Notations. Primary definition. Traditional name. Traditional notation. Mathematica StandardForm notation. Generalized hypergeometric function
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1 HyergeometricPFQ Notatios Traditioal ame Geeralied hyergeometric fuctio Traditioal otatio F a 1,, a ; b 1,, b ; Mathematica StadardForm otatio HyergeometricPFQa 1,, a, b 1,, b, Primary defiitio F a 1,, a ; b 1,, b ; a j k k b j k k Re b j a j 0 ; 1 I the cases 1 the series above does ot coverge but it (together with symbol) ca be used as asymtotic series, where, whe eeded a Borel summatio is imlicitly uderstood F a 1,, a ; b 1,, b ; a j k k b j k k ; a j a j For a i, b j m ; m beig oositive itegers ad ak a k a k bk b k m b k the fuctio F a 1,, a ; b 1,, b ; caot be uiuely defied by a limitig rocedure based o the above defiitio because the two variables a i, b j ca aroach oositive itegers, m; m at differet seeds. For the above coditios we defie: F a 1,, a i,, a ; b 1,, b j,, b ; a j k k b j k k ; a i b j m m m
2 htt://fuctios.wolfram.com Geeral characteristics Some abbreviatios a 1,, a a 1 a Domai ad aalyticity F a 1,, a ; b 1,, b ; is a aalytical fuctio of a 1,, a, b 1,, b ad which is defied i 1. I the cases for fixed a 1,, a, b 1,, b, it is a etire fuctio of. If arameters a k iclude egative itegers, the fuctio F a 1,, a ; b 1,, b ; degeerates to a olyomial i a 1 a b 1 b F a 1,, a ; b 1,, b ; Symmetries ad eriodicities Mirror symmetry F a 1,, a ; b 1,, b ; F a 1,, a ; b 1,, b ; ; 1, 1 Permutatio symmetry F a 1, a,, a k,, a j,, a ; b 1,, b ; F a 1, a,, a j,, a k,, a ; b 1,, b ; ; a k a j k j F a 1,, a ; b 1, b,, b k,, b j,, b ; F a 1,, a ; b 1, b,, b j,, b k,, b ; ; b k b j k j Periodicity No eriodicity Poles ad essetial sigularities With resect to For 1 ad fixed a l, b j i oolyomial cases (whe a 1 a ), the fuctio F a 1,, a ; b 1,, b ; does ot have oles ad essetial sigularities ig F a 1,, a ; b 1,, b ; ; 1 a 1,, a For ad fixed a l, b j i oolyomial cases (whe a 1 a ), the fuctio F a 1,, a ; b 1,, b ; has oly oe sigular oit at. It is a essetial sigular oit ig F a 1,, a ; b 1,, b ;, ; a 1,, a
3 htt://fuctios.wolfram.com 3 If arameters a k iclude r egative itegers Α k, the fuctio F a 1,, a ; b 1,, b ; is a olyomial ad has ole of order miα 1,, Α r at ig F a 1,, a ; b 1,, b ;, Α ; a 1,, a Α mia s1,, a sr a sk With resect to a l The fuctio F a 1,, a ; b 1,, b ; as a fuctio of a l, 1 l, has oly oe sigular oit at a l. It is a essetial sigular oit ig al F a 1,, a ; b 1,, b ;, ; 1 l With resect to b j The fuctio F a 1,, a ; b 1,, b ; as a fuctio of b j, 1 j, has a ifiite set of sigular oits: a) b j k ; k, are the simle oles with residues 1k k F a 1,, a ; b 1,, b j1, k, b j1,, b ; ; b) b j is the oit of accumulatio of oles, which is a essetial sigular oit ig b j F a 1,, a ; b 1,, b ; k, 1 ; k,, ; 1 j res b j F a 1,, a ; b 1,, b ; k 1k F a 1,, a ; b 1,, b j1, k, b j1,, b ; ; k 1 j k Brach oits With resect to The fuctio F a 1,, a ; b 1,, b ; does ot have brach oits for ad has two brach oits: 1, for 1 i oolyomial case (whe a 1 a ) F a 1,, a ; b 1,, b ; ; F a 1,, a 1 ; b 1,, b ; 1, ; a 1,, a F a 1,, a 1 ; b 1,, b ;, 1 log ; Ψ Ψ Ψ b j a j a 1,, a F a 1,, a 1 ; b 1,, b ;, 1 s ; 1 Ψ b j a j r s r s 1 gcdr, s 1 a 1,, a 1
4 htt://fuctios.wolfram.com F a 1,, a 1 ; b 1,, b ;, log ; ai,a j a i a j 1 i 1 1 j 1 i j a 1 a F a 1,, a 1 ; b 1,, b ;, lcms 1,, s 1 ; a l r l r l, s l s l 1 gcdr l, s l 1 1 l 1 a 1,, a 1 s l With resect to a l The fuctio F a 1,, a ; b 1,, b ; as a fuctio of a l, 1 l, does ot have brach oits al F a 1,, a ; b 1,, b ; ; 1 l With resect to b j The fuctio F a 1,, a ; b 1,, b ; as a fuctio of b j, 1 j, does ot have brach oits b j F a 1,, a ; b 1,, b ; ; 1 j Brach cuts With resect to For all oegative iteger a k, the fuctio F a 1,, a ; b 1,, b ; i the cases 1 is a sigle-valued fuctio o the -lae cut alog the iterval 1,, where it is cotiuous from below. I the cases this fuctio does ot have brach cuts F a 1,, a 1 ; b 1,, b ; 1,, ; a 1,, a F a 1,, a ; b 1,, b ; ; lim Ε0 1 F a 1,, a 1 ; b 1,, b ; x Ε 1 F a 1,, a 1 ; b 1,, b ; x ; x lim Ε0 1 F a 1,, a 1 ; b 1,, b ; x Ε b k 1 ak 1,1 G 1,1 Π 1 x 1, b 1,, b a 1,, a 1 ; x 1
5 htt://fuctios.wolfram.com lim Ε0 1 F a 1,, a 1 ; b 1,, b ; x Ε b k 1 ak 1 a k b j a k 1 a j a k Π a kx a k 1F a k, a k b 1 1,, a k b 1; 1 a 1 a k,, 1 a k1 a k, 1 a k1 a k,, 1 a 1 a k ; 1 x ; j,k,j,k 1 j 11 k 1 a j a k x 1 With resect to a l The fuctio F a 1,, a ; b 1,, b ; as a fuctio of a l, 1 l, does ot have brach cuts al F a 1,, a ; b 1,, b ; ; 1 l With resect to b j The fuctio F a 1,, a ; b 1,, b ; as a fuctio of b j, 1 j, does ot have brach cuts b j F a 1,, a ; b 1,, b ; ; 1 j Series reresetatios Geeralied ower series Exasios at geeric oit 0 For the fuctio itself k 0 F a 1,, a ; b 1,, b ; b j k 1F 1 1, a 1,, a ; 1 k, b 1,, b ; 0 0 k ; , F a 1,, a 1 ; b 1,, b ; b k 1 ak 1 1 a j a k b j a k 1 0 a k arg 0 Π 0 a k arg 0 1 Π a k j 0 j j j 0 1F a k b 1 1,, a k b 1, j a k ; 1a 1 a k,, 1 a k1 a k, 1 a k1 a k,, 1 a 1 a k ; j ; 0 1 j,k,j,k 1 j 11 k 1 a j a k Exasios o brach cuts for 1
6 htt://fuctios.wolfram.com 6 For the fuctio itself F a 1,, a 1 ; b 1,, b ; b k 1 k G 1,1 1,1 1 ak x argx Π Π 1 a 1 k,, 1 a 1 k 0, 1 b 1 k,, 1 b k xk ; x F a 1,, a 1 ; b 1,, b ; b k 1 ak 1 k G 1,1 1,1 argx Π 1 Π 1 x 1, k b 1,, k b k a 1,, k a 1 x k ; x 1 1F a 1,, a 1 ; b 1,, b ; b k 1 ak 1 1 x u u a k a j a k Π a k u 0 u b j a k argx 1Floor Π x a k 1F u a k, a k b 1 1,, a k b 1; 1 a 1 a k,, 1 a k1 a k, 1 a k1 a k,, 1 a 1 a k ; 1 x x u ; j,k,j,k 1 j 11 k 1 a j a k x 1 Exasios at F a 1,, a ; b 1,, b ; F a 1,, a ; b 1,, b ; a j b j a j k k b j k k a j a j 1 b j b j 1 ; 1 1 ; F a 1,, a ; b 1,, b ; 1 O Exasios at 1 for 1 The oit 1 is the ed oit of the brach cut for the fuctio 1 F a 1,, a 1 ; b 1,, b ;, where it has a 1 rather comlicated behavior. The corresodig geeral formula (for oiteger Ψ b j aj ) icludes two major terms - regular ad sigular which are a aalytical fuctios. Moreover, the sigular term has reresetatio of the form cost 1 Ψ 1 O 1 ad regular term is bouded ear oit 1. A more detailed descritio of this behavior is reseted below. At the sigular oit 1 the fuctio 1 F a 1,, a 1 ; b 1,, b ; is cotiuous for ReΨ 0, bouded for ReΨ 0, Ψ 0 ad has, i geeral, a logarithmic sigularity for Ψ 0 while for ReΨ 0 it has a ower sigularity of order Ψ to which for iteger Ψ a logarithmic sigularity ca also occur. The geeral formulas
7 htt://fuctios.wolfram.com F a 1,, a 1 ; b 1,, b ; b j F a 1,, a 1 ;, 1, b 1,, b ; F a 1,, a 1 ; b 1,, b ; b k 1 Ψ g k Ψ 1 k g k 0 1 k 1 ak ; 1 k k r a 1 k r a Ψ r k g k r k Ψ r k j j a1,, a 1, b 1,, b 1 k Ψ r k j 0 j a 1 Ψ j a Ψ k k r b j 1 k r a j 1 lim m Ψ r k 1F k r a 1, k r a,, r a 1, m; k r b 1, k r b,, r b, k m r Ψ 1; 1 g 0 Ψ Ψ k1 a1,, a 1, b 1,, b b 1 a 3 1 k1 b j k 1 j j 3 a j k 1 k 1 k 0 k 1 k 1 j b j j 3 a j a 3 b 1 k 1 1 k k b a 4 1 k b j k j 3 j 4 a j k k k 3 0 k k3 1 j 3 b j j 4 a j a 4 b k 1 k3 k3 b a 1 k b j k j a j k k k 1 0 k k1 b a 1 k1 b 1 a 1 k1 1 b j j a j a b k 1 k k1 k1 1 Ψ b j a j Ψ Rea 3 0 Rea 1 0 The logarithmic cases
8 htt://fuctios.wolfram.com F a 1,, a 1 ; b 1,, b ; b k Ψ 1 k j 1 j j 1 Ψ j j log1 1 j 1 ak j 0 j 0 ; 1 1 Ψ b j a j 1 1 k j 1 j j a 1 j a j k j Ψ 1 k a1,, a 1, b 1,, b ; Rea 3 j k a 1 Ψ k a Ψ 1 Ψ a 1 Ψ j a Ψ j k j 1 Rea 1 j j 1 j k a1,, a 1, b 1,, b j j j Ψ k j1 a 1 Ψ k a Ψ k j j k k a1,, a 1, b 1,, b Ψj k 1 Ψj Ψ 1 Ψj a 1 Ψ Ψj a Ψ ; a 1 Ψ k a Ψ k Rea 3 j Ψ Rea 1 j Ψ 1 Ψ1 a 1 Ψ j a Ψ j j j j Ψ j j k k a1,, a 1, b 1,, b Ψ a 1 Ψ k a Ψ k F a 1,, a 1 ; b 1,, b ; b k a 1 Ψ j a Ψ j 1 j j k j 1 1 Ψ k a1,, a 1, b 1,, b j j Ψ a 1 Ψ k a Ψ k 1 ak j 0 k j1 j j k log1 Ψj k 1 Ψj Ψ 1 Ψj a 1 Ψ Ψj a Ψ a 1 Ψ k a Ψ k k a1,, a 1, b 1,, b 1 j Ψ 1 j a 1 j a k j Ψ 1 k a1,, a 1, b 1,, b 1 j j k a 1 Ψ k a Ψ j 0 ; Ψ b j a j 1 Ψ
9 htt://fuctios.wolfram.com F a 1,, a 1 ; b 1,, b ; b k 1 ak Ψ 1 1 Ψ h j 1 j j u j v j log1 1 j j 0 j 0 ; j a 1 Ψ j a Ψ j j Ψ 1 j j k k a1,, a 1, b 1,, b Ψ b j a j 1 h j j a 1 Ψ k a Ψ k a 1 Ψ jψ a Ψ jψ u j 1 j j Ψ j j Ψ k jψ 1 k j Ψ 1 k a1,, a 1, b 1,, b a 1 Ψ k a Ψ k jψ Ψ j 1 Ψ k k a1,, a 1, b 1,, b Ψj 1 Ψj a 1 Ψj a Ψj k Ψ 1 ; a 1 Ψ k a Ψ k 1 Ψ1 a 1 Ψ jψ a Ψ jψ Rea 3 j Rea 1 j v j j j Ψ jψ Ψ j k k a1,, a 1, b 1,, b Ψ a 1 Ψ k a Ψ k F a 1,, a 1 ; b 1,, b ; b k 1 ak Ψ 1 j 1 k 1 Ψ k a1,, a 1, b 1,, b 1 j j Ψ 1 a 1 Ψ j a Ψ j j 0 j k a 1 Ψ k a Ψ k 1 j k a 1 Ψ 1 j k a k a1,, a 1, b 1,, b j Ψ j k k a 1 Ψ k a Ψ j Ψ log1 Ψj k 1 Ψj Ψ 1 Ψj k a 1 Ψj k a 1 j Ψ 1 j k a 1 Ψ1 1 j k a k a1,, a 1, b 1,, b 1 j j k k a 1 Ψ k a Ψ j k ; Ψ b j a j 1 Ψ
10 htt://fuctios.wolfram.com F a 1,, a 1 ; b 1,, b ; b k 1 ak a 1 j a j j j k log1 Ψj 1 Ψj a 1 Ψj a Ψj k 1 k a1,, a 1, b 1,, b j 0 j a 1 k a k 1 j j k j1 k j 1 a 1 k a k 1 1 Ψ b j a j 1 Ψ 0 1 k a1,, a 1, b 1,, b 1 j ; The major terms i the geeral formula for exasios of fuctio 1F a 1,, a 1 ; b 1,, b ; at F a 1,, a 1 ; b 1,, b ; 1 F a 1,, a 1 ; b 1,, b ; 11 O 1 1 Ψ b j a j Ψ F a 1,, a 1 ; b 1,, b ; Ψ 1 k 3 ak b k Ψ k k a1,, a 1, b 1,, b 1 O 1 k a 1 Ψ k a Ψ Ψ Ψ 1 ak 1 k 3 ak 1 Ψ b j a j 1 ReΨ 0 Rea 3 0 Rea F a 1,, a 1 ; b 1,, b ; 1 F a 1,, a 1 ; b 1,, b ; 11 O 1 1 Ψ b j a j Ψ 0 1 Exasios at for 1 b j 1 a j b k 1 Ψ 1 O 1 ; b k 1 Ψ 1 O 1 ; log1 1 O 1 ; The geeral formulas F a 1,, a 1 ; b 1,, b ; b k F a 1,, a ; b 1,, b ;,, ; 0, 1
11 htt://fuctios.wolfram.com F a 1,, a 1 ; b 1,, b ; b k ower a 1,, a ; F b 1,, b ;,, ; 0, 1 Case of simle oles F a 1,, a 1 ; b 1,, b ; b k 1 ak a 1 k 1 a j a k b j a k a k 1 1 j,k,j,k 1 j 11 k 1 a j a k F a 1,, a 1 ; b 1,, b ; b k 1 ak 1 j,k,j,k 1 j 11 k 1 a j a k F a 1,, a 1 ; b 1,, b ; a k a k b j 1 1 ak a j 1 a 1 k 1 a j a k b k 1 1 ak i 0 1 j,k,j,k 1 j 11 k 1 a j a k F a 1,, a 1 ; b 1,, b ; b k 1 ak b j a k a k a k 1 a k b j 1 a k b j a k 1 ak a j 1 a k a j i 0 Res 0; 1 a 1,, 1 a 1 ; ; 1 b 1,, 1 b ; a 1 k 1 a j a k b j a k a k a k i a k b j 1 i 1 i ak a j 1 i i 1 a k, 1, i; ; ; ; 1F a k, a k b 1 1,, a k b 1; 1 a 1 a k,, 1 a k1 a k, 1 a k1 a k,, 1 a 1 a k ; 1 ; 0, 1 j,k,j,k 1 j 11 k 1 a j a k Case of oles of order r i the oits a r k ; r, 3, 4 k F a 1,, a 1 ; b 1,, b ; b k ower a 1,, a ; F b 1,, b ;,, ; 0, 1 a k a k1 k r a k a 1 r 1 k 1 j,k,j,k r1 j 1r1 k 1 a j a k r, 3, 4 The major terms for exasios of fuctio 1 F a 1,, a 1 ; b 1,, b ; at
12 htt://fuctios.wolfram.com F a 1,, a 1 ; b 1,, b ; b k 1 ak j,k,j,k 1 j 11 k 1 a j a k a 1 k 1 a j a k b j a k a k 1 O 1 ; F a 1,, a 1 ; b 1,, b ; b k 1 ak 1 k r1 a 1 k a j a k b j a k a k 1 O 1 r1 Res 0; a 1,, a 1 ; a ; b 1,, b ; j1, j 1, 0; 1 O 1 j ; a k a k1 k r a k a 1 r 1 k 1 j,k,j,k r1 j 1r1 k 1 a j a k r, 3, F a 1,, a 1 ; b 1,, b ; b k 1 ak 1 a 1 k ak a 1 b k a 1 a 1 1 O 1 1 a a 1 1 a k 3 ak a a a 1 b k a a log Ψa a Ψa Ψa k a Ψb k a k 3 1 O 1 a 1 k 1 a j a k k 3 b j a k a k 1 O 1 ; a a 1 a k a 1 3 k 1 j,k,j,k 3 j 13 k 1 a j a k
13 htt://fuctios.wolfram.com F a 1,, a 1 ; b 1,, b ; b k 1 ak 1 a 1 k ak a 1 b k a 1 a 1 1 O 1 1 a a 1 1 a k 3 ak a a a 1 b k a a log Ψa a Ψa Ψa k a Ψb k a k 3 1 O 1 1 a a 1 1 a 3 k 4 ak a 3 a 3 a a 3 a 1 b k a 3 a 3 1 log Ψa 3 a 1 1 Ψa 3 a 1 Ψa 3 Ψa k a 3 Ψb k a 3 k 4 5 Π 6 Ψ1 a 3 a 1 1 Ψ 1 a 3 a 1 Ψ 1 a 3 Ψ 1 a k a 3 Ψ 1 b k a 3 1 k 4 O 1 1 a 1 k 1 a j a k a k k 4 b j a k 1 O 1 ; a a 1 a 3 a a k a 1 4 k 1 j,k,j,k 4 j 14 k 1 a j a k
14 htt://fuctios.wolfram.com F a 1,, a 1 ; b 1,, b ; b k 1 ak 1 a 1 k ak a 1 b k a 1 a 1 1 O 1 1 a a 1 1 a k 3 ak a a a 1 b k a a log Ψa a Ψa Ψa k a Ψb k a k 3 1 O 1 1 a a 1 1 a 3 k 4 ak a 3 a 3 a a 3 a 1 b k a 3 a 3 1 log Ψa 3 a 1 1 Ψa 3 a 1 Ψa 3 Ψa k a 3 Ψb k a 3 k 4 5 Π 6 Ψ1 a 3 a 1 1 Ψ 1 a 3 a 1 Ψ 1 a 3 Ψ 1 a k a 3 Ψ 1 b k a 3 1 k 4 O a 1 a a 3 a 1 4 a 4 k 5 ak a 4 6 a 4 a 3 a 4 a a 4 a 1 b k a 4 a log Ψa 4 a 1 1 Ψa 4 a 1 Ψa 4 a 3 1 Ψa 4 Ψa k a 4 Ψb k a 4 k Ψ 1 a 4 Ψ 1 a 4 a 1 1 Ψ 1 a 4 a 1 Ψ 1 a 4 a 3 1 Ψ 1 a k a 4 Ψ 1 b k a 4 k 5 7 Π log Ψa 4 a 1 1 Ψa 4 a 1 1 Ψa 4 a 3 1 Ψa 4 Ψa k a 4 Ψb k a 4 k 5 1 Ψ a 4 a 1 1 Ψ a 4 a 1 Ψ a 4 a 3 1 Ψ a 4 Ψ a k a 4 Ψ b k a 4 Ζ3 k 5 1 O 1 a 1 k 1 a j a k k 5 b j a k a k 1 O 1 ; a a 1 a 3 a a 4 a 3 a k a 1 5 k 1 j,k,j,k 5 j 15 k 1 a j a k Exasios at for olyomial cases
15 htt://fuctios.wolfram.com F, a, a 3,, a ; b 1,, b ; k a k 1F 1, 1 b 1,, 1 b ; 1 a, 1 a 3,, 1 a ; 11 b k ; Asymtotic series exasios Exasios for F a 1,, a ; b 1,, b ; b j ower a 1,, a ; F b 1,, b ;,, ex a 1,, a ; F b 1,, b ;,, ; b j F a 1,, a ; b 1,, b ; a j Χ 1 O 1 b j a j a k a j a k b j a k a k 1 O 1 ; j,k,j,k 1 j 1 k a j a k Exasios for F 1 a 1,, a ; b 1,, b 1 ; b j ower a 1,, a ; F b 1,, b 1 ;,, trig a 1,, a ; F,, b 1,, b 1 ; ; F 1 a 1,, a ; b 1,, b 1 ; 1 b j Π a k Χ Π Χ 1 O 1 Π Χ 1 O 1 1 b j a k a k a j a k 1 b j a k a k 1 O 1 ; j,k,j,k 1 j 1 k a j a k F 1 a 1,, a ; b 1,, b 1 ; 1 b j Π a k Χ cosπ Χ 1 O 1 c 1 siπ Χ 1 O 1 1 b j a k a k a j a k 1 b j a k a k 1 O 1 ; Χ 1 A B 1 1 c A B 1 A B A a k 1 s1 1 s1 B 1 b k a s a j b s b j j,k,j,k 1 j 1 k a j a k s s
16 htt://fuctios.wolfram.com 16 Exasios for F a 1,, a ; b 1,, b ; b j ower a 1,, a ; F b 1,, b ;,, ex a 1,, a ; F b 1,, b ; j,k,j,k 1 j 1 k a j a k,, ; F a 1,, a ; b 1,, b ; 1Β Π Β a k b j Χ exβ 1Β 1 O 1 1Β b j a k a k a j a k b j a k a k 1 O 1 ; Β 1 Χ 1 Β Β 1 a k b k j,k,j,k 1 j 1 k a j a k Exasios for 0 F F ; b 1, b ; b 1 b 3 3 Π 3 1 1b 3 1 b 1 3 b 1 3 b 1 b 1 3 b 3 b 1 9 b b b b 3 b 1 b b 3 3 b 17 b 4 b 1 9 b 4 1 b 3 3 b 1 b 4 ; F ; b 1, b ; b 1 b 3 3 Π 3 1 1b 3 1 b 1 O 1 3 ; Exasios for 0 F F 3 ; b 1, b, b 3 ; b 1 b b Π b 1 b b 3 1 b 1 8 b b 3 1 b 1 1 b 1 b 3 8 b 3 8 b b b b 3 b 3 1 b b 8 b 3 3 b 44 b 3 4 b 3 1 b b 3 4 b 3 3 b 4 b 3 40 b 3 1 b 1 b b 3 1 b 3 11 b b b b b b 3 64 b 3 3 b b 44 b 3 4 b b 1 b b 3 1 b ; F 3 ; b 1, b, b 3 ; b 1 b b Π b 4 1 b b 3 1 O 1 4 ;
17 htt://fuctios.wolfram.com 17 Geeral formulas of asymtotic series exasios F a 1,, a ; b 1,, b ; b j F a 1,, a ; b 1,, b ;,, ; F a 1,, a ; b 1,, b ; b j Θ,1 ex a 1,, a ; F b 1,, b ;,, ower a 1,, a ; F b 1,, b ;,,,1 trig a 1,, a ; F b 1,, b 1 ;,, ; 1 Mai terms of asymtotic exasios F a 1,, a ; b 1,, b ; b j res s a j s a j s b j s s a k 1 O 1,1 d 1 Χ cosπ Χ 1 O 1 Θ,1 d Χ Β 1Β 1 O 1 1Β ; Β 1 Χ 1 Β Β 1 a k b k d d 1 1Β Π Β F a 1,, a ; b 1,, b ; c k a k 1 O 1,1 e 1 Χ cosπ Χ 1 O Β 1 Χ 1 Β Β 1 a k e e 1 1Β Π Β a k 1 Θ,1 e Χ Β 1Β 1 O 1 b k c k b k j,k,j,k 1 j 1 k a j a k 1F a 1,, a 1 ; b 1,, b ; c 1 O 1 d 1 Ψ 1 O 1 ; a k 1 Ψ b j a j c 1 F a 1,, a 1 ; b 1,, b ; 1 d 1 1Β ; b j a j a k a j b j a k Ψ b k 1 ak Ψ Residue reresetatios
18 htt://fuctios.wolfram.com F a 1,, a ; b 1,, b ; b k a k res s j 0 s a k s b k s s j ; F a 1,, a 1 ; b 1,, b ; b k 1 ak 1 res s j 0 1 s ak s b k s s a k j ; 1 Cotiued fractio reresetatios F a 1,, a ; b 1,, b ; b k 1 a k 1 1 a j 1 b j 1 1 a j 1 b j 1 a j 3 b j a j 3 b j F a 1,, a ; b 1,, b ; 1 a k b k 1 k k a j k 1 k b j, k a j k 1 k b j 1 1 Differetial euatios Ordiary liear differetial euatios ad wroskias For the direct fuctio itself The differetial euatio for the fuctio F a 1,, a ; b 1,, b ; has the order max, 1. It has two ( 0, for ) or three ( 0, 1,, for 1) sigular oits. If, the the oit 0 is a regular sigular oit, while is a oregular (essetial) sigular oit; if 1, the all three sigular oits are regular. Reresetatio of fudametal system solutios ear oit 0 for 1 i the geeral case
19 htt://fuctios.wolfram.com w 1 1 b k w w 1 l 1 a l w 1 d d d d b k 1 w l 1 d d a 1 l w w 1 1 b k w w 1 1 a k w 1 w l 1 a l w a l 0 ; l 1 w c 1 F a 1,, a ; b 1,, b ; c, G,1 1 a 1,, 1 a 0, 1 b k, 1 b 1,, 1 b k1, 1 b k1,, 1 b c, G,1 1 1 a 1,, 1 a 0, 1 b 1,, 1 b k1, 1 b k1,, 1 b, 1 b k 1, c 1 G, a 1,, 1 a 0, 1 b 1,, 1 b w 1 1 b k w w 1 l 1 a l w 1 d d d d b k 1 w l 1 d d a 1 l w w 1 1 b k w w 1 1 a k w 1 w l 1 a l w a l 0 ; w c 1 F a 1,, a ; b 1,, b ; l 1 c k1 1b k F a 1 b k 1,, a b k 1; b k, b 1 b k 1,, b k1 b k 1, b k1 b k 1,, b b k 1; ; j,k,j,k 1 j 1 k b j b k b k W F a 1,, a ; b 1,, b ;, 1b 1 F a 1 b 1 1,, a b 1 1; b 1, 1 b 1 b,, 1 b 1 b ;,, 1b k F a 1 b k 1,, a b k 1; b k, b 1 b k 1,, b k1 b k 1, b k1 b k 1,, b b k 1;,, 1b F a 1 b 1,, a b 1; b, b 1 b 1,, b 1 b 1; 1 Π k1 siπ b k siπ b j b k 1 1 b k,1 1 l 1 1 al b k, Θ 1
20 htt://fuctios.wolfram.com 0 d d d d b d k 1 l 1 d a l w 0 ; w c 1 F a 1,, a ; b 1,, b ; c k1 1b k F a 1 b k 1,, a b k 1; b k, b 1 b k 1,, b k1 b k 1, b k1 b k 1,, b b k 1; ; j,k,j,k 1 j 1 k b j b k b k W F a 1,, a ; b 1,, b ;, 1b 1 F a 1 b 1 1,, a b 1 1; b 1, b 1 b 1,, b 1 b 1;,, 1b k F a 1 b k 1,, a b k 1; b k, b 1 b k 1,, b k1 b k 1, b k1 b k 1,, b b k 1;,, 1b F a 1 b 1,, a b 1; b, b 1 b 1,, b 1 b 1; cost 1 1 b k,1 1 l 1 1 al b k, Θ 1 ; t lim Ε0 Ε 1 b k W Ε F a 1,, a ; b 1,, b ; Ε, Ε 1b 1 F a 1 b 1 1,, a b 1 1; b 1, b 1 b 1,, b 1 b 1; Ε,, Ε 1b k F a 1 b k 1,, a b k 1; b k, b 1 b k 1,, b k1 b k 1, b k1 b k 1,, b b k 1; Ε,, Ε 1b F a 1 b 1,, a b 1; b, b 1 b 1,, b 1 b 1; Ε W F a 1,, a ; b 1,, b ;, 1b 1 F a 1 b 1 1,, a b 1 1; b 1, 1 b 1 b,, 1 b 1 b ;,, 1b k F a 1 b k 1,, a b k 1; b k, b 1 b k 1,, b k1 b k 1, b k1 b k 1,, b b k 1;,, 1b F a 1 b 1,, a b 1; b, b 1 b 1,, b 1 b 1; 1 k1 b k 1 b j b k 1 1 b k,1 1 l 1 1 al b k, Θ 1 Reresetatio of fudametal system solutios ear oit 1 for 1 i the geeral case 1,0 Below reresetatio icludes fuctios of two kids. The fuctio G 1,1 1 a 1,, 1 a 1 0, 1 b 1,, 1 b is the iecewise aalytical fuctio with a discotiuity o the uite circle 1. It has sigularity ear oit 1of the form cost 1 Ψ,3 1 O 1, whe 1. The fuctios G 3,3 aalytical fuctios ad are bouded ear oit 1. 1,0 The fuctio G 1,1 1 a 1,, 1 a 1 0, 1 b 1,, 1 b iside of 1ca be rereeted through hyergeometric fuctios defied for all comlex. 0, b k, 1 a 1,, 1 a 1 0, b k, 0, 1 b 1,, 1 b are the
21 htt://fuctios.wolfram.com 1 d d d d b d k 1 l 1 d a l w 0 ; 1,0 w c 1 G 1,1 1 a 1,, 1 a 1,3 c 0, 1 b 1,, 1 b k1 G 3,3 0, b k, 1 a 1,, 1 a 1 0, b k, 0, 1 b 1,, 1 b ; 1 1 Ψ b j a j Ψ j,k,j,k 1 j 1 k b j b k b k Reresetatio of fudametal system solutios ear oit for 1 i the geeral case d d d d b d k 1 l 1 d a l w 0 ; w c k a k 1 F 1 a k, a k b 1 1,, a k b 1; 1 a 1 a k,, 1 a k1 a k, 1 a k1 a k,, 1 a a k ; 11 j,k,j,k 1 j 1 k a j a k d d d d b d k 1 l 1 d a l w 0 ; w c k a k 1 F 1 a k, a k b 1 1,, a k b 1; 1 a 1 a k,, 1 a k1 a k, 1 a k1 a k,, 1 a a k ; 11 ; j,k,j,k 1 j 1 k a j a k Trasformatios Products, sums, ad owers of the direct fuctio Products of the direct fuctio F a 1,, a ; b 1,, b ; c r F s Α 1,, Α r ; Β 1,, Β s ; d c k k ; c k d k r Α j k s1 F r k, 1 Β 1 k,, 1 Β s k, a 1,, a ; 1 Α 1 k,, 1 Α r k, b 1,, b ; 1rs1 c s k Β j d k c k c k a j k r1 F s k, 1 b 1 k,, 1 b k, Α 1,, Α r ; 1 a 1 k,, 1 a r k, Β 1,, Β s ; 11 d k b j c k
22 htt://fuctios.wolfram.com F a 1,, a ; b 1,, b ; c r F s Α 1,, Α r ; Β 1,, Β s ; d k m 0 r a j m Α j km c m d km k s b j m Β j km mk m :;r : a 1,, a ; Α 1,, Α r ; F a 1,, a ; b 1,, b ; c r F s Α 1,, Α r ; Β 1,, Β s ; d F 0:;s c, d : b 1,, b ; Β 1,, Β s ; Idetities Recurrece idetities Distat eighbors with resect to F a 1,, a 1 ; b 1,, b ; b j k a1,, a 1, b 1,, b F 1 1a 1, a ; a 1 a Ψ k; ; Ψ b j 1 j 3 a j a j Fuctioal idetities Relatios betwee cotiguous fuctios b F a, b 1, a 3,, a ; b 1,, b ; a F a 1, b, a 3,, a ; b 1,, b ; a b F a, b, a 3,, a ; b 1,, b ; c F a, a,, a ; c, b,, b ; a F a 1, a,, a ; c 1, b,, b ; a c F a, a,, a ; c 1, b,, b ; d F a 1,, a ; c 1, d, b 3,, b ; c F a 1,, a ; c, d 1, b 3,, b ; c d F a 1,, a ; c 1, d 1, b 3,, b ; ca b F a, b, a 3,, a ; c, b,, b ; ac b F a 1, b, a 3,, a ; c 1, b,, b ; bc a F a, b 1, a 3,, a ; c 1, b,, b ; cd a F a, a,, a ; c, d 1, b 3,, b ; dc a F a, a,, a ; c 1, d, b 3,, b ; ac d F a 1, a,, a ; c 1, d 1, b 3,, b ; 0 b k F a, a,, a ; b 1,, b ; F a 1, a,, a ; b 1,, b ; a j F a 1, a 1,, a 1; b 1 1,, b 1; 0 j
23 htt://fuctios.wolfram.com cc 1 b k F a 1,, a ; c, b,, b ; F a 1,, a ; c 1, b,, b ; k a j F a 1 1,, a 1; c, b 1,, b 1; b k F a, b 1, a 3,, a ; b 1,, b ; F a 1, b, a 3,, a ; b 1,, b ; b a a j F a 1, b 1, a 3 1,, a 1; b 1 1,, b 1; 0 j cc 1 b k F a, a,, a ; c, b,, b ; F a 1, a,, a ; c 1, b,, b ; k c a a j F a 1, a 1,, a 1; c, b 1,, b 1; 0 j b k c F a, b, a 3,, a ; c, b,, b ; k a F a 1, b 1, a 3,, a ; c 1, b,, b ; c a F a, b 1, a 3,, a ; c 1, b,, b ; a a j F a 1, b 1, a 3 1,, a 1; c 1, b 1,, b 1; 0 j F a 1, b 1, a 3,, a ; c 1, d 1, e 1, b 4,, b ; c da eb e a b c ed e F a, b, a 3,, a ; c, d, e 1, b 4,, b ; c ea db d a b c de d F a, b, a 3,, a ; c, d 1, e, b 4,, b ; d ea cb c a b d ce c F a, b, a 3,, a ; c 1, d, e, b 4,, b ; a bd ce c F a, b, c, a 4,, a ; d, e, b 3,, b ; d e a cb c F a 1, b 1, c, a 4,, a ; d 1, e 1, b 3,, b ; a cd be b d e a bc b F a 1, b, c 1, a 4,, a ; d 1, e 1, b 3,, b ; b cd ae a d e b ac a F a, b 1, c 1, a 4,, a ; d 1, e 1, b 3,, b ; 0
24 htt://fuctios.wolfram.com a a j1 b j 1 F a, a,, a 1 ; b 1,, b ; b j a b j b j b k k j b j a k1 1 F a, a,, a 1 ; b 1,, b j1, b j 1, b j1,, b ; a1 1 F a 1, a,, a 1 ; b 1,, b ; Relatios of secial kid F a 1,, a ; c, 1 c, b 3,, b ; F a 1,, a ; c, 1 c, b 3,, b ; F a 1,, a ; 1 c, 1 c, b 3,, b ; F a, a,, a ; a, 1 a, b 3,, b ; F a, a,, a ; 1 a, 1 a, b 3,, b ; 1 F 1 a,, a ; 1 a, b 3,, b ; F a, a,, a ; 1 a, b,, b ; F a, a,, a ; 1 a, b,, b ; 1 F 1 a, a, a,, a ; 1 a, 1 a, b,, b ; F a, 1 a, a 3,, a ; b 1,, b ; F a, 1 a, a 3,, a ; b 1,, b ; 1 F 1 a, a, a 3,, a ; b 1,, b ; Divisio o eve ad odd arts ad geeraliatio F a 1,, a ; b 1,, b ; A A ; A 1 F a 1,, a ; b 1,, b ; F a 1,, a ; b 1,, b ; A 1 F a 1,, a ; b 1,, b ; F a 1,, a ; b 1,, b ; F a 1,, a ; b 1,, b ; A A ; a A 1 a F 1,,, a 1 1 a 1,, A a j a 1 1 F 1 b j a 1,, ; 1, b 1,, b, a 1,,, b 1 1 a b 1,, ; 4 1 ; 3, b 1 1,, b 1, b 1,, b ; F a 1,, a ; b 1,, b ; 1 k k a j k 1 F 1, a 1 k b j k,, a 1 k 1,, a k,, a k 1 ; k 1,, k, b 1 k,, b 1 k 1,, b k,, b k 1 ; 1 Case 1 F
25 htt://fuctios.wolfram.com F a 1,, a 1 ; b 1,, b ; b k 1 ak a 1 k 1 a j a k b j a k a k 1F a k, a k b 1 1,, a k b 1; 1 a 1 a k,, 1 a k1 a k, 1 a k1 a k,, 1 a 1 a k ; 1 ; m 0, 1 j,k,j,k 1 j 11 k 1 a j a k m b j b k 1 a j b k 1 j 1 b k 1 a j b k j m1 1 b j b k 1F 1 a 1 b k,, 1 a 1 b k ; 1 b 1 b k,, 1 b k1 b k, 1 b k1 b k,, 1 b 1 b k ; 1 m1 ak a j m 1 a k b j 1 j m1 a k 1 1 a k b j j 1 a j a k 1 1F 1 a k b 1,, 1 a k b 1 ; 1 a 1 a k,, 1 a k1 a k, 1 a k1 a k,, 1 a 1 a k ; 1m1 m m 1 1 m m 1, 0 ; a 1 1F a 1,, a 1 ; b 1,, b ; w 1 a k 1 1 F k, a,, a 1 ; b 1,, b ; w k 1 k Differetiatio Low-order differetiatio With resect to a F 1,0,,0,0,,0,0 a 1,, a ; b 1,, b ; F 1,0,,0,0,,0,0 a 1,, a ; b 1,, b ; Ψk a 1 a j k k j a j k b j k F b j Ψa 1 F a 1,, a ; b 1,, b ; ; 1 a 1 1,, a 1; 1; 1, a 1 ;, b 1 1,, b 1;; a 1 1;, With resect to b 1
26 htt://fuctios.wolfram.com F 0,,0,1,0,,0,0 a 1,, a ; b 1,, b ; Ψb 1 F a 1,, a ; b 1,, b ; F 0,,0,1,0,,0,0 a 1,, a ; b 1,, b ; a j F b 1 b j Ψk b 1 a j k k k b j k 1 a 1 1,, a 1; 1; 1, b 1 ;, b 1 1,, b 1;; b 1 1;, With resect to elemet of arameters With resect to elemet of arameters F a, a,, a ; a 1, b,, b ; a j a j a 1 1 F 1 a 1, a 1, a 1,, a 1; a, a, b 1,, b 1; j b j F a 1, a,, a ; a, b,, b ; a With resect to F a 1,, a ; b 1,, b ; F a 1,, a ; b 1,, b ; a j j a j a 1 F 1 a 1,, a 1; b 1,, b 1; j b j F a 1 1,, a 1; b 1 1,, b 1; b j a j a j 1 F a 1,, a ; b 1,, b ; b j b j 1 ; Symbolic differetiatio With resect to a F,0,,0,0,,0,0 a 1,, a ; b 1,, b ; j a j k a 1 k k b j a k 1 k ; 1 1 With resect to b F 0,,0,,0,,0,0 a 1,, a ; b 1,, b ; a j k k j b j k 1 b 1 k With resect to elemet of arameters With resect to elemet of arameters b 1 k ; 1 1 1
27 htt://fuctios.wolfram.com F a, a,, a ; a 1, b,, b ; a 1 1 j a j a 1 1 j b j F a 1, a,, a ; a, b,, b ; a F a 1,, a 1, a 1,, a 1; a,, a, b 1,, b 1; ; 1 a 1 F 1 a,, a ; b,, b ; a 1 j a j F 1 a,, a ; b,, b ; ; a With resect to F a 1,, a ; b 1,, b ; a j 1F 1 a 1,, a 1; b 1,, b 1; j b j F a 1,, a ; b 1,, b ; ; b j F a 1,, a ; b 1,, b ; b j 1 F 11, a 1,, a ; 1, b 1,, b ; ; Α F a 1,, a ; b 1,, b ; 1 Α Α 1F 1 Α 1, a 1,, a ; Α 1, b 1,, b ; ; a1 F a, a,, a ; b 1,, b ; a a1 F a, a,, a ; b 1,, b ; ; c1 F a 1,, a ; c, b,, b ; c c1 F a 1,, a ; c, b,, b ; ; F, a,, a ; 1, b,, b ; 1 F 1, 1, a,, a ; 1, 1, b,, b ; ; Α F, a,, a ; b 1,, b ; 1 Α Α 1 F 1, Α 1, a,, a ; Α 1, b 1,, b ; ;
28 htt://fuctios.wolfram.com Α F r, 1 r,, r1, a r r1,, a ; b 1,, b ; m 1 Α Α m F m 1 r 1,,,, Α 1 r r r m, Α Α m,, m m, a r1, Α 1 Α Α m, a ;,,,, b 1,, b ; m ; r m m m m F, a,, a ; b 1,, b ; 1 k k 1 F, k, a k,, a k; b 1 k,, b k; ; k b j k Fractioal itegro-differetiatio With resect to Α F a 1,, a ; b 1,, b ; Α b Α j 1 F 11, a 1,, a ; 1 Α, b 1,, b ; Itegratio Idefiite itegratio Ivolvig oly oe direct fuctio F a 1,, a ; b 1,, b ; b j 1 F a 1 1,, a 1; b 1 1,, b 1; a j 1 Ivolvig oe direct fuctio ad elemetary fuctios Ivolvig ower fuctio Α1 F a 1,, a ; b 1,, b ; Α Α 1F 1 Α, a 1,, a ; Α 1, b 1,, b ; Defiite itegratio For the direct fuctio itself
29 htt://fuctios.wolfram.com t Α1 F a 1,, a ; b 1,, b ; tt b k Α a k Α 0 ReΑ mirea 1,, Rea 1 0 ReΑ mi Rea 1,, Rea, Re a k b k Α ; a j b k 1 Summatio Fiite summatio a j k 1 F 1, a 1 k b j k,, a 1 k 1,, a k,, a k 1 ; k 1 k,,, b 1 k,, b 1 k 1, b k,, b k 1 ; 1 k k F a 1,, a ; b 1,, b ; Ifiite summatio a 1 k 1 k 1 F k, a,, a 1 ; b 1,, b ; w k 1 a 1 w 1F a 1,, a 1; b 1,, b ; 1 Oeratios Limit oeratio lim 1 Ψ 1 F a 1,, a 1 ; b 1,, b ; 1 lim log1 1 F a 1,, a 1 ; b 1,, b ; F a 1,, a ; b 1,, b ; lim b 1 b 1 Ψ b j 1 a j b j 1 a j ; Ψ b j a j ReΨ 0 1 ; Ψ b j a j Ψ 0 1 a j 1 F a 1 1,, a 1;, b 1,, b 1; ; lim a F a, a,, a ; b 1,, b ; a 1F a,, a ; b 1,, b ;
30 htt://fuctios.wolfram.com lim a F a 1,, a ; b, b,, b ; b F 1 a 1,, a ; b,, b ; lim a a 1 a 1 F a, a, a 3,, a 1 ; a 1, a 3 1,, a 1 1; S a a 3 a 1 a lim 1 1 F 1 m, a 1, a 3 1,, a 1 1; a, a 3,, a 1 ; 1 m1 m m 1 ; a a 3 a 1 1 m ; Reresetatios through more geeral fuctios Through hyergeometric fuctios Ivolvig F F a 1,, a ; b 1,, b ; b k F a 1,, a ; b 1,, b ; Through hyergeometric fuctios of two variables F a 1,, a ; b 1,, b ; F ; a 1,, a ;; 0 0 ; b 1,, b ;;, F a 1,, a ; b 1,, b ; b k F 0 0 ; a 1,, a ;; 0 0 ; b 1,, b ;;, 0 Through Meijer G Classical cases for the direct fuctio itself F a 1,, a ; b 1,, b ; b k a k 1, G,1 1 a 1,, 1 a 0, 1 b 1,, 1 b
31 htt://fuctios.wolfram.com F a 1,, a 1 ; b 1,, b ; b k 1 Π siψ Π ak 1 siπ b j a k siπ b j b k k j,1 G 1,1 1 a 1,, 1 a 1 0, 1 b j, 1 b 1,, 1 b j1, 1 b j1,, 1 b Π b k 1 siψ Π ak 1 Ψ 1 Ψ 0,1 G 1,1 1 a 1,, 1 a 1 1,0 0, 1 b 1,, 1 b G 1,1 1 a 1,, 1 a 1 0, 1 b 1,, 1 b ; Ψ b j a j 1, 0 Ψ F a 1,, a ; b 1,, b ; b k a k G 3,1 4,3 1 1, b 1,, b a 1,, a ; 0, Theorems Coectios betwee series ad cotiued fractio reresetatios Euler established that the coverget series a k ca be exressed i a cotiued fractio form a k a 1 1 CotiueFractio a k1 a k, 1 a 1 k1, k, 1,. a k I articular the followig reresetatio takes lace: F a 1, a,, a ; b 1, b,, b ; 1 a k b k 1 CotiueFractio k a j k 1 k b j, 1 k a j k 1 k b j, k, 1, 1. History J. F. Pfaff (1797) T. Clause (188); J. Thomae (1870, 1879) studied differetial euatio S. Picherle (1886, 1888) L. Pochhammer (1888) E. W. Bares ( ) T. W.Chaudy (1943) N. E. Nörlud (1955) A. P. Prudikov, Y.A. Brychkov ad O.I. Marichev (1986)
32 htt://fuctios.wolfram.com 3 Coyright This documet was dowloaded from fuctios.wolfram.com, a comrehesive olie comedium of formulas ivolvig the secial fuctios of mathematics. For a key to the otatios used here, see htt://fuctios.wolfram.com/notatios/. Please cite this documet by referrig to the fuctios.wolfram.com age from which it was dowloaded, for examle: htt://fuctios.wolfram.com/costats/e/ To refer to a articular formula, cite fuctios.wolfram.com followed by the citatio umber. e.g.: htt://fuctios.wolfram.com/ This documet is curretly i a relimiary form. If you have commets or suggestios, lease commets@fuctios.wolfram.com , Wolfram Research, Ic.
HermiteHGeneral. Notations. Primary definition. Specific values. Traditional name. Traditional notation. Mathematica StandardForm notation
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Σχολή Γεωτεχνικών Επιστημών και Διαχείρισης Περιβάλλοντος Μεταπτυχιακή διατριβή Κτίρια σχεδόν μηδενικής ενεργειακής κατανάλωσης :Αξιολόγηση συστημάτων θέρμανσης -ψύξης και ΑΠΕ σε οικιστικά κτίρια στην