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

Transcription

1 GmmRegulrized Nottions Trditionl nme Regulrized incomplete gmm function Trditionl nottion Q, z Mthemtic StndrdForm nottion GmmRegulrized, z Primry definition , z Q, z Specific vlues Specilized vlues For fixed Q, 0 ; Re Q, 0 ; Re Q, Subfctoril For fixed z Q0, z Q, z erfc z

2 Q n n, z erfc z n z n z n z k ; n nk Q, z erfc z z Π z Q n, z erfc z n z n n z ; n n k Q, z z n z k Qn, z z ; n k Qn, z 0 ; n z k Q n, z erfc z n z k z z k z n n k n n kn kn ; n Vlues t infinities Q, 0 Generl chrcteristics Domin nd nlyticity Q, z is n nlyticl function of nd z which is defined in. For fixed z, it is n entire function of zq, z Symmetries nd periodicities Mirror symmetry Q, z Q, z ; z, 0 Periodicity No periodicity

3 3 Poles nd essentil singulrities With respect to z For fixed, the function Q, z hs n essentil singulrity t z. At the sme time, the point z is brnch point for generic ing z Q, z, With respect to For fixed z, the function Q, z hs only one singulr point t. It is n essentil singulr point ing Q, z, Brnch points With respect to z For fixed, not being positive integer, the function Q, z hs two brnch points: z 0, z. At the sme time, the point z is n essentil singulrity z Q, z 0, ; z Q, z, 0 log ; z Q p q, z, 0 q ; p q gcdp, q z Q, z, log ; z Q p q, z, q ; p q gcdp, q With respect to For fixed z, the function Q, z does not hve brnch points Q, z Brnch cuts With respect to z For fixed, not being positive integer, the function Q, z hs one infinitely long brnch cut. It is single-vlued function on the z-plne cut long the intervl, 0, where it is continuous from bove.

4 z Q, z, 0, lim Q, x Ε Q, x ; x 0 Ε lim Q, x Ε Π Q, x ; x 0 Ε0 With respect to For fixed z, the function Q, z does not hve brnch cuts Q, z Series representtions Generlized power series Expnsions t generic point 0 For the function itself Q, z Q 0, z z F 0, 0 ; 0, 0 ; z Q 0, z logz Ψ 0 0 Q 0, z logz Ψ 0 Ψ 0 z F 3 0, 0, 0 ; 0, 0, 0 ; z 0 Ψ 0 logz F 0, 0 ; 0, 0 ; z 0 ; Q, z Q 0, z z F 0, 0 ; 0, 0 ; z Q 0, z logz Ψ 0 0 Q 0, z logz Ψ 0 Ψ 0 z F 3 0, 0, 0 ; 0, 0, 0 ; z 0 Ψ 0 logz F 0, 0 ; 0, 0 ; z 0 O 0 3

5 k Q, z ks 0 s 0 ks kis z 0 i 0 k s i k i s 0 kis log i z 4 F 4c, c,, c kis ; c, c,, c kis ; z s j s 0 j j k s s j Functionu, u j s 0 0 k ; c c c k 0 k j Q, z Q 0, z O 0 Expnsions of QΕ n, z t Ε 0 For the function itself QΕ n, z n n n, z Ε n n n, z H n logz G 3,0,3 z 3,0 z 6 n n 6 H n logz G,3, 0, 0, n 6 G 4,0 3,4 z,, 0, 0, 0, n, 0, 0, n Ε n, z 3 log z Π 3 Ψ 0 n Ψ 0 n logz 3 Ψ n Ε 3 OΕ 4 ; n k j QΕ n, z Ε n kj b ji,n c i,n j 0 i 0 Ε k ; k k Π k k k 0 b k,n k k n c k,n k k k,0 n, z k n QΕ n, z n n n, z Ε OΕ ; n Expnsions t generic point z z 0 For the function itself Q, z Π sinπ rgz z 0 rgz 0 Π z 0 rgzz 0 z 0 rgzz 0 z 0 z 0 z 0 z 0 z 0 Q, z 0 z z 0 z z 0 ; z z 0

6 Q, z Π sinπ rgz z 0 rgz 0 Π z 0 rgzz 0 z 0 rgzz 0 z 0 z 0 z 0 z 0 z 0 Q, z 0 z z 0 z z 0 Oz z k kj Q, z j k j j 0 z 0 rgzz 0 z 0 rgzz 0 Q j, z 0 Π rgz z 0 rgz 0 Π z z 0 k Q, z z 0 rgzz 0 rgzz 0 z 0 rgz z Π 0 sinπ k kj j z Q, z 0 z 0 0 k j k j j z z 0 k rgz 0 Π k j 0 Q, z Π sinπ rgz z 0 rgz 0 Π z 0 rgzz 0 z 0 rgzz 0 Q, z 0 Oz z 0 Expnsions on brnch cuts For the function itself Q, z Π sinπ rgz x rgzx Π Q, x x x z x x x x z x ; z x x x Q, z Π sinπ rgz x rgzx Π Q, x x x z x x x x z x Oz x 3 ; x x Q, z rgzx k kj x j Q, x x z xk k j k j j k j 0 Π sinπ rgz x ; x x Q, z Π sinπ Expnsions t z 0 rgz x rgzx Π Q, x Oz x ; x x 0 For the function itself

7 7 Generl cse Q, z z z ; z 0 z Q, z z z Oz3 z z Q, z z k k k Q, z z F ; ; z Q, z z Oz Q, z F z, ; F n z, z n z k k k Q, z n z n n F, n ; n, n ; z n Summed form of the truncted series expnsion. Specil cses Q, z z Oz Q, z z Oz3 Qn, z zn n Ozn ; n n z k Qn, z z ; n k Qn, z Oz n ; n Asymptotic series expnsions Expnsions t

8 Q, z z z z 88 z 3 z 88 O 3 ; z Q, z z k j z k j j j 0 k k ; Q, z z z O ; Expnsions t z Q, z z z O ; z z z z Q, z z z n k k z k O z n ; z n Q, z z z k k z k ; z Q, z z z F 0, ; ; z ; z Q, z z z O z ; z Residue representtions Q, z res s j 0 z s s s j Q, z res s s z s s 0 res s j 0 z s s s j Integrl representtions On the rel xis Of the direct function

9 Q, z t t t z Contour integrl representtions Q, z s s z s s s Γ s s Q, z z s s ; Re Γ rgz Π Γ s Q, z s szs s s Γ s sz s Q, z s ; mxre, 0 Γ rgz Π Γ s Continued frction representtions Q, z z z z z 3 z 3 z ; z, 0 Q, z z k k k z z k k k, z k ; z, Q, z z z z 3 z z z z z ; z, 0

10 z z Q, z ; z, 0 z k k k, k z Q, z z z z z z z 3 z 4 5 ; z, 0 Q, z k k z z k k z, k ; z, Q, z z z z z z z z 3z z 3 4 z z 4 z z z Q, z k k z, k z Q, z z z z z z z 3 z 4 5 Q, z z z k k k k k k z, k Differentil equtions Ordinry liner differentil equtions nd wronskins

11 For the direct function itself z w z z w z 0 ; wz c Q, z c W z, Q, z z z w z gz g z g z gz g z w z 0 ; wz c Q, gz c W z Q, gz, gz gz g z w z gz g z gz h z hz 0 ; wz c hzq, gz c hz g z g z w z h z g z h z hz gz hz g z h z hz g z g z h z h z hz wz W z hz Q, gz, hz gz gz hz g z z w z d r z r r s z w z s d r z r r s wz 0 ; wz c z s Q, d z r c z s W z z s Q, d z r, z s d zr r z s d z r w z d r z logr logs w z logs d r z logr logs wz 0 ; wz c s z Q, d r z c s z W z s z Q, d r z, s z d rz d r z s z logr Trnsformtions Trnsformtions nd rgument simplifictions Argument involving bsic rithmetic opertions z z Q, z Q, z Q, z Q, z z z

12 Q n, z Q, z n z z ; n k k Q n, z Q, z z k n z z n k n z k n k ; n Identities Recurrence identities Consecutive neighbors z z Q, z Q, z Q, z Q, z z z Distnt neighbors n z k Q, z Q n, z z z ; n k k n Q, z Q n, z z z ; n k z k Differentition Low-order differentition With respect to Q, z G 3,0,3 z, 0, 0, Q, z logz Ψ Q, z z F, ;, ; z Q, z, 0 logz Ψ

13 3 Q, z G 3, ,0 z,, 0, 0, 0, G 3,0,3 z, 0, 0, logz Ψ Q, z log z Ψ logz Ψ Ψ Q, z Q, z, 0 log z Ψ logz Ψ Ψ z 3 F 3,, ;,, ; z Ψ logz F, ;, ; z With respect to z Q, z z z z Q, z z z z z Symbolic differentition With respect to n Q, z n n nk n, k logz ; n k n k n Q, z n n nk nik n n z i 0 n k i n i k nik log i z nki F nki,,, nki ;,,, k j k j nki ; z j n k k j j 0 k j k ; n n j n Q n,0 m, z m n nj b ji,m c i,m ; k k Π k k k 0 b k,m k k m c k,m k j 0 i 0 k k,0 m, z k m n With respect to z n Q, z n nk n z k Q, z n z ; n z n k k k n

14 n Q, z n z n nk z n n k k n Q k, z ; n Frctionl integro-differentition With respect to Α, z Α Α t logt Α t QΑ, 0, logt t z With respect to z Α Q, z zα z Α Α zα F ; Α ; z ; Α Q, z z Α zα Α k Α exp z, kz kα k k Integrtion Indefinite integrtion Involving only one direct function Q, z z z Q, z Q, z Involving one direct function nd elementry functions Involving power function z Α Q, z z Α Α zα Q, z Q Α, z Involving only one direct function with respect to Q, z t t z logt t Integrl trnsforms Fourier cos trnsforms

15 c t Q, tx x sin tn x Π x Π x ; x Re Fourier sin trnsforms s t Q, tx Lplce trnsforms x cos tn x Π x ; x Re z t Q, tz z ; Rez 0 Re Representtions through more generl functions Through hypergeometric functions Involving F Q, z z F ; ; z ; Involving F z Q, z F ; ; z ; Involving hypergeometric U Q, z z U,, z Through Meijer G Clssicl cses for the direct function itself Q, z G,, z, Q, z G,0, z 0, Q, z Q, z z G,0,3 z 4 0,,

16 6 Clssicl cses involving exp sinπ z Q, z G,, z Π 0, sinπ z Q, z z Q, z G 3,,4 z Π 3 4,,, 0, sinπ z Q, z z Q, z z Π 3 z G 3,,4 z 4,,,, Π z Q, z Π sinπ z Q, z G 3,,3 z Π 3 4 0,, ; 0 rgz Π Π z Π Q, z z sinπ Q, z G 3,,3 z Π 3 4 0,, ; 0 rgz Π Clssicl cses for products of incomplete gmm functions Clssicl cses for products of incomplete gmm functions Q, z Q, z sinπ Π 3 G 4,,4 z 4, 0,,, Clssicl cses involving regulrized gmm , z Q, z sinπ G 4,,4 z Π 3 4, 0,,, Generlized cses for the direct function itself Π Π Q, z Q, z Π G,0,3 z, 0,, ; 0 rgz Π Π Π Q, z Q, z Π G,0,3 z, 0,, ; 0 rgz Π Generlized cses involving exp sin Π z Q, z z 3, Q, z G,4 Π 3 z,,,, 0,

17 z Q, z z Q, z z sin Π 3, Π 3 z G,4 z,,,,, Π z Q, z Π z sin Π 3, Q, z G,3 Π 3 z, 0,, ; 0 rgz Π Π z Q, z Π z sin Π 3, Q, z G,3 Π 3 z, 0,, ; 0 rgz Π Through other functions Involving some hypergeometric-type functions , z, 0 Q, z ; Re Q, z Q, z, 0 ; Re 0 Representtions through equivlent functions With inverse function Q, Q, z z With relted functions , z Q, z Q, z z E z

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