= yes | title= Special functions | publisher= Cambridge University Press | year= 1999 | series= Encyclopedia of Mathematics and its Applications | volume= 71 | isbn= 978-0-521-62321-6 | mr= 1688958 | ref= harv}}
* {{cite book | last=Bailey | first=W.N. | year=1935 | title=Generalized Hypergeometric Series | publisher=Cambridge University Press | url=http://plouffe.fr/simon/math/Bailey%20W.N.%20Generalized%20Hypergeometric%20Series%20%281964%29%28L%29%28T%29%2859s%29.pdf | ref=harv | access-date=2016-07-23 | archive-url=https://web.archive.org/web/20170624164738/http://plouffe.fr/simon/math/Bailey%20W.N.%20Generalized%20Hypergeometric%20Series%20(1964)(L)(T)(59s).pdf | archive-date=2017-06-24 | url-status=dead }}
* [[Frits Beukers|Beukers, Frits]] (2002), ''[http://www.math.uu.nl/people/beukers/MRIcourse93.ps Gauss' hypergeometric function] {{Webarchive|url=https://web.archive.org/web/20061014164742/http://www.math.uu.nl/people/beukers/MRIcourse93.ps |date=2006-10-14 }}''. (lecture notes reviewing basics, as well as triangle maps and monodromy)
* {{dlmf | first= Adri B. | last= Olde Daalhuis | id= 15}}
* {{cite book | last1= Erdélyi | first1= Arthur | author1-link= Arthur Erdélyi | last2= Magnus | first2= Wilhelm | author2-link= Wilhelm Magnus | last3= Oberhettinger | first3= Fritz | lastauthoramp= yes | last4= Tricomi | first4= Francesco G. | title= Higher transcendental functions | volume= Vol. I | location= New York – Toronto – London | publisher= McGraw–Hill Book Company, Inc. | year= 1953 | isbn= 978-0-89874-206-0 | mr= 0058756 | url= http://apps.nrbook.com/bateman/Vol1.pdf | ref= harv | access-date= 2020-07-08 | archive-date= 2011-08-11 | archive-url= https://web.archive.org/web/20110811153220/http://apps.nrbook.com/bateman/Vol1.pdf | dead-url= no }}
* Gasper, George & [[Mizan Rahman|Rahman, Mizan]] (2004). Basic Hypergeometric Series, 2nd Edition, [[Encyclopedia of Mathematics]] and Its Applications, 96, [[Cambridge University Press]], Cambridge. {{ISBN|0-521-83357-4}}.
* {{cite journal | last= Gauss | first= Carl Friedrich | authorlink= Carl Friedrich Gauss | title= Disquisitiones generales circa seriem infinitam <math> 1 + \tfrac {\alpha \beta} {1 \cdot \gamma} ~x + \tfrac {\alpha (\alpha+1) \beta (\beta+1)} {1 \cdot 2 \cdot \gamma (\gamma+1)} ~x~x + \mbox{etc.} </math> | language= Latin | url= https://books.google.com/books?id=uDMAAAAAQAAJ | location= Göttingen | journal= Commentationes societatis regiae scientarum Gottingensis recentiores | year= 1813 | volume= 2 | ref= harv}}
* {{cite book | last1= Gelfand | first1= I. M. | last2= Gindikin | first2= S.G. | lastauthoramp= yes | last3= Graev | first3= M.I. | title= Selected topics in integral geometry | origyear= 2000 | url= https://books.google.com/books?isbn=0821829327 | publisher= [[American Mathematical Society]] | location= Providence, R.I. | series= Translations of Mathematical Monographs | year= 2003 | volume= 220 | isbn= 978-0-8218-2932-5 | mr= 2000133 | ref= harv}}
* {{cite journal | last1= Gessel | first1= Ira | lastauthoramp= yes | last2= Stanton | first2= Dennis | title= Strange evaluations of hypergeometric series | journal= SIAM Journal on Mathematical Analysis | year= 1982 | volume= 13 | issue= 2 | pages= 295–308 | issn= 0036-1410 | doi= 10.1137/0513021 | mr= 647127 | ref= harv}}
* {{cite journal |last=Goursat |first=Édouard |authorlink=Édouard Goursat |title=Sur l'équation différentielle linéaire, qui admet pour intégrale la série hypergéométrique |language=French |url=http://www.numdam.org/item?id=ASENS_1881_2_10__S3_0 |accessdate=2008-10-16 |journal=Annales Scientifiques de l'École Normale Supérieure |volume=10 |year=1881 |pages=3–142 |ref=harv }}
* {{cite book | last1= Heckman | first1= Gerrit | last2= Schlichtkrull | first2= Henrik | lastauthoramp= yes | title= Harmonic Analysis and Special Functions on Symmetric Spaces | url= https://archive.org/details/harmonicanalysis0000heck | location= San Diego | publisher= Academic Press | year= 1994 | isbn= 0-12-336170-2 | ref= harv}} (part 1 treats hypergeometric functions on Lie groups)
* {{cite book | last=Hille | first=Einar | year=1976 | title=Ordinary differential equations in the complex domain | url=https://archive.org/details/ordinarydifferen00hill_0 | url-access=registration | publisher=Dover | ISBN=0-486-69620-0 | ref=harv}}
*{{cite book|last=Ince|first=E. L.|authorlink=E. L. Ince|title=Ordinary Differential Equations|url=https://archive.org/details/in.ernet.dli.2015.476224|publisher= Dover Publications|year= 1944}}
Baris 35 ⟶ 63:
* {{cite journal | last= Riemann | first= Bernhard | author-link= Bernhard Riemann | year= 1857 | title= Beiträge zur Theorie der durch die Gauss'sche Reihe ''F(α, β, γ, x)'' darstellbaren Functionen | journal= Abhandlungen der Mathematischen Classe der Königlichen Gesellschaft der Wissenschaften zu Göttingen | language= German | volume= 7 | pages= 3–22 | publisher= Verlag der Dieterichschen Buchhandlung | location= Göttingen | url= http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=GDZPPN002018691 | ref= harv}} (a reprint of this paper can be found in {{cite web|url= http://www.emis.de/classics/Riemann/PFunct.pdf |title=All publications of Riemann }})
* {{cite book | last= Slater | first= Lucy Joan | authorlink= Lucy Joan Slater | title= Confluent hypergeometric functions | url= https://archive.org/details/confluenthyperge0000slat | url-access= registration | location= Cambridge, UK | publisher= Cambridge University Press | year= 1960 | mr= 0107026 | ref= harv}}
* {{cite book | last= Slater | first= Lucy Joan |authorlink=Lucy Joan Slater| title= Generalized hypergeometric functions | url= https://archive.org/details/generalizedhyper0000unse_g0b6 | location= Cambridge, UK | publisher= Cambridge University Press | year= 1966 | isbn= 0-521-06483-X | mr= 0201688 | ref= harv}} (there is a 2008 paperback with {{ISBN|978-0-521-09061-2}})
* {{cite journal | last= Vidunas | first = Raimundas | title = Transformations of some Gauss hypergeometric functions | year = 2005 | journal = Journal of Symbolic Computation | volume = 178 | pages = 473–487 | doi=10.1016/j.cam.2004.09.053 | ref=harv| arxiv = math/0310436 }}
* {{cite book | last= Wall | first= H.S. | title= Analytic Theory of Continued Fractions | url= https://archive.org/details/dli.ernet.16804 | publisher= D. Van Nostrand Company, Inc. | year= 1948 | ref= harv}}
* {{cite book | last1= Whittaker | first1= E.T. | last2= Watson | first2= G.N. | lastauthoramp= yes | title= A Course of Modern Analysis | location= Cambridge, UK | publisher= Cambridge University Press | year= 1927 | ref= harv}}
* {{cite book | last= Yoshida | first= Masaaki | title= Hypergeometric Functions, My Love: Modular Interpretations of Configuration Spaces | location= Braunschweig – Wiesbaden | publisher= Friedr. Vieweg & Sohn | year= 1997 | isbn= 3-528-06925-2 | mr= 1453580 | ref= harv}}
Baris 43 ⟶ 71:
==Pranala luar==
* {{springer|title=Hypergeometric function|id=p/h048450}}
* John Pearson, [http://people.maths.ox.ac.uk/porterm/research/pearson_final.pdf Computation of Hypergeometric Functions] {{Webarchive|url=https://web.archive.org/web/20210507013830/http://people.maths.ox.ac.uk/porterm/research/pearson_final.pdf |date=2021-05-07 }} ([[University of Oxford]], MSc Thesis)
* Marko Petkovsek, Herbert Wilf and Doron Zeilberger, [https://web.archive.org/web/20060129095451/http://www.cis.upenn.edu/~wilf/AeqB.html The book "A = B"] (freely downloadable)
* {{MathWorld |title=Hypergeometric Function |urlname= HypergeometricFunction}}
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