查看“︁阿佩尔函数”︁的源代码

'''阿佩尔函数'''是法国数学家（Paul Apell)在1880年为推广[[超几何函数|高斯超几何函数]]而创建的一组雙变数函数，定义如下
[[File:Appell function F1.gif|thumb|300px|阿佩尔函数——F1]]


:<math>
F_1(a,b_1,b_2,c;x,y) = \sum_{m,n=0}^\infty \frac{(a)_{m+n} (b_1)_m (b_2)_n} {(c)_{m+n} \,m! \,n!} \,x^m y^n ~,
</math>


:<math>
F_2(a,b_1,b_2,c_1,c_2;x,y) = \sum_{m,n=0}^\infty \frac{(a)_{m+n} (b_1)_m (b_2)_n} {(c_1)_m (c_2)_n \,m! \,n!} \,x^m y^n ~,
</math>
 

:<math>
F_3(a_1,a_2,b_1,b_2,c;x,y) = \sum_{m,n=0}^\infty \frac{(a_1)_m (a_2)_n (b_1)_m (b_2)_n} {(c)_{m+n} \,m! \,n!} \,x^m y^n ~,
</math>



:<math>
F_4(a,b,c_1,c_2;x,y) = \sum_{m,n=0}^\infty \frac{(a)_{m+n} (b)_{m+n}} {(c_1)_m (c_2)_n \,m! \,n!} \,x^m y^n ~.
</math>


其中的符号<math>:(a)_{m+n}</math>是[[遞進階乘與遞降階乘|阶乘幂]]

阿佩尔函数是[[嫪丽切拉函数]]和[[Kampé_de_Fériet函数]]的特例。

==归递关系==
:<math>
(a-b_1-b_2) F_1(a,b_1,b_2,c; x,y) - a \,F_1(a+1,b_1,b_2,c; x,y) + b_1 F_1(a,b_1+1,b_2,c; x,y) + b_2 F_1(a,b_1,b_2+1,c; x,y) = 0 ~,
</math>

:<math>
c \,F_1(a,b_1,b_2,c; x,y) - (c-a) F_1(a,b_1,b_2,c+1; x,y) - a \,F_1(a+1,b_1,b_2,c+1; x,y) = 0 ~,
</math>

:<math>
c \,F_1(a,b_1,b_2,c; x,y) + c(x-1) F_1(a,b_1+1,b_2,c; x,y) - (c-a)x \,F_1(a,b_1+1,b_2,c+1; x,y) = 0 ~,
</math>

:<math>
c \,F_1(a,b_1,b_2,c; x,y) + c(y-1) F_1(a,b_1,b_2+1,c; x,y) - (c-a)y \,F_1(a,b_1,b_2+1,c+1; x,y) = 0 ~.
</math>

其它式子<ref>例如：<math>(y-x) F_1(a, b_1+1, b_2+1,c,x,y) = y \, F_1(a,b_1,b_2+1,c,x,y) - x \, F_1(a,b_1+1,b_2,c,x,y)</math></ref>可從這四個關係導出。


:<math>
c \,F_3(a_1,a_2,b_1,b_2,c; x,y) + (a_1+a_2-c) F_3(a_1,a_2,b_1,b_2,c+1; x,y) - a_1 F_3(a_1+1,a_2,b_1,b_2,c+1; x,y) - a_2 F_3(a_1,a_2+1,b_1,b_2,c+1; x,y) = 0 ~,
</math>

:<math>
c \,F_3(a_1,a_2,b_1,b_2,c; x,y) - c \,F_3(a_1+1,a_2,b_1,b_2,c; x,y) + b_1 x \,F_3(a_1+1,a_2,b_1+1,b_2,c+1; x,y) = 0 ~,
</math>

:<math>
c \,F_3(a_1,a_2,b_1,b_2,c; x,y) - c \,F_3(a_1,a_2+1,b_1,b_2,c; x,y) + b_2 y \,F_3(a_1,a_2+1,b_1,b_2+1,c+1; x,y) = 0 ~,
</math>

:<math>
c \,F_3(a_1,a_2,b_1,b_2,c; x,y) - c \,F_3(a_1,a_2,b_1+1,b_2,c; x,y) + a_1 x \,F_3(a_1+1,a_2,b_1+1,b_2,c+1; x,y) = 0 ~,
</math>

:<math>
c \,F_3(a_1,a_2,b_1,b_2,c; x,y) - c \,F_3(a_1,a_2,b_1,b_2+1,c; x,y) + a_2 y \,F_3(a_1,a_2+1,b_1,b_2+1,c+1; x,y) = 0 ~.
</math>

==导数与微分方程==

:<math>
\frac {\partial} {\partial x} F_1(a,b_1,b_2,c; x,y) = \frac {a b_1} {c} F_1(a+1,b_1+1,b_2,c+1; x,y) ~,
</math>

:<math>
\frac {\partial} {\partial y} F_1(a,b_1,b_2,c; x,y) = \frac {a b_2} {c} F_1(a+1,b_1,b_2+1,c+1; x,y) ~.
</math>


:<math>
\left( x(1-x) \frac {\partial^2} {\partial x^2} + y(1-x) \frac {\partial^2} 
{\partial x \partial y} + [c - (a+b_1+1) x] \frac {\partial} {\partial x} - b_1 y 
\frac {\partial} {\partial y} - a b_1 \right) F_1(x,y) = 0 ~,
</math>

:<math>
\left( y(1-y) \frac {\partial^2} {\partial y^2} + x(1-y) \frac {\partial^2} 
{\partial x \partial y} + [c - (a+b_2+1) y] \frac {\partial} {\partial y} - b_2 x 
\frac {\partial} {\partial x} - a b_2 \right) F_1(x,y) = 0 ~.
</math>


:<math>
\frac {\partial} {\partial x} F_3(a_1,a_2,b_1,b_2,c; x,y) = \frac {a_1 b_1} {c} F_3(a_1+1,a_2,b_1+1,b_2,c+1; x,y) ~,
</math>

:<math>
\frac {\partial} {\partial y} F_3(a_1,a_2,b_1,b_2,c; x,y) = \frac {a_2 b_2} {c} F_3(a_1,a_2+1,b_1,b_2+1,c+1; x,y) ~.
</math>



:<math>
\left( x(1-x) \frac {\partial^2} {\partial x^2} + y \frac {\partial^2} 
{\partial x \partial y} + [c - (a_1+b_1+1) x] \frac {\partial} {\partial x} - 
a_1 b_1 \right) F_3(x,y) = 0 ~,
</math>

:<math>
\left( y(1-y) \frac {\partial^2} {\partial y^2} + x \frac {\partial^2} 
{\partial x \partial y} + [c - (a_2+b_2+1) y] \frac {\partial} {\partial y} - 
a_2 b_2 \right) F_3(x,y) = 0 ~.
</math>

==积分关系==
:<math>
F_1(a,b_1,b_2,c; x,y) = \frac{\Gamma(c)} {\Gamma(a) \Gamma(c-a)} 
\int_0^1 t^{a-1} (1-t)^{c-a-1} (1-xt)^{-b_1} (1-yt)^{-b_2} \,\mathrm{d}t, 
\quad \real \,c > \real \,a > 0 ~.
</math>
==特例==
:<math>
F(\phi,k) = \int_0^\phi \frac{\mathrm{d} \theta} 
{\sqrt{1 - k^2 \sin^2 \theta}} = \sin \phi \,F_1(\tfrac 1 2, \tfrac 1 2, \tfrac 1 2, \tfrac 3 2; \sin^2 \phi, k^2 \sin^2 \phi), \quad |\real \,\phi| < \frac \pi 2 ~,
</math>

:<math>
E(\phi, k) = \int_0^\phi \sqrt{1 - k^2 \sin^2 \theta} \,\mathrm{d} \theta = \sin \phi \,F_1(\tfrac 1 2, \tfrac 1 2, -\tfrac 1 2, \tfrac 3 2; \sin^2 \phi, k^2 \sin^2 \phi), \quad |\real \,\phi| < \frac \pi 2 ~,
</math>

:<math>
\Pi(n,k) = \int_0^{\pi/2} \frac{\mathrm{d} \theta} {(1 - n \sin^2 \theta) 
\sqrt{1 - k^2 \sin^2 \theta}} = \frac {\pi} {2} \,F_1(\tfrac 1 2, 1, \tfrac 1 2, 1; 
n,k^2) ~.
</math>

==参见==
[[Q阿佩尔函数]]

==参考文献==
<references/>
* {{cite journal | last= Appell | first= Paul | authorlink= Paul Émile Appell | title= Sur les séries hypergéométriques de deux variables et sur des équations différentielles linéaires aux dérivées partielles | language= fr | journal= Comptes rendus hebdomadaires des séances de l'Académie des sciences | year= 1880 | volume= 90 | pages= 296–298 and 731–735 | jfm= 12.0296.01 | ref= harv}} (see also "Sur la série F<sub>3</sub>(α,α',β,β',γ; x,y)" in ''C. R. Acad. Sci.'' '''90''', pp. 977–980)
* {{cite journal | last= Appell | first= Paul | title= Sur les fonctions hypergéométriques de deux variables | language= fr | url= http://portail.mathdoc.fr/JMPA/afficher_notice.php?id=JMPA_1882_3_8_A8_0 | journal= [[Journal de Mathématiques Pures et Appliquées]] | series= (3ème série) | year= 1882 | volume= 8 | pages= 173–216 | ref= harv | access-date= 2015-04-04 | archive-date= 2013-04-12 | archive-url= https://archive.today/20130412204752/http://portail.mathdoc.fr/JMPA/afficher_notice.php?id=JMPA_1882_3_8_A8_0 }}
* {{cite book | last1= Appell | first1= Paul | last2= Kampé de Fériet | first2= Joseph | author2-link= Joseph Kampé de Fériet | title= Fonctions hypergéométriques et hypersphériques; Polynômes d'Hermite | language= fr | location= Paris | publisher= Gauthier–Villars | year= 1926 | jfm= 52.0361.13 | ref= harv}} (see p.&nbsp;14)
* {{dlmf | id= 16.13 | first= R. A. | last= Askey | first2= Adri B. Olde | last2= Daalhuis}}
* {{cite book | first1= H. | last1= Bateman | author1-link= Harry Bateman | first2= A. | last2= Erdélyi | author2-link= Arthur Erdélyi | title= Higher Transcendental Functions, Vol. I | url= http://apps.nrbook.com/bateman/Vol1.pdf | format= PDF | location= New York | publisher= McGraw–Hill | year= 1953 | ref= harv | access-date= 2015-04-04 | archive-date= 2011-08-11 | archive-url= https://web.archive.org/web/20110811153220/http://apps.nrbook.com/bateman/Vol1.pdf | dead-url= no }} (see p.&nbsp;224)
* {{cite book | last1= Gradshteyn | first1= Izrail' Solomonovich | last2= Ryzhik | first2= Iosif Moiseevich | title= Tablitsy integralov, summ, ryadov i proizvedeniy [Tables of integrals, sums, series and products] | language= ru | edition= 5th | location= Moscow | publisher= Nauka | year= 1971 | ref= harv}} (see Chapter 9.18)
* {{cite journal | last= Humbert | first= Pierre | authorlink= Pierre Humbert (mathematician) | title= Sur les fonctions hypercylindriques | language= fr | journal= Comptes rendus hebdomadaires des séances de l'Académie des sciences | year= 1920 | volume= 171 | pages= 490–492 | jfm= 47.0348.01 | ref= harv}}
* {{cite journal | last= Lauricella | first= Giuseppe | authorlink= Giuseppe Lauricella | title= Sulle funzioni ipergeometriche a più variabili | language= it | journal= [[Rendiconti del Circolo Matematico di Palermo]] | year= 1893 | volume= 7 | pages= 111–158 | doi= 10.1007/BF03012437 | jfm= 25.0756.01 | ref= harv}}
* {{cite journal | last= Picard | first= Émile | authorlink= Charles Émile Picard | title= Sur une extension aux fonctions de deux variables du problème de Riemann relativ aux fonctions hypergéométriques | language= fr | url= http://www.numdam.org/item?id=ASENS_1881_2_10__305_0 | journal= Annales scientifiques de l'École Normale Supérieure | series= (2ème série) | year= 1881 | volume= 10 | pages= 305–322 | jfm= 13.0389.01 | ref= harv | access-date= 2015-04-04 | archive-date= 2015-01-21 | archive-url= https://web.archive.org/web/20150121134005/http://www.numdam.org/item?id=ASENS_1881_2_10__305_0 | dead-url= no }} (see also ''C. R. Acad. Sci.'' '''90''' (1880), pp. 1119–1121 and 1267–1269)
* {{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)


[[Category:超幾何函數]]