Factorial. Notations. Specific values. Traditional name. Traditional notation. Mathematica StandardForm notation. Specialized values

Μέγεθος: px

Εμφάνιση ξεκινά από τη σελίδα:

Download "Factorial. Notations. Specific values. Traditional name. Traditional notation. Mathematica StandardForm notation. Specialized values"

Γολγοθά Κούνδουρος
5 χρόνια πριν
Προβολές:

1 Factorial Notatios Traditioal ame Factorial Traditioal otatio Mathematica StadardForm otatio Factorial Specific values Specialized values k ; k ; p q q p q p k q ; p q p q k p q q k p k q q p q ; p q p q Values at fixed poits

2 Values at ifiities 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

3 Symmetries ad periodicities Mirror symmetry 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. ; ig k, ; k,, res k k ; k k Brach poits The fuctio does ot have brach poits Brach cuts The fuctio does ot have brach cuts Series represetatios Geeralized power series Expasios at 0 ; 0 m Ψ 0 0 Ψ 0 Ψ Ψ 0 Ψ 0 Ψ 0 Ψ 0 0 ; 0 0

4 k 0 0 k ; 0 0 k Ψ 0 0 O 0 ; 0 0 Expasios at m m O m ; m m m m m m m m Ψm m 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 Π ; Stirlig's formula Π O 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 ; 0 ; arg Π Π O ; arg Π

5 5 a b ab k b a k Bk, a b, a k k t Α t z ; arga Π B, Α, z t t Α a b a b a b ab O ; arga Π Product represetatios k ; k k k k k k Π k siπ k ; k Ζ k exp k exp k Ζk k k k Limit represetatios lim x x k x k m m lim m m lim m m, m ; w lim w F ; ; w

6 6 lim m m t m m t t ; Re Trasformatios Trasformatios ad argumet simplificatios Argumet ivolvig basic arithmetic operatios Π cscπ m m m m ; m m Multiple argumets Π Π m m m Π m m k ; m m Products, sums, ad powers of the direct fuctio Products of the direct fuctio Π siπ Π csc Π

7 Π csc Π Π csc Π m m m m m m m m m m m m m m, m m m m, m Idetities Recurrece idetities Cosecutive eighbors Distat eighbors m m m m m ; m Fuctioal idetities

8 8 Relatios of special kid 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 Ψ Ψ Ψ Symbolic differetiatio m z Rm, ; Rm, z Ψz Rm, z m R0, m, z R0, z m 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 t logt Α k QΑ, logt t t Α F, ; Α; k k k Summatio Fiite summatio o o m m k, 0 k m, 0 i j k i,j 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

9 9 Ifiite summatio Parameter-free sums k k k 6 6 Π k k k Π k k k k k 4 k k k 5 k k 0 Π k Π Π k Π k Π k Π k

10 k Π k k 9 k Ζ 4 k 5 78 Ζ 077 k k Ζ k k Ζ k k 4 Π 4 k k 4 k Π Π k k k Π log

11 k k k Π log log k 5 k 56 k k k 4 k k Ζ k 5 88 k 56 k k k 6 0 Π k 6 k 5 k k k 6 k 4 Parameter-cotaiig sums k k ; k k k log k k 4 k j j ; j j 5Π log 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 k ; 4 j 4 4 j 4 4 j k k 4 Π log4 4 k 4 4 j 4 j 40 j 8 j j 9 ; 4 j j 4 j 4 4

12 k k log 4 k j j 40 j 40 j 40 ; 4 j 4 j 4 4 j k k log log j ; k k j k k 4 6 log log 4 j ; k k j k k 8 j 9 j 9 7 k k j j Π log log4 ; k k 8 Π k k 8 j 9 j 9 4 ; j j k k k 8 j 9 j 9 k j j Π log log4 6 6 ; k k p k p p ; p p k k k p k p p ; p p

13 k k k p k p log p p p p p j j ; k k 6 4 log 6 j 5 j 5 4 j ; k 4 j j j k k j 0 j 0 j 7 Π 6 4 ; k 4 j j k Π j j 4 ; k 6 4 j j 4 j k k log 4 ; j k p logp logp p k k p p k k k cos j p j p ; j j ; j k cosh p p p k j p ; j

14 k 4 a Π 4 k 4 ; a 90 a Π Π 4 a 7 a a ; k a ; a k k Π log a Π log log a a a ; k k j 56 j j j 56 5 j 4 j j Ζ ; k cos Π 8 a ; a 0 a 6 si 7 k k k Π 8 a 6 7 a 9 a 4 ; k k k 0 Π j 8 j 7 j j j 6 6 j 6 j 5 6 j 6 j 6 j j ;

15 k 4 j z j z j 4 z 4 k k z k j z z ; k k Π log 6 j 5 j 5 4 j 4 j j j 4 j 4 j ; k k k k 4 j 4 4 j 4 Π 4 j 0 0 j j 7 Π j j ; k k z k z log z z z z z log z z z z j z j z j ; k 4 k 4 j j 0 40 j 9 j 0 j Π j 4 j 4 4 j log 4 4 j 4 4 j ;

16 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 j 4 4 j ; k 4 k 4 4 Π 4 4 log4 4 j 4 j 40 j 8 j j 9 j j 4 j log j 4 j ; k 4 k 4 4 log j j 40 j 40 j 40 j 4 j 4 4 j log j 4 4 j ; k 4 k 9 k log 4 8 j 9 j 9 7 j j Π log log4 log 4 4 j j j ;

17 k 4 k 9 k log 4 8 j 9 j 9 4 j j 8 Π log 4 4 j j j ; k 4 k 9 k log j 9 j 9 j j Π log log4 6 6 log 4 4 j j j ; 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 a ba lim b Represetatios through more geeral fuctios

18 8 Through other fuctios Ivolvig some hypergeometric-type fuctios , 0 ; Re Represetatios through equivalet fuctios With related fuctios cos Π Π si Π Iequalities k k k k k ; k k ; k k ; k Zeros ; Theorems Taylor's formula f k a z a k f z. k Derivative of compositio (Faà di Bruo's formula)

19 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

20 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 Please cite this documet by referrig to the fuctios.wolfram.com page from which it was dowloaded, for example: To refer to a particular formula, cite fuctios.wolfram.com followed by the citatio umber. e.g.: This documet is curretly i a prelimiary form. If you have commets or suggestios, please commets@fuctios.wolfram.com , Wolfram Research, Ic.

Σχετικά έγγραφα

HermiteHGeneral. Notations. Primary definition. Specific values. Traditional name. Traditional notation. Mathematica StandardForm notation

HermiteHGeneral. Notations. Primary definition. Specific values. Traditional name. Traditional notation. Mathematica StandardForm notation HermiteHGeeral Notatios Traditioal ame Hermite fuctio Traditioal otatio H Mathematica StadardForm otatio HermiteH, Primary defiitio 07.0.0.000.0 H F ; ; F ; 3 ; Specific values Specialied values For fixed