Gmm3 Nottios Trditiol me Geerlized icomplete gmm fuctio Trditiol ottio, z, z Mthemtic StdrdForm ottio Gmm, z, z Primry defiitio 06.07.0.000.0, z, z z z t t t Specific vlues Specilized vlues 06.07.03.000.0, z, 0, z ; Re 0 06.07.03.000.0, 0, z, z ; Re 0 Vlues t ifiities 06.07.03.0003.0, z,, z 06.07.03.0004.0,, z, z 06.07.03.0005.0, 0, ; Re 0 Geerl chrcteristics Domi d lyticity
http://fuctios.wolfrm.com, z, z is lyticl fuctio of, z, z which is defied i 3. For fixed z, z, it is etire fuctio of. 06.07.04.000.0 z z, z, z Symmetries d periodicities Mirror symmetry 06.07.04.000.0, z, z, z, z ; z, 0 z, 0 Permuttio symmetry 06.07.04.0003.0, z, z, z, z Periodicity No periodicity Poles d essetil sigulrities With respect to z k For fixed, the fuctio, z, z hs essetil sigulrity t z (for fixed z ) d t z (for fixed z ). At the sme time, the poits z k ; k, re brch poits for geeric. 06.07.04.0004.0 ig zk, z, z, ; k, With respect to For fixed z, z, the fuctio, z, z hs oly oe sigulr poit t. It is essetil sigulr poit. 06.07.04.0005.0 ig, z, z, Brch poits With respect to z The fuctio, z, z hs for fixed, z or fixed, z (with ) two brch poits with respect to z or z : z k 0, z k, k,. At the sme time, the poits z k ; k, re essetil sigulrities. 06.07.04.0006.0 zk, z, z 0, ; k, 06.07.04.0007.0 zk, z, z, 0 log ; k, 06.07.04.0008.0 zk p q, z, z, 0 q ; p q gcdp, q k,
http://fuctios.wolfrm.com 3 06.07.04.0009.0 zk, z, z, log ; k, 06.07.04.000.0 zk p q, z, z, q ; p q gcdp, q k, With respect to For fixed z, z, the fuctio, z, z does ot hve brch poits. 06.07.04.00.0, z, z Brch cuts With respect to z For fixed z d, the fuctio, z, z is sigle-vlued fuctio o the z -ple cut log the itervl, 0, where it is cotiuous from bove. 06.07.04.00.0 z, z, z, 0, 06.07.04.003.0 lim, z, x Ε, z, x ; x 0 Ε0 06.07.04.004.0 lim, z, x Ε, z, x Π, x, 0 ; x 0 Ε0 With respect to z For fixed z d, the fuctio, z, z is sigle-vlued fuctio o the z -ple cut log the itervl, 0, where it is cotiuous from bove. 06.07.04.005.0 z, z, z, 0, 06.07.04.006.0 lim, x Ε, z, x, z ; x 0 Ε0 06.07.04.007.0 lim, x Ε, z Π, 0, x, x, z ; x 0 Ε0 With respect to For fixed z, z, the fuctio, z, z does ot hve brch cuts. 06.07.04.008.0, z, z Series represettios Geerlized power series
http://fuctios.wolfrm.com 4 Expsios t z, z 0, 0 For the fuctio itself Geerl cse, z, z z 06.07.06.000.0 z z z z z ; z 0 z 0, z, z z 06.07.06.0008.0 z z Oz 3 z z z Oz 3 06.07.06.000.0 z k, z, z z k k z z k k k 06.07.06.0003.0, z, z z F ; ; z z F ; ; z 06.07.06.0004.0, z, z z Oz z Oz Specil cses 06.07.06.0009.0, z, z z z Oz Oz 06.07.06.000.0, z, z z z Oz Oz ; 06.07.06.0005.0 z k, z, z z z k k z k ; 06.07.06.0006.0, z, z k z k z k logz logz ; k k 06.07.06.00.0 0, z, z logz logz z z 4 z 3 8 Oz 4 z z 4 z 3 8 Oz 4 06.07.06.00.0 k, z, z logz logz z z z z 7 Oz 4 z z z z 7 Oz 4 3 3
http://fuctios.wolfrm.com 5 06.07.06.003.0, z, z k z k z k logz logz ; k k k 06.07.06.0007.0 k z k z k, z, z k k z F, ;, ; z z F, ;, ; z logz logz ; Itegrl represettios O the rel xis Of the direct fuctio 06.07.07.000.0 z t, z, z t t z Differetil equtios Ordiry lier differetil equtios d wroskis For the direct fuctio itself With respect to z 06.07.3.000.0 z w z z w z 0 ; wz c, z, z c 06.07.3.000.0 W z,, z, z z z With respect to z 06.07.3.0003.0 z w z z w z 0 ; wz c, z, z c 06.07.3.0004.0 W z,, z, z z z Trsformtios Trsformtios d rgumet simplifictios Argumet ivolvig bsic rithmetic opertios
http://fuctios.wolfrm.com 6 06.07.6.000.0, z, z z z z z, z, z 06.07.6.000.0, z, z, z, z z z z z 06.07.6.0003.0 k z k, z, z, z, z z z z k k k k ; 06.07.6.0004.0, z, z, z, z z k z k k z k z k k ; Idetities Recurrece idetities Cosecutive eighbors 06.07.7.000.0, z, z, z, z z z z z 06.07.7.000.0, z, z, z, z z z z z Distt eighbors 06.07.7.0003.0 z k, z, z, z, z z k k z k z k k ; 06.07.7.0004.0, z, z, z, z z k z k k z k z k k ; Fuctiol idetities Reltios of specil kid 06.07.7.0005.0, z, z k 0, z z 0, z z z k k k k z k ; Differetitio Low-order differetitio With respect to
http://fuctios.wolfrm.com 7 06.07.0.000.0, z, z F, ;, ; z z F, ;, ; z z, 0, z logz, 0, z logz 06.07.0.000.0, z, z, z log z, z log z log z log z z 3F 3,, ;,, ; z logz F, ;, ; z 3 z 3F 3,, ;,, ; z logz F, ;, ; z 3 With respect to z 06.07.0.0003.0, z, z z z z 06.07.0.0004.0, z, z z z z z With respect to z 06.07.0.0005.0, z, z z z z 06.07.0.0006.0, z, z z z z z Symbolic differetitio With respect to 06.07.0.0007.0, z, z, k logz, k logz ; k k 06.07.0.0008.0, z, z z j j 0 j k j j log j z j F j,,, j ;,,, j ; z z j j 0 j j j log j z j F j,,, j ;,,, j ; z ;
http://fuctios.wolfrm.com 8 With respect to z 06.07.0.006.0, z, z, z, z z z k k z k ; 06.07.0.0009.0, z, z z k z k k k k, z ; With respect to z 06.07.0.007.0, z, z, z, z z z k k z k ; 06.07.0.000.0, z, z, z z z k k k, z ; Frctiol itegro-differetitio With respect to 06.07.0.00.0 Α, z, z Α z t Α logt Α t QΑ, 0, logt t z With respect to z 06.07.0.00.0 Α, z, z z Α Α z Α, z z Α F ; Α ; z ; Α, z, z z Α 06.07.0.003.0 z Α Α, z k Α kα exp z, k z k k With respect to z 06.07.0.004.0 Α, z, z Α z Α F Α z ; Α ; z z Α, z ; Α, z, z z Α 06.07.0.005.0 z Α Α, z k Α kα exp z, k z k k Itegrtio Idefiite itegrtio
http://fuctios.wolfrm.com 9 Ivolvig oly oe direct fuctio 06.07..000.0, z, z z, z z, z, z Ivolvig oe direct fuctio d elemetry fuctios Ivolvig power fuctio 06.07..000.0 z Α, z, z z Α, z, z z Α Α, z Ivolvig expoetil fuctio 06.07..0003.0 b z c, 0, z z b zc c, b z b z c b z c b z c, z 06.07..0004.0 z z c, 0, z b z c b b z c c, b b z b b z c z c z c, b z Ivolvig fuctios of the direct fuctio d elemetry fuctios Ivolvig elemetry fuctios of the direct fuctio d elemetry fuctios Ivolvig powers of the direct fuctio d expoetil fuctio 06.07..0005.0 z, 0, z z, z z z z, z z z, z, z Ivolvig oly oe direct fuctio with respect to z 06.07..0006.0, z, z z z, z, z, z Ivolvig oe direct fuctio d elemetry fuctios with respect to z Ivolvig power fuctio 06.07..0007.0 z Α, z, z z Α z Α, z, z Α, z Ivolvig oly oe direct fuctio with respect to
http://fuctios.wolfrm.com 0 06.07..0008.0, z, z z z t t logt t Represettios through more geerl fuctios Through hypergeometric fuctios Ivolvig F 06.07.6.000.0, z, z z F ; ; z z F ; ; z ; Ivolvig F 06.07.6.000.0, z, z z F ; ; z z Ivolvig hypergeometric U F ; ; z ; 06.07.6.0003.0, z, z z U,, z z U,, z Through Meijer G Clssicl cses for the direct fuctio itself 06.07.6.0004.0, z, z G,, z, 0 G,, z, 0 06.07.6.0005.0, z, z G,0, z 0, G,0, z 0, 06.07.6.0006.0, 0, z G,, z, 0, 0, 06.07.6.0007.0 z Π, G z,3 4,, 0 Clssicl cses ivolvig exp 06.07.6.0008.0 z, 0, z Π cscπ G,,3 z, 0, 0, 0 06.07.6.0009.0 z, 0, z z G,, z 0 0, Geerlized cses for the direct fuctio itself
http://fuctios.wolfrm.com 06.07.6.000.0, 0, z Π, G z,3,,, 0 Represettios through equivlet fuctios With iverse fuctio 06.07.7.000.0, z, Q, z, z z With relted fuctios 06.07.7.000.0, z, z, z, z 06.07.7.0003.0, z, z Q, z, z 06.07.7.0004.0, z, z z E z z E z
http://fuctios.wolfrm.com Copyright This documet ws dowloded from fuctios.wolfrm.com, comprehesive olie compedium of formuls ivolvig the specil fuctios of mthemtics. For key to the ottios used here, see http://fuctios.wolfrm.com/nottios/. Plese cite this documet by referrig to the fuctios.wolfrm.com pge from which it ws dowloded, for exmple: http://fuctios.wolfrm.com/costts/e/ To refer to prticulr formul, cite fuctios.wolfrm.com followed by the cittio umber. e.g.: http://fuctios.wolfrm.com/0.03.03.000.0 This documet is curretly i prelimiry form. If you hve commets or suggestios, plese emil commets@fuctios.wolfrm.com. 00-008, Wolfrm Reserch, Ic.