MeijerG Nottions Trditionl nme Generlied Meijer G-function Trditionl nottion Mthemtic StndrdForm nottion MeijerG,, n, n,,, b,, b m, b m,, b,, r Primry definition 07.5.0.000.0 m k n r r 0 m n m n n b k s k s k s k m b k s s r s ; The infinite contour of integrtion sertes the oles of k st s k j, j from the oles of b i s t s b i l, l. Such contour lwys exists in the cses k b i. There re three ossibilities for the contour : (i) runs from Γ- to Γ+ (where ImΓ 0) so tht ll oles of b i s, i,..., m, re to the left, nd ll the oles of i s, i,..., n, to the right, of. This contour cn be stright line Γ, Γ if Reb i k (then Reb i Γ Re k ). (In this cse the integrl converges if m n, Arg m n. If m n 0, then must be rel nd ositive nd dditionl condition Γ ReΜ 0, Μ l b l k, should be dded.) (ii) is left loo, strting nd ending t - nd encircling ll oles of b i s, i,..., m, once in the ositive direction, but none of the oles of i s, i,..., n. (In this cse the integrl converges if nd either or nd or nd nd m n 0 nd ReΜ 0.)
htt://functions.wolfrm.com (iii) is right loo, strting nd ending t + nd encircling ll oles of i s, i,..., n, once in the negtive direction, but none of the oles of b i s, i,..., m. (In this cse the integrl converges if nd either or nd or nd nd m n 0 nd ReΜ 0.) Secific vlues Secilied vlues Generl cse with restrictions on nd r 07.5.0.000.0 G m,n, r ; r r r rg r 07.5.0.000.0 G m,n,, n, n,,,, G m,n, 07.5.0.000.0 G m,n,, n, n,,,, r Ξ m r,n G r r, r r r,, r,, n,, nr r r r r b,, b r,, b m,, b mr r r r r, n r, b m r m n r Θ k b k r,, n r,,,, r r r r,, b m r,, b,, b r r r r ; Cse m, n,, 0,,, 0 G 0,,0, G 0,,0, G 0,,0, 07.5.0.0004.0, c 07.5.0.0005.0, 07.5.0.000.0, c J c sin cos Cse m, n,, 0,,, 07.5.0.0007.0 G 0,,,, c, c I c
htt://functions.wolfrm.com 07.5.0.0008.0 G 0,,, 4,, 4 4 4 C 07.5.0.0009.0 G 0,,, 4,, 4 4 4 erf erfi G 0,,, 4 07.5.0.000.0,, 4 4 4 S 07.5.0.00.0 G 0,,, 4,, 4 4 4 erfi erf Cse m, n,, 0,,, 0 07.5.0.00.0 G 0,,0, cscc, c c c I c c I c G 0,,0, 07.5.0.00.0, c c K c ;, 0 07.5.0.004.0 G 0,,0,, Ai G 0,,0, 07.5.0.005.0, Ai Cse m, n,, 0,,, G 0,,, 07.5.0.00.0, b 07.5.0.007.0 4 b ex H b G 0,,,, erfc 07.5.0.008.0 G 0,,,, ex Ai
htt://functions.wolfrm.com 4 07.5.0.009.0 G 0,,,, 4 ex Ai Cse m, n,, 0,,, G 0,,, 07.5.0.000.0, c b, b b c Θ b c b c b c bc b c C bc b c G 0,,, 07.5.0.00.0, c b, b c b c Θ bc b b C bc b c G 0,,, 07.5.0.00.0, b, c c b c b c c bc C bc c Θ 07.5.0.00.0 G 0,,,, b, b Θ b U b 07.5.0.004.0 G 0,,,, b, b P b Θ G 0,,, 07.5.0.005.0, b, b Θ T b G 0,,, 07.5.0.00.0, b, b bθ 0 C b 07.5.0.007.0 G 0,,,, b b, b b Θ T b 07.5.0.008.0 G 0,,,, b b, b b P b Θ
htt://functions.wolfrm.com 5 07.5.0.009.0 G 0,,,, b b, b Θ b b U b Cse m, n,, 0,,, 07.5.0.000.0 G 0,,,, c, c Y c G 0,,, 07.5.0.00.0,, Ci ;, 0 Cse m, n,, 0,,, 07.5.0.00.0 G 0,,,, 4, 5, 5 ex Bi Cse m, n,, 0,, 4, 07.5.0.00.0 G 0, 4,,,,,, Chi ;, 0 G 0, 4,, 07.5.0.004.0,,,, Shi G 0, 4,, 07.5.0.005.0,,,, Bi 07.5.0.00.0 G 0, 4,,,,, 7, cosh Ai 07.5.0.007.0 G 0, 4,,,,, 5, 5 Ai sinh 07.5.0.008.0 G 0, 4,,,,, 5, 5 Bi
htt://functions.wolfrm.com Cse m, n,, 0,,, 07.5.0.009.0 G 0,,,,, Ai ;, 0 G 0,,, 07.5.0.0040.0,, 4 Ai Ai ;, 0 07.5.0.004.0 G 0,,,,, 4 4 Ai ;, 0 Cse m, n,, 0, 4, 4, 07.5.0.004.0 G 0,4 4,,,, b, b b, b b Ai K b ; Re 0 Cse m, n,,, 0, 0, 07.5.0.004.0 G,0 0,, b, c bc J bc G,0 0,, 07.5.0.0044.0 b, b 07.5.0.0045.0 b sin G,0 0,, b, b b cos Cse m, n,,, 0, 0, 4 07.5.0.057.0 G,0 0,4, 4 b, b 4, b, b 4 4 b 4 cos cosh Cse m, n,,, 0,, 07.5.0.058.0 G,0,, b,, b sin b I b
htt://functions.wolfrm.com 7 07.5.0.004.0 G,0,,,, b b I b G,0,, 4 07.5.0.0047.0 4,, 4 4 C G,0,, 4 07.5.0.0048.0 4,, 4 4 erf 4 4 erfi G,0,, 4 07.5.0.0049.0 4,, 4 4 S G,0,, 4 07.5.0.0050.0 4,, 4 4 erfi erf Cse m, n,,,,, 07.5.0.005.0 G,,,, erf Cse m, n,,,,, G,,, 07.5.0.005.0, b, b H b 07.5.0.005.0 G,,,,, Si G,,, 4 07.5.0.059.0 4,, 4 4 4 C G,,, 4 07.5.0.00.0 4,, 4 4 4 S Cse m, n,,,,, 07.5.0.0054.0 G,,,, erf Cse m, n,,,,, 4
htt://functions.wolfrm.com 8 07.5.0.0055.0 G,,4,, b, b,, b b sinb L b Cse m, n,,,,, G,,, 07.5.0.005.0,, c c H c 07.5.0.0057.0 G,,,,, Si Cse m, n,,,, 4, G, 4,, 07.5.0.0058.0, b,, b, b Cse m, n,,,,, b sin b L b ;, 0 G,,, 07.5.0.0059.0, b 07.5.0.000.0 b H b G,,,, ex Ai G,,, 07.5.0.00.0, 4 4 ex Ai Cse m, n,,,,, 07.5.0.00.0 G,,,,, sinh G,,, 07.5.0.00.0,, 07.5.0.004.0 sinh G,,,,, tn Cse m, n,,,,,
htt://functions.wolfrm.com 9 G,,, 07.5.0.005.0,, Ai Bi ;, 0 G,,, 07.5.0.00.0,, Ai Bi ;, 0 G,,, 07.5.0.007.0,, Ai Bi Ai Bi ;, 0 07.5.0.008.0 G,,,,, Ai Bi ;, 0 G,,, 07.5.0.009.0, 4, 4 Ai Bi ;, 0 Cse m, n,,,,, 07.5.0.0070.0 G,,,,,, ex Bi 07.5.0.007.0 G,,,, 4, 5, 5 ex Bi Cse m, n,,,,, 5 G,,5, 07.5.0.0.0,, b b, c, b, b, c I bc I bc G,,5, 07.5.0.0.0,, b b, b, b, b, b b b b F b ; b; F b ; b; b G,,5, 07.5.0.0.0,, b b b, b, b,, b b b b F b ; b; F b ; b; b
htt://functions.wolfrm.com 0 Cse m, n,,,, 5, 07.5.0.007.0 G, 5,,,,,,,, Bi Ai Bi 07.5.0.007.0 G, 5,,,,,,,, Ai Bi Bi G, 5,, 07.5.0.0074.0,,,, 4,, Bi Ai Bi G, 5,, 07.5.0.0075.0,,,,,, Bi Bi Ai Cse m, n,,,, 4, G, 4,, 07.5.0.007.0,, 4, 4, 4 Ai Ai Cse m, n,,, 0, 0, 07.5.0.0077.0 G,0 0,, b, c bc K cb 07.5.0.04.0 G,0 0,, b, c csc b c bc I cb I bc 07.5.0.0078.0 G,0 0,, b, b b Ai 07.5.0.0079.0 G,0 0,, b, b b G,0 0,, 07.5.0.0080.0 b, b b Ai Cse m, n,,, 0,,
htt://functions.wolfrm.com 07.5.0.008.0 G,0,, b, b b b H b 07.5.0.008.0 G,0,, 7, 7 4 ex Ai G,0,, 07.5.0.008.0, erfc G,0,, 07.5.0.0084.0 5, 5 4 ex Ai Cse m, n,,, 0,, G,0,, 07.5.0.0085.0 b, b, Y b G,0,, 07.5.0.008.0,, Ci Cse m, n,,, 0,, 07.5.0.0087.0 G,0,,, c b c Θ b, b b bc C bc b c b c b c b c G,0,, 07.5.0.0088.0, c b, b c b c Θ bc b b C bc c b G,0,, 07.5.0.0089.0, b, c c Θ b c 4 b c c bc bc C bc c G,0,, 07.5.0.0090.0, b, b Θ b U b 07.5.0.009.0 G,0,,, b, b P b Θ
htt://functions.wolfrm.com G,0,, 07.5.0.009.0, b, b Θ T b 07.5.0.009.0 G,0,,, b b, b Θ b T b ;, 0 07.5.0.0094.0 G,0,,, b b, b b b Θ 0 C b ;, 0 07.5.0.0095.0 G,0,,, b b, b b P b Θ ;, 0 07.5.0.009.0 G,0,,, b b, b Θ b U b ;, 0 b 07.5.0.0097.0 G,0,,, b b, b Θ b b U b Cse m, n,,, 0,, G,0,, 07.5.0.0098.0,, 5, Cse m, n,,, 0,, 4 ex Bi G,0,4, 07.5.0.05.0, b, b,, I b I b G,0,4, 07.5.0.0.0, b, b,, I b I b 07.5.0.07.0 G,0,4,, b, b, b, b I cosh I sinh b b G,0,4, 07.5.0.0099.0, Chi,,,
htt://functions.wolfrm.com 07.5.0.000.0 G,0,4,,,,, Bi 07.5.0.00.0 G,0,4,,,,, Shi G,0,4, 07.5.0.00.0,, 5,, Bi 07.5.0.00.0 G,0,4,, 5,, 5, 5 4 cosh Ai G,0,4, 07.5.0.004.0,,,, Bi 07.5.0.005.0 G,0,4,,, 5,, 5 5 4 sinh Ai 07.5.0.00.0 G,0,4,,, 5,, Bi Cse m, n,,,,, 07.5.0.007.0 G,,, b, b b b H b G,,, 07.5.0.008.0 7, 4 7 4 ex Ai G,,, 07.5.0.009.0 5, 5 4 ex Ai Cse m, n,,,,, G,,, 07.5.0.00.0 7,, 4 7 Ai Bi
htt://functions.wolfrm.com 4 G,,, 07.5.0.0.0 5,, 5 Ai Bi G,,, 07.5.0.0.0,, Ai Bi G,,, 07.5.0.0.0,, Ai Bi G,,, 07.5.0.04.0,, Ai Bi Ai Bi Cse m, n,,,,, 07.5.0.05.0 G,,,, csch, 07.5.0.0.0 G,,,,, cot G,,, 07.5.0.07.0,, csch Cse m, n,,,,, 07.5.0.08.0 G,,,, 7,, 7 4 ex Bi 07.5.0.09.0 G,,,, 5,, 5 4 ex Bi Cse m, n,,,,, 5 07.5.0.00.0 G,,5,, 5, 7,, 5,, 7 Bi Bi Ai
htt://functions.wolfrm.com 5 07.5.0.0.0 G,,5,,, 5,,,, 5 Bi Ai Bi 07.5.0.0.0 G,,5,,,,, 5,, 5 Bi Bi Ai 07.5.0.0.0 G,,5,,,,, 7,, 7 Bi Ai Bi Cse m, n,,,,, 4 07.5.0.04.0 G,,4,, b, b, b, b b b Ai I b 07.5.0.05.0 G,,4,, b, b, b b b Ai, b I b 07.5.0.0.0 G,,4,, 7,,, 7 4 7 4 cosh Ai 07.5.0.07.0 G,,4,,,,, Ai Ai Bi 07.5.0.08.0 G,,4,,, 7, 7, 4 7 4 sinh Ai 07.5.0.09.0 G,,4,, 7,,, Ai Bi Cse m, n,,,,, 07.5.0.00.0 G,,,, b, b, b, b tn b sinh b sinh cot b cosh b sinh
htt://functions.wolfrm.com 07.5.0.0.0 G,,,,, b b, b, b csc b b 07.5.0.0.0 b 4 b 4 4 b cos b 4 G,,,,, 4, 4, cos ;, 0 07.5.0.0.0 G,,,,, 4 4, 4, 4 cos ;, 0 Cse m, n,,,, 4, 07.5.0.04.0 G, 4,,,, b, b b, b b Ai I b G, 4,, 07.5.0.05.0,, b, b b, b b 4 5 Ai I b 07.5.0.0.0 G, 4,,,,, 5, Ai Bi 07.5.0.07.0 G, 4,,,,, 5, 4 cosh Ai G, 4,, 07.5.0.08.0,,,, Ai Ai Bi 07.5.0.09.0 G, 4,,,,, 7, 7 4 sinh Ai Cse m, n,,,, 4,
htt://functions.wolfrm.com 7 07.5.0.040.0 G, 4,,,, b, b b, b, b, b, b, b b b Bi I b 07.5.0.04.0 G, 4,,,, 4, 4 7,, 7, 4,, 4 7 4 cosh Bi 07.5.0.04.0 G, 4,,,, 4, 4 5,, 4,, 4, 5 5 4 Bi cosh 07.5.0.04.0 G, 4,,,, 4, 4, 5, 5, 4,, 4 5 4 Bi sinh 07.5.0.044.0 G, 4,,,, 4, 4, 7, 4, 4,, 7 7 4 sinh Bi 07.5.0.045.0 G, 4,,,, b, b 5 b, b, b, b 5, b, b b b Bi I b Cse m, n,,,,, 4 07.5.0.04.0 G,,4,,, b, b,, b, b,, b Bi I b 07.5.0.047.0 G,,4,,, b, b,, b, b,, b Bi I b 07.5.0.048.0 G,,4,,,,,, 7,,, Bi cosh 07.5.0.049.0 G,,4,,,,,, 5, 5,, Bi sinh
htt://functions.wolfrm.com 8 07.5.0.050.0 G,,4,,,,,, 5,,, cosh Bi 07.5.0.05.0 G,,4,,,,,, 7, 7,, sinh Bi Cse m, n,,, 0,, G,0,, 07.5.0.05.0, 7, 4 7 Ai G,0,, 07.5.0.05.0 5,, 5 Ai G,0,, 07.5.0.054.0,, 4 Ai Ai Cse m, n,,, 0,, 07.5.0.08.0 G,0,, 4, sec b 4 b I b, b, b,, 4 4 b J 4 b4 J 4 4 b, b 07.5.0.09.0 G,0,, 4, b, b, b,,, b sec b 4 b I 4 4 b J 4 4 b J 4 4 b Cse m, n,,,,, 07.5.0.070.0 G,,,, b, b 5 csc b J J Y Y b b b b Cse m, n,,,,, 4 07.5.0.055.0 G,,4,, 7,, 4, 4 Ai Ai Cse m, n,, 4, 0,, 4
htt://functions.wolfrm.com 9 07.5.0.05.0 G 4,0,4,, b, b, b, b Ai K ; Re 0 b 07.5.0.07.0 G 4,0,4,, b, b, b, b Ai K b Generl chrcteristics Brnch cuts Brnch cut loctions: When, then there is brnch cut long, 0. If, however, then the existence of the brnch cut deends on the rmeters of MeijerG; if the brnch cut exists, it will be long, 0. In cses where nd m n 0, there will be jum discontinuity on the unit circle. Sets of discontinuity hs discontinuity on the unit circle in the cse m n. Series reresenttions Generlied ower series Exnsions t 0 07.5.0.000.0 m m j j k n j j b k m sin b j b k j n j b k b k r F b k,, b k ; b b k,, b k b k, b k b k,, b b k ; mn r ; j,k,j,k j k j m k m b j b k Exnsions t 07.5.0.000.0 n n j j k m j k b j n sin k j j m k b j k r F k b,, k b ; k,, k k, k k,, k ; mn j,k,j,k j k j n k n j k r ;
htt://functions.wolfrm.com 0 Residue reresenttions 07.5.0.000.0 m res s j 0 m k n n s b k k s s k k m b k s s r b k j ; 07.5.0.0004.0 n res s j 0 m k n n s b k k s s k k m b k s s r k j ; Integrl reresenttions Contour integrl reresenttions 07.5.07.000.0 m k n r r 0 m n m n n s b k k s s k k m b k s s r s ; Trnsformtions Trnsformtions nd rgument simlifictions Argument involving bsic rithmetic oertions 07.5..000.0 Α G m,n,, Α n, Α n,, Α,, r Α b,, Α b m, Α b m,, Α b Αr 07.5..000.0 n,m G,, r b,, b m, b m,, b G m,n,, n, n,,,, r ; 07.5..000.0 m,n G,, r ; G n,m,, r 07.5..0004.0 b,, b m, b m,, b G m,n,, n, n,,,, r ; 07.5..0005.0 G m,n, t,, n, n,,, r t ; t t Identities
htt://functions.wolfrm.com Functionl identities 07.5.7.000.0 G m,n,, n, n,,,, r Α r Α,, Α n, Α n,, Α Α b,, Α b m, Α b m,, Α b 07.5.7.000.0 G m,n,, n, n,,,, r n,m G,, r b,, b m, b m,, b ;,, n, n,, 07.5.7.0004.0 G m,n,, n, n,,,, r m,n G,, r ; 07.5.7.0005.0 G m,n,, n, n,,,, r G n,m,, r b,, b m, b m,, b,, n, n,, ; 07.5.7.000.0 G m,n,, n, n,,,, r Ξ r Θ m u,n G u u, u u r, v,, u,, n,, nu u u u u b,, b u,, b m,, b mu u u u u, n u, b m u,, n u,,,, u u u u,, b m u,, b,, b u u u u ; r u v u v Ξ m n u Θ k b k Differentition Low-order differentition With resect to 07.5.0.000.0 r G m,n,, r r, r,, n r, n r,, r b r,, b m r, r, b m r,, b r Symbolic differentition With resect to 07.5.0.000.0 u G m,n,, n, n,,,, r r u m,nu G u,u u, r r, r,, u r, u r,, n u r, n u r,, u r b u r,, b m u r, r, r,, u r, b m u r,, b u r ; u
htt://functions.wolfrm.com Frctionl integro-differentition With resect to u 07.5.0.000.0 u s m r Α s r k n r r 0 m n m n n s b k k s s k k m b k s s r Α s ; Integrtion Indefinite integrtion Involving only one direct function 07.5..000.0 G m,n,, n, n,, m,n,, r r G b,, b m, b m,, b,, r, r,, r n, r n,, r r b,, r b m, 0, r b m,, r b Involving one direct function nd elementry functions Involving ower function 07.5..000.0 Α G m,n,, n, n,, m,n,, r r G b,, b m, b m,, b,, r, r Α,, r Α n, r Α n,, r Α r Α b,, r Α b m, 0, r Α b m,, r Α b Integrl trnsforms Mellin trnsforms t 07.5..000.0 r m k n n r s b k k r s r s k k m b k r s Reresenttions through more generl functions Through hyergeometric functions Hyergeometric functions re not more generl thn Meijer G function. Involving F
htt://functions.wolfrm.com 07.5..000.0 m m j j k n j j b k m sin b j b k j n j b k b k r F b k,, b k ; b b k,, b k b k, b k b k,, b b k ; mn r ; j,k,j,k j k j m k m b j b k Involving F 07.5..000.0 m m b n j b k j j b k j j k j n b k r j b k j m b j b k F b k,, b k ; b b k,, b k b k, b k b k,, b b k ; mn r ; j,k,j,k j k j m k m b j b k Through Fox H 07.5..000.0 G m,n,, n, n,,,, r r H m,n,, r,, n, r, n, r,,, r b, r,, b m, r, b m, r,, b, r Through Meijer G Clssicl cses Clssicl Meijer G function is rticulr cse of Meijer G function with rmeter r. 07.5..0004.0 G m,n, r ; r r r rg r 07.5..0005.0 Ξ m r,n G r r, r r r,, r,, n,, nr r r r r b,, b r,, b m,, b mr r r r r, n r, b m r m n r Θ k b k r,, n r,,,, r r r r,, b m r,, b,, b r r r r ; History J. Keier nd O.I. Mrichev (99) extended MeijerG function s rt of the Mthemtic system develoment effort.
htt://functions.wolfrm.com 4 Coyright This document ws downloded from functions.wolfrm.com, comrehensive online comendium of formuls involving the secil functions of mthemtics. For key to the nottions used here, see htt://functions.wolfrm.com/nottions/. Plese cite this document by referring to the functions.wolfrm.com ge from which it ws downloded, for exmle: htt://functions.wolfrm.com/constnts/e/ To refer to rticulr formul, cite functions.wolfrm.com followed by the cittion number. e.g.: htt://functions.wolfrm.com/0.0.0.000.0 This document is currently in reliminry form. If you hve comments or suggestions, lese emil comments@functions.wolfrm.com. 00-008, Wolfrm Reserch, Inc.