Factorial2. Notations. Primary definition. Specific values. Traditional name. Traditional notation. Mathematica StandardForm notation

Transcription

1 Factorial Notations Traditional name Double factorial Traditional notation n Mathematica StandardForm notation Factorialn Primary definition n cos n 4 n n Specific values Specialized values k k j ; k j k k k ; k k ; k k k j ; k j k k k ; k

2 Values at fixed points Values at infinities

3 3 General characteristics Domain and analyticity n is an analytical function of n which is defined in the whole complex n-plane with the exception of countably many points n k ; k. n is an entire function nn Symmetries and periodicities Mirror symmetry n n Periodicity No periodicity Poles and essential singularities The function n has an infinite set of singular points: a) n k ; k are the simple poles with residues k k b) n is the point of convergence of poles, which is an essential singular point ing n n k, ; k,, ; res n nk k ; k k Branch points The function n does not have branch points n n Branch cuts The function n does not have branch cuts n n Series representations

4 4 Generalized power series Expansions at n n 0 ; n 0 m n 0 4 log4 Ψ n log 4 Ψ n 0 log sinn 0 n n 0 4 Ψ n 0 4 Ψ n 0 log log sinn 0 log 4 cosn 0 sinn 0 4 log log sinn 0 n n log3 8 Ψ n Ψ n 0 6 Ψ n 0 4 log 4 Ψ n 0 log 4 cosn 0 sinn 0 log6 log sinn 0 log 4 cosn 0 log sinn 0 log 6 log sinn 0 log64 log sinn 0 6 log sin n 0 Ψ n n 0 4 log4 Ψ n 0 Expansions at n m Ψ n 0 log4 log sin n 0 n n 0 3 ; n n 0 n 0 log sinn 0 n n 0 On n 0 ; n n 0 n 0 m m On m ; n m m m m n m m m n m log Ψm n m On m ; n m m

5 m m m m n log Ψm n m 4 3 log 3 Ψm log8 3 log log64 Ψm 3 Ψ m n m 48 log3 3 log log Ψm 3 log8 Ψm log8 log log8 Ψ m Ψm 3 log log8 3 log 3 Ψ m Ψ m n m log4 90 log log 30 log 90 log log 30 4 log 60 Ψm 3 log 90 4 log log 60 Ψ m log Ψm log 45 Ψ m 30 4 log 30 3 log log8 3 log Ψ m 30 Ψm 3 log log8 3 log 3 Ψ m 60 Ψm log 3 3 log log log8 log log8 Ψ m Ψ m 5 Ψ 3 m n m 4 On m 5 ; n m m Asymptotic series expansions cos n 4 n n n ; n cos n 4 n n n 6 n n 6480 n n n n O n n n 9 n 0 ; argn n cos n 4 n n n k j P j k, j n k j j k k j argn n Pm, j m m Pm 3, j Pm, j P0, 0 Pm, m Pm, j 0 ; m 3 j ; cos n 4 n n n O n ; argn n Product representations k k j ; k j

6 k k j ; k j Transformations Transformations and argument simplifications Argument involving basic arithmetic operations n n n n n n n n n n cos n csc n cos n csc n cos n csc n n n ; n n n n n m n m n m n ; m n n n n m m m n ; m n m Multiple arguments n n sin n n n

7 n 3 3n 4 cos3 n n n 3 n m n m m n n m mcosm n m 4 k 0 k n ; m m Products, sums, and powers of the direct function Products of the direct function n n n n n n ; n n n n n n cos n csc n cos n csc n n m mn n mn cosm cosn 4 m n n m nm cosm cosn m nm n m nm cosm cosn 4 n m n nm m n m n cosm cosn cos mn 4 m, n Identities Recurrence identities Consecutive neighbors

8 n n n n n n Distant neighbors n m m n ; m n m n m m n n m ; m m Functional identities Relations of special kind f n n f n ; f n n gn gn gn f Differentiation Low-order differentiation n n n log Ψ n log n n 4 n log Ψ n log sinn sinn cosn log Ψ n Summation Finite summation n n k k n k k k k 0 n n n n n n ; n Representations through more general functions Through other functions Involving some hypergeometric-type functions

9 9 n cos n 4 n n, 0 ; Ren Representations through equivalent functions With related functions n n n n n n cos n n cos n 4 n cos n n z 4 zcos z3 cos z z 3 C z Inequalities n n ; n n Zeros n 0 ; n History J. Keiper and O.I. Marichev (994) extended n to arbitrary complex n

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