Fungsi hipergeometris

Fungsi hipergeometris
Fungsi hipergeometris biasa	₂F₁(a,b;c;z)
Deret hipergeometris	: ${}_{2}F_{1}(a,b;c;z)=\sum _{n=0}^{\infty }{\frac {(a)_{n}(b)_{n}}{(c)_{n}}}{\frac {z^{n}}{n!}}.$
Rumus Antiturunan	: ${\frac {d}{dz}}\ {}_{2}F_{1}(a,b;c;z)={\frac {ab}{c}}\ {}_{2}F_{1}(a+1,b+1;c+1;z)$ dan lebih umum ${\frac {d^{n}}{dz^{n}}}\ {}_{2}F_{1}(a,b;c;z)={\frac {(a)_{n}(b)_{n}}{(c)_{n}}}\ {}_{2}F_{1}(a+n,b+n;c+n;z)$ In the special case that $c=a+1$ , we have ${\frac {d}{dz}}\ {}_{2}F_{1}(a,b;a+1;z)={\frac {d}{dz}}\ {}_{2}F_{1}(b,a;a+1;z)={\frac {a((1-z)^{-b}-{}_{2}F_{1}(a,b;1+a;z))}{z}}$
Persamaan turunan Fungsi hipergeometris	: $z(1-z){\frac {d^{2}w}{dz^{2}}}+\left[c-(a+b+1)z\right]{\frac {dw}{dz}}-ab\,w=0.$
Pecahan berlanjut Gauus	: ${\frac {{}_{2}F_{1}(a+1,b;c+1;z)}{{}_{2}F_{1}(a,b;c;z)}}={\cfrac {1}{1+{\cfrac {{\frac {(a-c)b}{c(c+1)}}z}{1+{\cfrac {{\frac {(b-c-1)(a+1)}{(c+1)(c+2)}}z}{1+{\cfrac {{\frac {(a-c-1)(b+1)}{(c+2)(c+3)}}z}{1+{\cfrac {{\frac {(b-c-2)(a+2)}{(c+3)(c+4)}}z}{1+{}\ddots }}}}}}}}}}$

Dalam matematika, Fungsi hipergeometris biasa atau Gaussia ₂F₁(a,b;c;z) adalah sebuah fungsi istimewa yang diwakili oleh rangkaian hipergeometris, yang meliputi sebagian besar fungsi istimewa lainnya sebagai kasus spesifik atau pembatasan. Fungsi tersebut adalah solusi dari persamaan diferensial biasa (ODE) linear urutan kedua. Setiap ODE liberal urutan kedua dengan tiga titik tinggal reguler dapat bertransformasi menjadi persamaan tersebut.

Sejarah sunting

Deret hipergeometrik sunting

Rumus diferensiasi sunting

Kasus khusus sunting

Persamaan diferensial hipergeometrik sunting

Rumus integral sunting

Hubungan berdekatan Gauss sunting

Rumus transformasi sunting

Nilai pada poin khusus $z$ sunting

Referensi sunting

Andrews, George E.; Askey, Richard & Roy, Ranjan (1999). Special functions. Encyclopedia of Mathematics and its Applications. 71. Cambridge University Press. ISBN 978-0-521-62321-6. MR 1688958.
Bailey, W.N. (1935). Generalized Hypergeometric Series (PDF). Cambridge University Press. Diarsipkan dari versi asli (PDF) tanggal 2017-06-24. Diakses tanggal 2016-07-23.
Beukers, Frits (2002), Gauss' hypergeometric function Diarsipkan 2006-10-14 di Wayback Machine.. (lecture notes reviewing basics, as well as triangle maps and monodromy)
Olde Daalhuis, Adri B. (2010), "Fungsi hipergeometris", dalam Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W., NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, MR 2723248
Erdélyi, Arthur; Magnus, Wilhelm; Oberhettinger, Fritz & Tricomi, Francesco G. (1953). Higher transcendental functions (PDF). Vol. I. New York – Toronto – London: McGraw–Hill Book Company, Inc. ISBN 978-0-89874-206-0. MR 0058756. Diarsipkan (PDF) dari versi asli tanggal 2011-08-11. Diakses tanggal 2020-07-08.
Gasper, George & Rahman, Mizan (2004). Basic Hypergeometric Series, 2nd Edition, Encyclopedia of Mathematics and Its Applications, 96, Cambridge University Press, Cambridge. ISBN 0-521-83357-4.
Gauss, Carl Friedrich (1813). "Disquisitiones generales circa seriem infinitam $1+{\tfrac {\alpha \beta }{1\cdot \gamma }}~x+{\tfrac {\alpha (\alpha +1)\beta (\beta +1)}{1\cdot 2\cdot \gamma (\gamma +1)}}~x~x+{\mbox{etc.}}$ ". Commentationes societatis regiae scientarum Gottingensis recentiores (dalam bahasa Latin). Göttingen. 2.
Gelfand, I. M.; Gindikin, S.G. & Graev, M.I. (2003) [2000]. Selected topics in integral geometry. Translations of Mathematical Monographs. 220. Providence, R.I.: American Mathematical Society. ISBN 978-0-8218-2932-5. MR 2000133.
Gessel, Ira & Stanton, Dennis (1982). "Strange evaluations of hypergeometric series". SIAM Journal on Mathematical Analysis. 13 (2): 295–308. doi:10.1137/0513021. ISSN 0036-1410. MR 0647127.
Goursat, Édouard (1881). "Sur l'équation différentielle linéaire, qui admet pour intégrale la série hypergéométrique". Annales Scientifiques de l'École Normale Supérieure (dalam bahasa French). 10: 3–142. Diakses tanggal 2008-10-16.
Heckman, Gerrit & Schlichtkrull, Henrik (1994). Harmonic Analysis and Special Functions on Symmetric Spaces. San Diego: Academic Press. ISBN 0-12-336170-2. (part 1 treats hypergeometric functions on Lie groups)
Hille, Einar (1976). Ordinary differential equations in the complex domain . Dover. ISBN 0-486-69620-0.
Ince, E. L. (1944). Ordinary Differential Equations. Dover Publications.
Klein, Felix (1981). Vorlesungen über die hypergeometrische Funktion. Grundlehren der Mathematischen Wissenschaften (dalam bahasa German). 39. Berlin, New York: Springer-Verlag. ISBN 978-3-540-10455-1. MR 0668700.
Koepf, Wolfram (1995). "Algorithms for m-fold hypergeometric summation". Journal of Symbolic Computation. 20 (4): 399–417. doi:10.1006/jsco.1995.1056. ISSN 0747-7171. MR 1384455.
Kummer, Ernst Eduard (1836). "Über die hypergeometrische Reihe $1+{\tfrac {\alpha \cdot \beta }{1\cdot \gamma }}~x+{\tfrac {\alpha (\alpha +1)\beta (\beta +1)}{1\cdot 2\cdot \gamma (\gamma +1)}}x^{2}+{\tfrac {\alpha (\alpha +1)(\alpha +2)\beta (\beta +1)(\beta +2)}{1\cdot 2\cdot 3\cdot \gamma (\gamma +1)(\gamma +2)}}x^{3}+{\mbox{etc.}}$ ". Journal für die reine und angewandte Mathematik (dalam bahasa German). 15: 39–83, 127–172. ISSN 0075-4102.
Lavoie, J. L.; Grondin, F.; Rathie, A.K. (1996). "Generalizations of Whipple's theorem on the sum of a ₃F₂". J. Comput. Appl. Math. 72: 293–300.
Press, W.H.; Teukolsky, S.A.; Vetterling, W.T. & Flannery, B.P. (2007). "Section 6.13. Hypergeometric Functions". Numerical Recipes: The Art of Scientific Computing (edisi ke-3rd). New York: Cambridge University Press. ISBN 978-0-521-88068-8.
Rakha, M.A.; Rathie, Arjun K. (2011). "Extensions of Euler's type-II transformation and Saalschutz's theorem". Bull. Korean Math. Soc. 48 (1): 151–156.
Rathie, Arjun K.; Paris, R.B. (2007). "An extension of the Euler's-type transformation for the 3F2 series". Far East J. Math. Sci. 27 (1): 43–48.
Riemann, Bernhard (1857). "Beiträge zur Theorie der durch die Gauss'sche Reihe F(α, β, γ, x) darstellbaren Functionen". Abhandlungen der Mathematischen Classe der Königlichen Gesellschaft der Wissenschaften zu Göttingen (dalam bahasa German). Göttingen: Verlag der Dieterichschen Buchhandlung. 7: 3–22. (a reprint of this paper can be found in "All publications of Riemann" (PDF). )
Slater, Lucy Joan (1960). Confluent hypergeometric functions . Cambridge, UK: Cambridge University Press. MR 0107026.
Slater, Lucy Joan (1966). Generalized hypergeometric functions. Cambridge, UK: Cambridge University Press. ISBN 0-521-06483-X. MR 0201688. (there is a 2008 paperback with ISBN 978-0-521-09061-2)
Vidunas, Raimundas (2005). "Transformations of some Gauss hypergeometric functions". Journal of Symbolic Computation. 178: 473–487. arXiv:math/0310436  . doi:10.1016/j.cam.2004.09.053.
Wall, H.S. (1948). Analytic Theory of Continued Fractions. D. Van Nostrand Company, Inc.
Whittaker, E.T. & Watson, G.N. (1927). A Course of Modern Analysis. Cambridge, UK: Cambridge University Press.
Yoshida, Masaaki (1997). Hypergeometric Functions, My Love: Modular Interpretations of Configuration Spaces. Braunschweig – Wiesbaden: Friedr. Vieweg & Sohn. ISBN 3-528-06925-2. MR 1453580.