Factorial Notatios Traditioal ame Factorial Traditioal otatio Mathematica StadardForm otatio Factorial Specific values Specialized values 06.0.0.000.0 k ; k 06.0.0.000.0 ; 06.0.0.000.0 p q q p q p k q ; p q p q k 06.0.0.0004.0 p q q k p k q q p q ; p q p q Values at fixed poits 06.0.0.0005.0 06.0.0.0006.0 0 06.0.0.0007.0
http://fuctios.wolfram.com 6 06.0.0.0008.0 06.0.0.0009.0 06.0.0.000.0 06.0.0.00.0 4 4 06.0.0.00.0 5 0 06.0.0.00.0 6 70 06.0.0.004.0 7 5040 06.0.0.005.0 8 40 0 06.0.0.006.0 9 6 880 06.0.0.007.0 0 68 800 Values at ifiities 06.0.0.008.0 06.0.0.009.0 06.0.0.000.0 0 06.0.0.00.0 0 06.0.0.00.0 Geeral characteristics Domai ad aalyticity is a aalytical fuctio of which is defied i the whole complex -plae with the exceptio of coutably may poits k ; k. is a etire fuctio. 06.0.04.000.0
http://fuctios.wolfram.com Symmetries ad periodicities Mirror symmetry 06.0.04.000.0 Periodicity No periodicity Poles ad essetial sigularities The fuctio has a ifiite set of sigular poits: a) k ; k are the simple poles with residues k k b) is the poit of covergece of poles, which is a essetial sigular poit. ; 06.0.04.000.0 ig k, ; k,, 06.0.04.0004.0 res k k ; k k Brach poits The fuctio does ot have brach poits. 06.0.04.0005.0 Brach cuts The fuctio does ot have brach cuts. 06.0.04.0006.0 Series represetatios Geeralized power series Expasios at 0 ; 0 m 06.0.06.000.0 0 Ψ 0 0 Ψ 0 Ψ 0 0 6 Ψ 0 Ψ 0 Ψ 0 Ψ 0 0 ; 0 0
http://fuctios.wolfram.com 4 06.0.06.000.0 k 0 0 k ; 0 0 k 06.0.06.000.0 0 Ψ 0 0 O 0 ; 0 0 Expasios at m 06.0.06.0004.0 m O m ; m m m m 06.0.06.0005.0 m m m m Ψm m 06.0.06.0006.0 m m m m m O m ; m m Ψm 6 Ψm Π Ψ m m 6 Ψm Π Ψ m Ψm Ψ m m 60 5 Ψm4 Π Ψ m Ψm 4 Ψ m Ψm Ψ m Π Ψ m 7 Π 4 5 Ψ m m O m 4 ; m m Asymptotic series expasios 06.0.06.0007.0 Π ; Stirlig's formula 06.0.06.0008.0 Π 88 9 5 840 57 488 0 4 6 879 09 08 880 5 5 46 89 75 46 796 800 6 54 70 5 4 48 59 4 6 9 6 7 O 90 96 56 600 7 86 684 09 9 600 8 54 904 800 886 784 000 9 06.0.06.0009.0 Π k j P j k, j k jk j k k j arg Π Pm, j m m Pm, j Pm, j P0, 0 Pm, m Pm, j 0 ; m j 06.0.06.000.0 ; 0 ; arg Π Π O ; arg Π
http://fuctios.wolfram.com 5 a 06.0.06.00.0 b ab k b a k Bk, a b, a k k t Α t z ; arga Π B, Α, z t t Α 06.0.06.00.0 a b a b a b ab O ; arga Π Product represetatios 06.0.08.000.0 k ; k 06.0.08.000.0 k k k 06.0.08.000.0 k k Π k 06.0.08.0004.0 siπ k ; k Ζ k exp k 06.0.08.0005.0 exp k Ζk k k k Limit represetatios lim x 06.0.09.000.0 x k x k 06.0.09.000.0 m m lim m m 06.0.09.000.0 lim m m, m 06.0.09.0004.0 ; w lim w F ; ; w
http://fuctios.wolfram.com 6 lim m 0 06.0.09.0005.0 m t m m t t ; Re Trasformatios Trasformatios ad argumet simplificatios Argumet ivolvig basic arithmetic operatios 06.0.6.000.0 Π cscπ 06.0.6.000.0 06.0.6.000.0 m m 06.0.6.0004.0 06.0.6.0005.0 m m ; m m Multiple argumets 06.0.6.0006.0 Π 06.0.6.0007.0 Π 06.0.6.0008.0 m m m Π m m k ; m m Products, sums, ad powers of the direct fuctio Products of the direct fuctio 06.0.6.0009.0 Π siπ 06.0.6.000.0 Π csc Π
http://fuctios.wolfram.com 7 06.0.6.00.0 Π csc Π 06.0.6.00.0 Π csc Π 06.0.6.00.0 m m m 06.0.6.004.0 m m m 06.0.6.005.0 m m m 06.0.6.006.0 m m m 06.0.6.007.0 m m, m 06.0.6.008.0 m m m, m Idetities Recurrece idetities Cosecutive eighbors 06.0.7.000.0 06.0.7.000.0 Distat eighbors 06.0.7.000.0 m m 06.0.7.0004.0 m m m ; m Fuctioal idetities
http://fuctios.wolfram.com 8 Relatios of special kid 06.0.7.0005.0 f f ; f g g g f is the uique ozero solutio of the fuctioal equatio f f which is logarithmically covex for all real 0; that is, for which log f is a covex fuctio for 0. Differetiatio Low-order differetiatio 06.0.0.000.0 Ψ 06.0.0.000.0 Ψ Ψ Symbolic differetiatio 06.0.0.000.0 m z Rm, ; Rm, z Ψz Rm, z m R0, m, z R0, z m 06.0.0.0004.0 m m t log m t t t m m mf m z, z,, z m ; z, z,, z m ; ; m z z z m m Fractioal itegro-differetiatio Α Α Α 06.0.0.0005.0 t logt Α k QΑ, logt t t Α F, ; Α; k k k Summatio Fiite summatio 06.0..000.0 o o m m k, 0 k m, 0 i j k i,j 06.0..000.0 k k k H. J. Brothers m i p a i j b j ; j m k i,j a i k i,j b i a i b j p o Maxk,,, k m, i m i j
http://fuctios.wolfram.com 9 Ifiite summatio Parameter-free sums 06.0..000.0 k k k 6 6 Π 06.0..0004.0 k k k 4 7 9 Π 06.0..0005.0 k k k 6 06.0..0006.0 k k 4 k 8 06.0..0007.0 k k 5 k 0 06.0..0008.0 k 0 Π k 06.0..0009.0 4 8 7 Π 88 000 480 45 79 Π 66 679 00 5 k 67 60 Π k 4 664 06.0..000.0 49 750 66 5 5 00 Π 06 79 9 600 k 9 760 Π k 6 0 400 000
http://fuctios.wolfram.com 0 06.0..00.0 k 680 498 75 7 59 0 Π k 8 8 78 450 790 400 000 06.0..00.0 k 9 k 06.0..00.0 Ζ 4 k 5 78 Ζ 077 k 5 7 04 06.0..004.0 k 7 89 467 4 000 Ζ k 7 50 884 800 000 06.0..005.0 k 9 559 000 Ζ 5 09 07 k 9 7 000 76 07 0 640 000 06.0..006.0 k 4 Π 4 k 4 90 06.0..007.0 k 4 k 4 45 575 50 Π Π 4 06.0..008.0 k k k Π log
http://fuctios.wolfram.com 06.0..009.0 k k k Π log log 06.0..000.0 k 5 k 56 k k k 4 k k Ζ 06.0..00.0 k 5 88 k 56 k k k 6 0 Π k 6 k 5 k k k 6 k 4 Parameter-cotaiig sums 06.0..00.0 k k ; 06.0..00.0 k k k log k k 4 k 06.0..004.0 j j ; j j 5Π log4 4 06.0..005.0 j j 0 40 j 9 j 0 j 9 0 ; j j 4 4 j 4 k k 4 Π j j 0 40 j 9 j 0 j 9 4 4 k ; 4 j 4 4 j 4 4 j 4 06.0..006.0 k k 4 Π log4 4 k 4 4 j 4 j 40 j 8 j 40 40 j 9 ; 4 j j 4 j 4 4
http://fuctios.wolfram.com 06.0..007.0 k k log 4 k j j 40 j 40 j 40 ; 4 j 4 j 4 4 j 4 4 06.0..008.0 k k log log j ; k k j 06.0..009.0 k k 4 6 log log 4 j ; k k j 06.0..000.0 k k 8 j 9 j 9 7 k k j j Π log log4 ; 06.0..00.0 k k 8 Π k k 8 j 9 j 9 4 ; j j 06.0..00.0 8 k k k 8 j 9 j 9 k j j Π log log4 6 6 ; 06.0..00.0 k k p k p p ; p p 06.0..004.0 k k k p k p p ; p p
http://fuctios.wolfram.com 06.0..005.0 k k k p k p log p p p p p j j ; 06.0..006.0 k k 6 4 log 6 j 5 j 5 4 j ; k 4 j j j 06.0..007.0 k k j 0 j 0 j 7 Π 6 4 ; k 4 j j 06.0..008.0 k Π j j 4 ; k 6 4 j j 4 j 06.0..009.0 k k log 4 ; j 06.0..0040.0 k p logp logp p k k p p k 06.0..004.0 k k cos j p j p ; j j ; j 06.0..004.0 k cosh p p p k j p ; j
http://fuctios.wolfram.com 4 06.0..004.0 k 4 a Π 4 k 4 ; a 90 a 45 575 50 Π Π 4 a 7 a 5 5 0 4 4 4 a ; 4 06.0..0044.0 k a ; a k k Π log a Π log log a a a ; 06.0..0045.0 k k j 56 j j j 56 5 j 4 j j Ζ ; 06.0..0046.0 k cos Π 8 a ; a 0 a 6 si 7 k k k Π 8 a 6 7 a 9 a 4 ; k 06.0..0047.0 k k 0 Π 6 4 6 j 8 j 7 j 7 5 6 j j 6 6 j 6 j 5 6 j 6 j 6 j j ;
http://fuctios.wolfram.com 5 06.0..0048.0 k 4 j z j z j 4 z 4 k k z k j z z ; 06.0..0049.0 k k Π 4 4 6 4 log 6 j 5 j 5 4 j 4 j j j 4 j 4 j 4 06.0..0050.0 ; k k k k 4 j 4 4 j 4 Π 4 j 0 0 j j 7 Π 6 4 4 j j ; 06.0..005.0 k k z k z log z z z z z log z z z z j z j z j ; 06.0..005.0 4 k 4 k 4 j j 0 40 j 9 j 0 j 9 4 4 Π j 4 j 4 4 j 4 4 4 4 log 4 4 j 4 4 j ;
http://fuctios.wolfram.com 6 06.0..005.0 4 k 4 k 4 j j 0 40 j 9 j 0 j 9 0 j j 4 j 4 4 5Π 4 log4 4 log 4 4 4 j 4 4 j ; 06.0..0054.0 4 k 4 k 4 4 Π 4 4 log4 4 j 4 j 40 j 8 j 40 40 j 9 j j 4 j 4 4 4 log 4 4 4 4 j 4 j ; 06.0..0055.0 4 k 4 k 4 4 log j j 40 j 40 j 40 j 4 j 4 4 j 4 4 4 log 4 4 4 j 4 4 j ; 06.0..0056.0 4 k 4 k 9 k log 4 8 j 9 j 9 7 j j Π log log4 log 4 4 j j j ;
http://fuctios.wolfram.com 7 06.0..0057.0 4 k 4 k 9 k log 4 8 j 9 j 9 4 j j 8 Π log 4 4 j j j ; 06.0..0058.0 4 k 4 k 9 k log 4 8 8 j 9 j 9 j j Π log log4 6 6 log 4 4 j j j ; 06.0..0059.0 k cos Π 8 a ; a 0 a 6 si 7 k k k Π 8 a 6 7 a 9 a 4 ; Operatios Limit operatio 06.0.5.000.0 a ba lim b Represetatios through more geeral fuctios
http://fuctios.wolfram.com 8 Through other fuctios Ivolvig some hypergeometric-type fuctios 06.0.6.000.0, 0 ; Re Represetatios through equivalet fuctios With related fuctios 06.0.7.000.0 06.0.7.000.0 06.0.7.000.0 4 cos Π Π si Π 06.0.7.0004.0 Iequalities 06.0.9.000.0 k k k k k ; k k 06.0.9.000.0 ; 06.0.9.000.0 k k ; k Zeros 06.0.0.000.0 0 ; Theorems Taylor's formula f k a z a k f z. k Derivative of compositio (Faà di Bruo's formula)
http://fuctios.wolfram.com 9 f gx x m k k k m k k k f m gx j j k j k j j g j x k j m m m j m j gx mj gxj x f m gx. Compositio of two series b m a k z k m 0 k m c z ; c 0 b 0 c a b c a b a b c a b a a b a b 0 c 4 a 4 b a a b a b a a b a 4 b 4 c k,k,,k 0 k k k a j k j m b m j k j. m j k j Maxfield theorem ad Castell cojecture J. E. Maxfield proved that the base 0 digits of ay positive iteger occur i m as the first digits for some iteger m (J. E. Maxfield. Math. Mag. 4, 64, (970)). Castell's cojecture states that the digits to b of the base b expasio of are asymptotically equally distributed (S. P. Castell. Eureka, 6, 44 97). History J. Stirlig (70) foud his famous asymptotic formula L. Euler (75) C. Kramp (808, 86) itroduced the otatio
http://fuctios.wolfram.com 0 Copyright This documet was dowloaded from fuctios.wolfram.com, a comprehesive olie compedium of formulas ivolvig the special fuctios of mathematics. For a key to the otatios used here, see http://fuctios.wolfram.com/notatios/. Please cite this documet by referrig to the fuctios.wolfram.com page from which it was dowloaded, for example: http://fuctios.wolfram.com/costats/e/ To refer to a particular formula, cite fuctios.wolfram.com followed by the citatio umber. e.g.: http://fuctios.wolfram.com/0.0.0.000.0 This documet is curretly i a prelimiary form. If you have commets or suggestios, please email commets@fuctios.wolfram.com. 00-008, Wolfram Research, Ic.