GegenbauerC3General. Notations. Primary definition. Specific values. Traditional name. Traditional notation. Mathematica StandardForm notation

Transcript

1 GegenbauerC3General Notations Traditional name Gegenbauer function Traditional notation C Ν Λ z Mathematica StandardForm notation GegenbauerCΝ, Λ, z Primary definition C Λ Ν z Λ Π Ν Λ F Ν, Ν Λ; Λ Ν Λ ; z Specific values Specialized values For fixed Ν, Λ C Ν Λ Ν Π Λ Ν Ν Ν Λ Λ Ν C Λ Ν Λ Ν cosπ Λ Ν Λ Ν secπ Λ C Λ Ν ; ReΛ Λ Ν C Ν Λ ; ReΛ For fixed Ν, z

2 C Ν 0 z C Ν m z 0 ; m CΝ z P Ν z C Ν z U Ν z Ν CΝ z ; For fixed Λ, z C 0 Λ z C Λ z Λ z C Λ z Λ Λ z Λ C 3 Λ z 4 3 Λ Λ Λ z3 Λ Λ z C Λ 4 z 3 Λ Λ Λ Λ 3 z4 Λ Λ Λ z Λ Λ C 5 Λ z 4 5 Λ Λ Λ Λ 3 Λ 4 z5 4 3 Λ Λ Λ Λ 3 z3 Λ Λ Λ z C 6 Λ z 4 45 Λ Λ Λ Λ 3 Λ 4 Λ 5 z6 3 Λ Λ Λ Λ 3 Λ 4 z4 Λ Λ Λ Λ 3 z Λ Λ Λ C 7 Λ z 8 35 Λ Λ Λ Λ 3 Λ 4 Λ 5 Λ 6 z7 4 5 Λ Λ Λ Λ 3 Λ 4 Λ 5 z5 3 Λ Λ Λ Λ 3 Λ 4 z3 Λ Λ Λ Λ 3 z C Λ 8 z 35 Λ Λ Λ Λ 3 Λ 4 Λ 5 Λ 6 Λ 7 z Λ Λ Λ Λ 3 Λ 4 Λ 5 Λ 6 z6 3 Λ Λ Λ Λ 3 Λ 4 Λ 5 z4 3 Λ Λ Λ Λ 3 Λ 4 z Λ Λ Λ Λ 3 4

3 Λ Λ Λ Λ 3 Λ 4 Λ 5 Λ 6 Λ 7 Λ 8 C Λ z9 9 z Λ Λ Λ Λ 3 Λ 4 Λ 5 Λ 6 Λ 7 z7 5 Λ Λ Λ Λ 3 Λ 4 Λ 5 Λ 6 z5 9 Λ Λ Λ Λ 3 Λ 4 Λ 5 z3 Λ Λ Λ Λ 3 Λ 4 z Λ Λ Λ Λ 3 Λ 4 Λ 5 Λ 6 Λ 7 Λ 8 Λ 9 C Λ z0 0 z Λ Λ Λ Λ 3 Λ 4 Λ 5 Λ 6 Λ 7 Λ 8 z8 45 Λ Λ Λ Λ 3 Λ 4 Λ 5 Λ 6 Λ 7 z6 9 Λ Λ Λ Λ 3 Λ 4 Λ 5 Λ 6 z4 Λ Λ Λ Λ 3 Λ 4 Λ 5 z Λ Λ Λ Λ 3 Λ C n Λ z n Λ n z n 0 n ; n C Λ n z 0 ; n Λ C Λ z ; General characteristics Domain and analyticity C Λ Ν z is an analytical function of Ν, Λ, z which is defined in 3. For integer Ν, C Λ Ν z degenerates to a polynomial in z Ν Λ zc Λ Ν z Symmetries and periodicities Parity C Λ n z n C Λ n z ; n Mirror symmetry C Λ Ν z C Λ Ν z ; z,

4 4 Periodicity No periodicity Poles and essential singularities With respect to z For fixed Ν ; Ν, Λ, the function C Ν Λ z does not have poles and essential singularities ing z C Ν Λ z ; Ν For positive integer Ν, the function C Ν Λ z is polynomial and has pole of order Ν at z ing z C Ν Λ z, Ν ; Ν For nonpositive integer Ν, the function C Ν Λ z is constant (0 for Ν 0 or for Ν 0). With respect to Λ For fixed Ν, z, the function C Ν Λ z has an infinite set of singular points: a) Λ Νj ; j, are the simple poles with residues j jν b) Λ is an essential singular point ing Λ C Ν Λ z Ν j, ; j,, Π Ν j Ν j F j, Ν; Νj ; z ; res Λ C Ν Λ z Ν j j jν Π Ν j F Ν j Νj j, Ν; ; z ; j With respect to Ν For fixed Λ, z, the function C Ν Λ z has an infinite set of singular points: a) Ν j Λ ; j, are the simple poles with residues Λ Π j j j Λ Λ F j, j Λ; Λ ; z ; b) Ν is the point of convergence of poles, which is an essential singular point ing Ν C Ν Λ z j Λ, ; j,, res Ν C Λ Ν z j Λ j Λ Π j j Λ Λ F j, j Λ; Λ ; z ; j Branch points

5 5 With respect to z For fixed Ν ; Ν, Λ, the function C Ν Λ z has two branch points: z, z. For fixed Λ and integer Ν, the function C Ν Λ z does not have branch points z C Λ Ν z, ; Ν z C Λ Ν z ; Ν z C Λ Ν z, log ; Λ Λ Ν Λ Ν z C Ν Λ z, s ; Λ r s r s gcdr, s Ν Λ Ν z C Λ Ν z, log ; Λ Ν Λ Ν z C Λ Ν z, lcms, u ; Ν r s Λ Ν t r, s, t, u s u gcdr, s gcdt, u Ν Λ Ν u With respect to Λ For fixed Ν, z, the function C Ν Λ z does not have branch points Λ C Λ Ν z With respect to Ν For fixed Λ, z, the function C Λ Ν z does not have branch points. Ν C Ν Λ z Branch cuts With respect to z For fixed Ν ; Ν, Λ, the function C Λ Ν z is a single-valued function on the z-plane cut along the interval, where it is continuous from above. For fixed Λ and integer Ν, the function C Λ Ν z does not have branch cuts z C Ν Λ z,, ; Ν

6 z C Ν Λ z ; Ν lim C Λ Ν x Ε C Λ Ν x ; x Ε lim C Λ Ν x Ε Π Λ cosπ Ν Λ C Λ Ν x Π Λ C Λ Ν x ; x Ε0 With respect to Λ For fixed Ν, z, the function C Ν Λ z does not have branch cuts Λ C Λ Ν z With respect to Ν For fixed Λ, z, the function C Λ Ν z does not have branch cuts. Ν C Ν Λ z Series representations Generalized power series Expansions at generic point z z 0 For the function itself

7 C Ν Λ z Λ cosπ Λ Ν sinπ Ν Π 3 Λ Π 3 cscπ Ν4 Λ Λ Π Λ argzz 0 Π argz z 0 Π argz 0 Π Π C Ν Λ z 0 z 0 Λ argzz 0 Π z 0 Λ argzz 0 Π, G, z 0 Ν, Λ Ν 0, Λ z 0 Λ argzz 0 Π z 0 Λ argzz 0 Π, G, z 0 Ν, Λ Ν 0, Λ Π Π Λ argzz 0 Π argz z 0 Π argz 0 Π Π Ν Λ Ν F Ν, Λ Ν ; Λ 3 ; z 0 z z 0 8 Π Π Λ argzz 0 Π argz z 0 Π argz 0 Π Π Ν Λ Ν F Ν, Λ Ν ; Λ 5 ; z 0 z 0 Λ argzz 0 Π z 0 Λ argzz 0 Π, G, z 0 Ν, Λ Ν 0, Λ 3 z z 0 ; z z 0

8 C Ν Λ z Λ cosπ Λ Ν sinπ Ν Π 3 Λ Π 3 cscπ Ν4 Λ Λ Π Λ argzz 0 Π argz z 0 Π argz 0 Π Π C Ν Λ z 0 z 0 Λ argzz 0 Π z 0 Λ argzz 0 Π, G, z 0 Ν, Λ Ν 0, Λ z 0 Λ argzz 0 Π z 0 Λ argzz 0 Π, G, z 0 Ν, Λ Ν 0, Λ Π Π Λ argzz 0 Π argz z 0 Π argz 0 Π Π Ν Λ Ν F Ν, Λ Ν ; Λ 3 ; z 0 z z 0 8 Π Π Λ argzz 0 Π argz z 0 Π argz 0 Π Π Ν Λ Ν F Ν, Λ Ν ; Λ 5 ; z 0 z 0 Λ argzz 0 Π z 0 Λ argzz 0 Π, G, z 0 Ν, Λ Ν 0, Λ 3 z z 0 Oz z C Λ Ν z Λ cosπ Λ Ν sinπ Ν Π 3 Λ Π 0 Π Λ argzz 0 Π argz z 0 Π argz 0 Π Π Ν Λ Ν F Ν, Λ Ν; Λ ; z 0 z 0 Λ argzz 0 Π z 0 Λ argzz 0 Π G z, 0, Ν, Λ Ν 0, Λ z z 0

9 C Λ Ν z Λ sinπ Ν Π Λ 0 Λ Π secπ Λ z 0 Λ Λ argzz 0 Π z 0 Λ argzz 0 Π F Λ Ν, Λ Ν ; Λ 3 ; z 0 cosπ Λ Ν Ν Λ Ν Π Λ argzz 0 Π argz 0 Π Π argz z 0 Π secπ Λ z 0 Λ argzz 0 Π z 0 Λ argzz 0 Π F Ν, Λ Ν; Λ ; z 0 z z 0 ; Λ C Ν Λ z Λ cosπ Λ Ν sinπ Ν Π 3 Λ Π 3 cscπ Ν4 Λ Λ Π Λ argzz 0 Π argz z 0 Π argz 0 Π Π C Ν Λ z 0 z 0 Λ argzz 0 Π z 0 Λ argzz 0 Π, G, z 0 Ν, Λ Ν 0, Λ Oz z 0 Expansions on branch cuts For the function itself C Ν Λ z Λ cosπ Λ Ν sinπ Ν Π 3 4 Λ Π Π Π 3 Λ 8 Π Π argzx Λ argz x Π cscπ ΝΛ Π argzx Λ argz x Π Π Π argzx Π Λ Π argzx Λ Π argzx Π Λ Π, x G, argz x Π, x G, C Ν Λ x argzx Π Λ Π, x G, Ν Λ Ν F Ν, Λ Ν ; Λ 3 ; x Ν, Λ Ν 0, Λ z x Ν Λ Ν F Ν, Λ Ν ; Λ 5 ; x Ν, Λ Ν 0, Λ 3 z x ; z x x x Ν, Λ Ν 0, Λ

10 C Ν Λ z Λ cosπ Λ Ν sinπ Ν Π 3 4 Λ Π Π Π 3 Λ 8 Π Π argzx Λ argz x Π cscπ ΝΛ Π argzx Λ argz x Π Π Π argzx Π Λ Π argzx Λ Π argzx Π Λ Π, x G, argz x Π, x G, C Ν Λ x argzx Π Λ Π, x G, Ν Λ Ν F Ν, Λ Ν ; Λ 3 ; x Ν, Λ Ν 0, Λ z x Ν Λ Ν F Ν, Λ Ν ; Λ 5 ; x Ν, Λ Ν 0, Λ 3 z x Oz x 3 ; x x Ν, Λ Ν 0, Λ C Ν Λ z Λ cosπ Λ Ν sinπ Ν 0 C Ν Λ z Π 3 Λ Π Π argzx Π Λ Π Λ sinπ Ν Π Λ 0, x G, argzx Λ Π argz x Π Ν, Λ Ν 0, Λ Λ Π secπ Λ x Λ cosπ Λ Ν Ν Λ Ν Π F Ν, Λ Ν; Λ ; x Ν Λ Ν F Ν, Λ Ν; Λ ; x argzx Π Λ Π argzx Λ Π z x ; x x F Λ Ν, Λ Ν ; Λ 3 ; x argz x Π secπ Λ z x ; Λ x x argzx Π Λ Π

11 C Λ Ν z Λ Π Π argzx Λ Π secπ Λ sinπ Ν Λ x Λ cosπ Λ Ν 0 argz x Π F Λ Ν, Λ Ν; 3 x Λ; Π x Λ Λ Ν Ν cscπ Ν cscπ Λ Ν argz x Π Π argzx Λ Π cscπ Ν cscπ Λ Ν cos Π Λ Ν secπ Λ cos Π Λ Ν F Λ Ν, Λ Ν; Λ ; x z x ; Λ x x C Ν Λ z Λ cosπ Λ Ν sinπ Ν Π 3 Λ argzx Π Λ Π, x G, Π 3 4 Λ Π Ν, Λ Ν argzx Λ argz x Π cscπ ΝΛ Π 0, Λ Oz x ; x x C Ν Λ x Expansions at z 0 For the function itself General case C Λ Ν z Λ Π Λ Ν Ν Λ Π Ν Λ Ν Π z Λ Λ Ν Ν z ; z 0 Λ Ν Ν Π Ν Λ Ν C Λ Ν z Λ Π Λ Ν Ν Λ Π Ν Λ Ν Π z Λ Λ Ν Ν z Oz 3 Λ Ν Ν Π Ν Λ Ν C Λ Ν z Λ Π Λ Ν Ν j Λ Ν j z j ; z Ν Λ j 0 0 j Λ j j C Ν Λ z Λ Π Λ Ν Ν Λ F Ν, Λ Ν;;; Λ ;;;, z

12 C Λ Ν z Λ Π Λ Ν Ν Λ Ν z j ; z Ν Λ 0 j 0 Λ j j Ν Π Λ Ν Ν Λ Ν Ν Π z Λ Ν C Λ Ν z z Λ Ν Ν 0 Λ Ν Ν 0 Ν Ν Λ z ; z 3 C Ν Λ z cos Π Ν Λ Ν Ν Λ F Ν, Λ Ν ; ; z Ν Π Λ Ν z Λ Ν Ν F Ν, Ν Λ; 3 ; z cos Π Ν Λ Ν C Λ Ν z Oz Ν Λ C Ν Λ z F z, Ν, Λ ; F m z, Ν, Λ Ν Π Λ Ν m Ν Λ Ν Λ Ν Ν 0 Ν Π z Λ Ν m Ν Ν Λ z Λ Ν Ν z 0 3 Ν Π z m Λ Ν Λ Ν Ν C Λ m m Ν z m Λ Ν Ν m 3 F, m Ν, m Λ Ν ; m 3, m ; z Ν Π z m3 Λ Ν Ν Ν Λ m m 3 F, m Ν 3, m Λ Ν 3 ; m, m 5 ; z m m Λ Ν Ν 3 m Summed form of the truncated series expansion. Special cases C n Λ z n Λ n z n 0 n ; n C n Λ z n Λ n n n zn n n n Oz ; z 0 n Generic formulas for main term

13 3 C Ν Λ z Ν Λ Ν Ν Λ Ν Ν Ν Ν Ν cos Π Ν Λ Ν True Ν Λ Expansions at z For the function itself Ν Ν z Ν ; z 0 General case C Λ Ν z Λ Π Λ Ν Ν Λ Λ Ν Λ Νz Ν Ν Λ Ν Λ Ν z ; Λ 3 8 Λ 5 z Λ C Λ Ν z Λ Π Λ Ν Ν Λ Λ Ν Λ Νz Ν Ν Λ Ν Λ Ν z Oz 3 ; Λ 3 8 Λ 5 Λ C Λ Ν z Λ Π Λ Ν Ν Λ Ν Ν Λ 0 Λ z ; z Λ Ν Ν C Λ Λ Ν Ν z Λ Ν 0 Λ z ; Λ C Λ Ν z Λ Π Λ Ν F Ν, Λ Ν; Λ Ν Λ ; z Λ Ν C Λ Ν z Λ Ν F Ν, Λ Ν; Λ ; z ; Λ Λ Ν C Λ Ν z Λ Ν Oz ; z Λ

14 C Ν Λ z F z, Ν, Λ ; Λ Ν Ν Λ Ν F m z, Ν, Λ Λ Ν 0 Λ z C Ν Λ z m Λ Ν Ν m Λ Ν m z m m Λ Ν Λ m 3F, m Ν, m Λ Ν ; m, m Λ 3 ; z m Λ Summed form of the truncated series expansion. Special cases C Λ n z Λ Π Λ n n n Λ n n Λ 0 Λ z ; n C Λ Ν z Π z Λ secπ Λ Ν sinπ Ν Λ 3 Λ Λ Ν Λ Ν z 4 3 Λ Λ Ν 3 Λ Ν Λ Ν Λ Ν 3 z ; z Λ 3 3 Λ 5 Λ C Λ Ν z Π z Λ secπ Λ Ν sinπ Ν Λ 3 Λ Λ Ν Λ Ν z 4 3 Λ Λ Ν 3 Λ Ν Λ Ν Λ Ν 3 z Oz 3 ; Λ 3 3 Λ 5 Λ C Λ Ν z Π z Λ secπ Λ Ν sinπ Ν Λ 0 Λ Ν Λ Ν Λ 3 z ; Λ C Λ Ν z Π z Λ secπ Λ Ν sinπ Ν Λ 3 Λ 0 Λ Ν Λ Ν 3 Λ z ; Λ C Λ Ν z Π z Λ secπ Λ Ν sinπ Ν F Λ Ν Λ, Λ Ν ; 3 z Λ; ; Λ

15 C Λ Ν z Π z Λ secπ Λ Ν sinπ Ν F Λ 3 Λ Λ Ν, Λ Ν ; 3 z Λ; ; Λ C Λ Ν z Π z Λ secπ Λ Ν sinπ Ν Oz ; z Λ Λ 3 Λ Generic formulas for main term C Ν Λ z Ν Λ Ν Π Λ secπ ΛΝ sinπ Ν z Λ Λ 3 Λ Λ ΛΝ Λ Ν True ; z Expansions at z For the function itself General case cosπ Λ Ν secπ Λ Λ Ν C Λ Ν z Ν Λ Ν Λ Ν Ν Ν Λ Ν Λ Ν z z Λ Λ Λ 3 Λ sinν Π Λ z Λ Λ Ν Λ Ν z Π Λ 43 Λ Λ Ν 3 Λ Ν Λ Ν 3 Λ Ν z ; z Λ 3 3 Λ 5 Λ cosπ Λ Ν secπ Λ Λ Ν C Λ Ν z Ν Λ Ν Λ Ν Ν Ν Λ Ν Λ Ν z z Oz 3 Λ Λ Λ 3 Λ sinν Π Λ z Λ Λ Ν Λ Ν z Π Λ 43 Λ Λ Ν 3 Λ Ν Λ Ν 3 Λ Ν z Oz 3 ; Λ 3 3 Λ 5 Λ

16 cosπ Λ Ν secπ Λ Λ Ν Ν C Λ Λ Ν Ν z Ν Λ 0 Λ z Λ sinν Π Λ z Λ Λ Ν Λ Ν Π Λ 0 3 Λ z ; z Λ cosπ Λ Ν secπ Λ Λ Ν C Λ Ν z F Ν, Λ Ν; Λ Ν Λ ; z Λ sinν Π Λ z Λ F Λ Ν Π Λ, Λ Ν; 3 z Λ; ; Λ cosπ Ν Λ secπ Λ Ν Λ Λ sinν Π Λ C Λ Ν z Oz z Λ Oz ; Ν Λ Π Λ z Λ C Ν Λ z F z, Ν, Λ ; cosπ Λ Ν secπ Λ Λ Ν m Ν Λ Ν F m z, Ν, Λ Ν Λ 0 Λ z Λ sinν Π Λ z Λ Π Λ m Λ Ν Λ Ν 0 3 Λ z C Λ Ν z m secπ Λ cosπ Λ Ν Λ Ν Ν m Λ Ν m m Λ Ν Λ m z m 3 F, m Ν, m Λ Ν ; m, m Λ 3 ; z mλ sinπ Ν Λ Λ Ν Λ Ν m m z mλ 3 Π m Λ 3 Λ m 3F, m Λ Ν 3, m Λ Ν 3 ; m, m Λ 5 ; z m Λ Summed form of the truncated series expansion. Special cases

17 sinν Π Λ Ν C Λ Ν z Π Ν Λ log z F Ν, Λ Ν; Λ ; z Λ sinν Π Λ Λ 3 Λ Ν Λ Ν z Λ Π Λ 0 3 Λ z Λ sinν Π Ν Λ Π Ν Λ Ν Λ Ν Ψ Ψ Λ 0 Λ Ψ Λ Ν Ψ Ν z ; Λ 3 Ν sinν Π Λ Ν C Λ Ν z Π Ν Λ log z Ψ Λ ΨΝ Ψ Λ Ν Oz Λ sinν Π Λ z Λ Oz ; z Λ 3 Ν Π Λ CΝ sinν Π z log z Π F Ν, Ν ; ; z Ν Ν Ψ Ψ Ν Ψ Ν 0 z ; Ν sinν Π z CΝ z log Π ΨΝ ΨΝ Oz ; z Ν C Λ Ν z Λ cosπ Λ Ν log Λ Π Λ z z Λ F Λ Ν, Λ Ν; 3 z Λ; Λ cosπ Λ Ν z Λ Π Λ 0 Λ Λ Ν Λ Ν Ψ Ψ Λ 3 Ψ Λ Ν Ψ Λ Ν z Λ cosπ Λ Ν Λ Λ Ν Λ Ν Λ Ν Π Λ Ν 0 Λ z ; Λ Ν C Ν Λ z Λ cosπ Λ Ν Λ Λ Ν Π Λ Ν Oz Λ cosπ Λ Ν z Λ Π Λ Λ log z Ψ 3 Λ Ψ Λ Ν Ψ Λ Ν Oz ; Λ Ν C Λ n z n n Λ Oz ; n n Λ

18 8 Generic formulas for main term C Ν Λ z Ν Λ Ν sinν Π ΛΝ z log ΨΛ Λ sinπ Ν Λ ΨΝ Ψ Λ Ν z Λ Π Ν Λ Π Λ Λ 3 Ν sinν Π log z ΨΝ ΨΝ Λ Ν Π Λ cosπ ΛΝ Λ ΛΝ Π Λ Ν Λ cosπ ΛΝ Λ ΛΝ Π Λ Ν Λ cosπ ΛΝ z Λ Λ Π Λ log z Ψ 3 Λ ΨΛ Ν ΨΛ Ν Λ Λ sinπ Ν Λ z Λ Π Λ True Ν C Ν Λ z Ν Λ Ν Λ sinπ Ν Λ z Λ sinν Π logz Π Π Λ ReΛ Ν Λ Ν ; z Λ cosπ ΛΝ Λ ΛΝ Π Λ Ν Λ cosπ ΛΝ Λ ΛΝ Π Λ Ν Λ sinπ Ν Λ z Λ Π Λ ReΛ True Expansions at z For the function itself Expansions in z C Ν Λ z Ν Λ Ν z Ν Ν Ν 3 Ν Ν Ν Ν Λ Ν ΛΝ sinπ Ν Λ Ν Λ Ν z ΛΝ 4 Λ Νz 3 Λ Ν Λ Νz 4 Π Λ Λ Ν Λ Ν Λ Ν Λ Ν Λ Ν 3 Λ Ν ; z Λ Ν 4 Λ Νz 3 Λ Ν Λ Νz 4

19 C Λ Ν z Ν Λ Ν z Ν Λ Ν ΛΝ ΛΝ sinπ Ν Λ Ν Λ Ν z Π Λ Ν Ν 3 Ν Ν Ν Ν O 4 Λ Νz 3 Λ Ν Λ Νz 4 z 6 Λ Ν Λ Ν Λ Ν Λ Ν Λ Ν 3 Λ Ν O ; Λ Ν 4 Λ Νz 3 Λ Ν Λ Νz 4 z C Λ Ν z Ν Λ Ν z Ν Ν Ν Λ Ν 0 Λ Ν z ΛΝ sinπ Ν Λ Ν Λ Ν z ΛΝ Π Λ z Λ Ν C Ν Λ z Ν Λ Ν z Ν Λ Ν F Ν, Ν ; Λ Ν ; z ΛΝ ΛΝ sinπ Ν Λ Ν Λ Ν z Π Λ C Λ n z n z n n Λ n n n n n Λ z ; Λ n C n Λ z n z n Λ n n F n, n ; n Λ ; z ; Λ n 0 Λ Ν Ν Λ Λ Ν F Λ Ν Ν, Λ ; Λ Ν ; ; Λ Ν z, 0 z z ; C Ν Λ z Ν Λ Ν z Ν Λ Ν O z ΛΝ sinπ Ν Λ Ν Λ Ν z ΛΝ O Π Λ z ; Λ Ν C Λ Ν z n z n n Λ O ; n Λ n z

20 C Ν Λ z F z, Ν, Λ ; F m z, Ν, Λ Ν Λ Ν z Ν m Λ Ν 0 Ν Ν Λ Ν z ΛΝ sinπ Ν Λ Ν Λ Νz ΛΝ Π Λ m 0 Λ Ν Ν Λ Λ Ν z C Λ Ν z mλν mλν cscπ Λ Ν m Λ Ν sinπ Ν z 3 F, m Λ Ν, m Λ m Λ Ν m Λ Ν 3 ; m, m Λ Ν ; z mν mν cscπ Λ Ν sinπ Ν m Ν z m Λ m Λ Ν 3F, m Ν, m Ν 3 ; m, m Λ Ν ; z m Ν Summed form of the truncated series expansion. Expansions in z C Ν Λ z C Ν Λ z Ν Λ Ν Ν Λ z Ν Ν ΛΝ z Λ Ν z z Ν Λ Ν Ν Λ z Ν Ν ΛΝ z Λ Ν z Ν Ν 3 Λ Ν 4 Λ Ν z ΛΝ sinν ΠΛ Ν Λ Ν Π Λ 3 Λ Ν Λ Ν Λ Ν ; z Λ Ν 4 Λ Ν z z Ν Ν 3 Λ Ν O ΛΝ sinν ΠΛ Ν Λ Ν 4 Λ Ν z z 3 Π Λ 3 Λ Ν Λ Ν Λ Ν 4 Λ Ν z O z 3 ; Λ Ν C Λ Ν z Ν Λ Ν Ν Λ Ν Ν Λ z Ν 0 Λ Ν z ΛΝ sinν ΠΛ Ν Λ Ν Λ Ν Λ Ν Π Λ z ΛΝ Λ Ν 0 z ; z Λ Ν C Ν Λ z Ν Λ Ν Ν Λ z Ν F Ν, Λ Ν ; Λ Ν ; z ΛΝ sinν ΠΛ Ν Λ Ν z ΛΝ F Λ Ν, Λ Ν Π Λ ; Λ Ν ; z ; z, Λ Ν C Ν Λ z Ν Λ Ν z Ν Ν Λ O z ΛΝ ΛΝ sinν Π Λ Ν Λ Ν z O Π Λ z ; z Λ Ν

21 C Λ Ν z ΛΝ ΛΝ sinν Π Λ Ν log Π Λ Λ Ν z ΛΝ ΛΝ sinν Π Λ Ν z ΛΝ Π Λ Λ Ν 0 z ΛΝ F Λ Ν, Λ Ν ; Λ Ν ; Λ Ν Λ Ν Ψ Ψ Λ Ν Ψ Λ Ν Λ Ν Ψ Λ Ν z z ΛΝ Π z Ν Λ Ν Λ Ν Ν Ν Λ 0 Λ Ν z ; Λ Ν Ν C Λ Ν z ΛΝ ΛΝ sinν Π Λ Ν z ΛΝ log z Π Λ Λ Ν Ν Λ Ν z Ν Λ Ν O z ; z Λ Ν Ν Ψ Λ Ν Ψ Λ Ν Ψ Λ Ν O z C Ν Ν z sinπ Ν Ν z Ν log logz ΨΝ O Π z ; z Ν C Λ Ν z Ν ΛΝ Λ Ν sinν Π z Π Λ Λ Ν Λ Ν Λ Ν 0 Λ Ν z ΛΝ Ν Ν Λ Ν Λ log z z Ν F Ν, Λ Ν; Λ Ν; z ΛΝ Ν Ν Λ Ν Λ z Ν Ν 0 Λ Ν Λ Ν Ψ Ψ Λ Ν Ψ Λ Ν Ψ Ν z ; Λ Ν Ν ΛΝ Ν z Ν C Λ Ν z Ν Λ Ν Λ log z ΨΝ Ψ Λ Ν ΛΝ ΛΝ sinπ Ν Λ Ν Λ Νz O Π Λ z Ψ Λ Ν ; z Λ Ν Ν O z C Λ Ν z Ν Λ Ν z Ν Ν Λ ΛΝ Ν Λ Ν 0 z Λ Ν ΛΝ ΛΝ sinν Π Λ Ν z ΛΝ F Λ Ν, Λ Ν Λ Λ Ν ; Λ Ν ; z ; Λ Ν Ν

22 C Ν Λ z Ν Λ Ν Ν Λ zν O z ΛΝ ΛΝ sinν Π Λ Ν z Ν Λ O Λ Λ Ν z ; z Λ Ν C Ν Λ z ΛΝ Ν Π Ν Λ Ν Λ z Ν F Ν, Λ Ν; Λ Ν; z ΛΝ Λ Ν Λ Ν sinν Π Π Λ ΛΝ z ΛΝ Λ Ν Λ Ν Λ Ν 0 z ; Λ Ν Ν C Ν Λ z ΛΝ Ν Π Ν Λ Ν Λ zν z Λ Ν Ν O z ΛΝ Λ Ν Λ Ν sinν Π ΛΝ z Π Λ O z ; C Ν Λ z F z, Ν, Λ ; F m z, Ν, Λ Ν Λ Ν m Ν Λ Ν Ν Λ z Ν 0 Λ Ν z ΛΝ sinν ΠΛ Ν Λ Ν Π Λ m z ΛΝ 0 Λ Ν Λ Ν Λ Ν z m ΛΝ m sinπ Ν Λ Ν Λ ΝΛ Ν Λ Ν C Λ m m Ν z z m ΛΝ Π m Λ Λ Ν m 3F, m Λ Ν 3, m Λ Ν ; m, m Λ Ν ; z mν m Λ Ν Λ Ν Ν m m z mν m Λ Ν Λ Ν m 3F, m Ν, m Λ Ν 3 ; m, m Λ Ν ; z m Λ Ν Summed form of the truncated series expansion. Generic formulas for main term

23 C Λ Ν z 0 Ν Λ Λ Λ Ν Λ Ν ΛΝ z Ν ΛΝ cosπ ΛΝ ΛΝ sinπ Ν z ΛΝ loglogzψ ΛΝΨ ΛΝΨ Λ Ν Λ Ν Π Λ ΛΝ Ν z Ν sinπ Ν loglogzψν Π ΛΝ ΛΝ ΛΝ sinπ Ν z ΛΝ Π Λ Ν ΛΝ z Ν Λ Ν Ν ΛΝ z Ν ΛΝ ΛΝ ΛΝ sinπ Ν z ΛΝ Λ Ν Π Λ Ν cosπ ΛΝ zν loglogzψνψλν Ψ Λ Ν Λ ΛΝ Ν Λ Ν Λ Ν Λ Ν 0 Λ Ν Λ Ν Λ Ν Λ Ν True ; z C Ν Λ z Ν Λ Λ Ν Λ Ν Ν Λ Ν Λ Λ Ν Λ Ν Ν z Ν ΛΝ Λ Ν Ν sinπ Νlogz z Ν Π ΛΝ ΛΝ ΛΝ sinπ Ν z ΛΝ Π Λ Ν ΛΝ z Ν ΛΝ ΛΝ ΛΝ sinπ Ν z ΛΝ Λ Ν Π Λ ReΛ Ν 0 Λ Ν 0 ReΛ Ν 0 True ; z Integral representations On the real axis Of the direct function C Λ Ν z Λ Ν Λ Π Ν Λ 0 z z cost Ν sin Λ tt ; ReΛ 0 Rez 0 Integral representations of negative integer order Rodrigues-type formula.

24 4 C n Λ z n Λ n Λ z Λ n z nλ n n Λ n Λ z n ; n Generating functions C n Λ z t n t z t Λ ; n z Differential equations Ordinary linear differential equations and wronsians For the direct function itself z w z Λ z w z Ν Ν Λ wz 0 ; wz c C Λ Ν z c z W z C Λ Ν z, z 4 Λ Q ΝΛ Λ Λ Π z Λ z Λ 4 Λ Q ΛΝ Λ z w z Λ gz g z g z gz g z w z Ν Λ Ν g z Λ wz 0 ; wz c C Λ Ν gz c gz gz 4 Q ΛΝ Λ gz Λ W z C Λ Ν gz, gz Λ Λ Π gz Λ g z gz Λ 4 Q ΛΝ w z Λ gz g z h z g z gz hz g z Ν Λ Ν g z Λ gz h z g z gz gz hz w z h z hz h z g z hz g z h z hz wz 0 ; Λ wz c hzc Λ Ν gz c hz gz Λ W z hz C Λ Ν gz, hz gz 4 Q ΛΝ 4 Q ΛΝ Λ gz Λ Λ Π gz Λ g z hz gz Λ

25 w z a s r Λ z r r s z a z r Λ wz c z s C Λ Ν a z r c z s a z r Λ W z z s C Λ Ν a z r, z s a z r w z a Λ logr r z 4 Q ΛΝ w z a z r s r Ν s r Λ Ν s r s wz 0 ; z a z r 4 Q ΛΝ Λ a z r Λ Λ a Π r z r s a z r Λ a z r Λ a r z logr logs w z a r z Ν logr logs logs Λ Ν logr logs logr logs wz 0 ; a r z Λ wz c s z C Λ Ν a r z c s z a r z Λ W z s z C Λ Ν a r z, s z a r z 4 Q ΛΝ 4 Q ΛΝ Λ a r z Λ Λ a Π r z a r z Λ s z logr a r z Λ Transformations Transformations and argument simplifications Argument involving basic arithmetic operations C Λ n z n C Λ n z ; n Identities Recurrence identities Consecutive neighbors With respect to Ν Λ Ν z C Λ Λ Ν z C Ν z Ν Λ Ν Λ Ν C Λ Νz Λ Ν z Λ Ν C Λ Λ Ν z C Ν z Ν Ν Λ C Ν z

26 6 With respect to Λ Λ Λ z Λ Ν 4 Λ Λ z C Λ Ν z C Λ Ν z Λ Ν Λ Ν Λ Ν Λ Ν C Ν Λ z Λ z Ν z 4 Λ 5 Ν Λ 4 Ν Λ 3 C Λ Ν z C Λ Ν z C Λ Ν z z Λ 4 z Λ 3 Λ Distant neighbors With respect to Ν C Λ Λ Ν z n Ν, Λ, z C Νn z Ν n n Λ Ν Λ nν, Λ, zc Νn z ; 0 Ν, Λ, z Λ Ν z z n Λ Ν Ν, Λ, z n Ν, Λ, z Λ Ν n Λ Ν Ν n nν, Λ, z n Λ Ν nν, Λ, z n C Λ Λ Ν z n Ν, Λ, z C Νn z Λ Ν n Ν n Λ n Ν, Λ, z C Νn z ; 0 Ν, Λ, z Λ Ν z z Λ Ν n Λ Ν n Ν, Λ, z n Ν, Λ, z n Ν, Λ, z Ν Ν n Ν n nν, Λ, z n Functional identities Relations between contiguous functions Recurrence relations Λ Λ Ν C Ν Λ z Ν C Ν z Λ Ν z C Λ Ν z C Λ Ν z Λ Ν z Λ Ν C Λ Λ Νz Ν C Ν z Λ Ν z C Λ Λ Ν z C Ν z Ν Λ Ν Λ Ν C Λ Νz C Λ Λ Ν z z C Ν z Λ Ν Λ C Ν Λ z Ν z Λ z C Λ Ν z Λ Ν C Λ Νz C Λ Ν z Λ Ν

27 7 Normalized recurrence relation Ν Ν Λ Ν z pν, z pν, z pν, z ; pν, z C Λ 4 Ν Λ Ν Λ Ν Ν z Λ Ν Relations of special ind sinπ Ν C Λ Ν z sinπ Λ Ν C Λ ΛΝz Λ Ν Ν C Λ Ν z z Λ Λ C Λ Ν z 4 z Λ Λ C Λ Ν z 0 Complex characteristics Real part ReC n Λ x y n j j Λ j j 0 j jλ C n jx y j ; x y Λ n Imaginary part ImC n Λ x y n j j Λ j j 0 j jλ C n jx y j ; x y Λ n Differentiation Low-order differentiation With respect to Ν C Λ Ν z z Λ Ν Ν Λ Ν Λ Ν F 0 Ν, Λ Ν ; ;, Λ Ν;, Λ 3 ;; Λ Ν ; z, z Λ Ν F 0 Ν, Λ Ν ; ;, Ν;, Λ 3 ;; Ν; z, z Λ Π Λ Ν ΨΝ Ψ Λ Ν F Ν, Λ Ν; Λ Ν Λ ; z With respect to Λ

28 C Λ Ν z Ν z Λ Ν Λ F 0 Λ Λ Ν Λ Ν, Λ Ν ; ;, Λ Ν;, Λ 3 ;; Λ Ν ; z, z Ν, Λ Ν ; ;, Λ ; Λ Ν F 0 z, Λ 3 ;; Λ 3 ;, z Λ Π Λ Ν Ψ Λ Ψ Λ Ν F Ν, Λ Ν; Λ Ν Λ ; z With respect to z Forward shift operator: C Ν Λ z z C Ν Λ z z Λ C Λ Ν z Bacward shift operator: 4 Λ Λ C Λ Ν z z C Ν Λ z Ν Ν Λ z Λ C Λ Ν z C Λ Ν z z Λ z Λ C Ν Λ z z Ν Ν Λ z Λ 3 C Λ Ν z Λ Symbolic differentiation With respect to z m C Ν Λ z z m m Λ m C mλ Νm z ; m m C Ν Λ z Λ Π z m Λ Ν z m Ν Λ 3 F, Ν, Λ Ν; m, Λ ; z ; m Fractional integro-differentiation With respect to z Α C Ν Λ z Λ Π Λ Ν z Α Ν Λ z Α F 0 0 Ν, Λ Ν; ; ; Λ ; Α; ; z,

29 9 Integration Indefinite integration Involving only one direct function C Λ Ν z z Λ C Λ Νz CΝ z z P Ν z C Ν z z U Νz Involving one direct function and elementary functions Involving power function z Α C Ν Λ z z Λ Π Λ Ν z Α Α Ν Λ F 0 0 Ν, Λ Ν; Α; ; Λ ; Α ; ; z, Involving algebraic functions z Λ z Λ Λ C Λ Ν z z C Λ Ν z Ν Ν Λ z Ν3 C Λ Ν z z z Ν Ν Λ z Ν Λ3 z C Λ Ν z z Definite integration Orthogonality: Involving the direct function Λ C Ν z Ν Λ t Λ C m Λ t C n Λ tt Π Λ n Λ Ν n n Λ Λ Λ C Ν z m,n ; m n ReΛ Λ 0

30 30 Summation Finite summation n 4 n Λ Λ n Λ 0 4 Λ Π Λ Λ C Λ n z z z z Α z z C Λ n z C Λ Λ n z C Α Infinite summation C Λ n z w n w z w Λ ; z w n Λ n C Λ n z w n Λ n n 0 Λ w z w w z w z w Λ ; z w C Λ n z w n z w 0F ; Λ Λ n ; 4 z w ; z w n C Λ n z w n 0 F ; Λ n 0 Λ n Λ ; z w 0F ; Λ ; z w ; z w n Γ n Λ Γ n C Λ n z w n F Γ, Λ Γ; Λ n 0 Λ n Λ ; n w z w w F Γ, Λ Γ; Λ ; w z w w ; z w Λ n C Λ n z w n w z Λ F Λ n n n n Λ n 0 n Λ C n Λ x C Λ n y Π Λ Λ Λ, Λ ; Λ z w ; ; z w w z Λ Λ x 4 y 4 x y ; ReΛ Λ 0 x y Operations Limit operation

31 lim Λ0 Λ C Ν Λ z C 0 Ν z lim Λ0 Λ C Ν Λ z Ν T Νz lim Λ Ν z Λ C Ν Λ Λ Ν H Νz ; z lim z zν C Λ Ν z Λ Ν Ν Orthogonality, completeness, and Fourier expansions The set of functions C n Λ x, n 0,,, forms a complete, orthogonal (with weight system on the interval,. n nλ Λ Π Λ n Λ x Λ ) n n Λ Λ Π Λ n Λ n 0 Λ x 4 C Λ n x n n Λ Λ Π Λ n Λ Λ y 4 C Λ n y x y ; ReΛ Λ 0 x y m m Λ Λ Λ t 4 C Λ m t Π Λ m Λ n n Λ Λ Π Λ n Λ Λ t 4 C Λ n t t m,n ; ReΛ Λ 0 Any sufficiently smooth function f x can be expanded in the system C n Λ x n 0,, as a generalized Fourier series, with its sum converging to f x almost everywhere f x c n Ψ n x ; c n Ψ n t f tt Ψ n x n 0 n n Λ Λ Π Λ n Λ Λ x 4 C Λ n x x Representations through more general functions Through hypergeometric functions Involving 0 F

32 C Λ Ν z Λ Π Λ Ν F Ν, Λ Ν; Λ Ν Λ ; z C Λ Ν z Π z Λ secπ Λ Ν sinπ Ν F Λ Ν Λ, Λ Ν ; 3 z Λ; ; Λ Involving F Ν Λ C Λ Ν z Λ Ν F Ν, Ν Λ; Λ ; z ; Λ cosπ Λ Ν secπ Λ Λ Ν C Λ Ν z F Ν, Λ Ν; Λ Ν Λ ; z Λ sinν Π Λ z Λ F Λ Ν Π Λ, Λ Ν; 3 z Λ; ; Λ C Ν Λ z Ν Λ Ν Ν Λ z Ν F Ν, Λ Ν ; Λ Ν ; z ΛΝ sinν ΠΛ Ν Λ Ν z ΛΝ F Λ Ν, Λ Ν Π Λ ; Λ Ν ; z ; Λ Ν Through hypergeometric functions of two variables C Ν Λ z Λ Π Λ Ν Ν Λ F Ν, Λ Ν;;; Λ ;;;, z Through Meijer G Classical cases for the direct function itself C Ν Λ z Λ sinπ Ν Π Λ, G z, Ν, Λ Ν 0, Λ ; Ν C n Λ z Λ Π Λ, lim sinπ m G z, mn m, m Λ 0, Λ ; n C Ν Λ z Λ sinπ Ν Π Λ G,, z Ν, Λ Ν 0, Λ ; Ν Classical cases involving algebraic functions

33 z Λ cosλ Ν Π Λ C Λ Ν Ν z G, cosλ Π Λ Ν, z Λ Ν, Λ Ν 0, Λ z Λ C Λ Ν cosλ Ν Π Λ Ν G, z cosλ Π Λ Ν, z Λ, Λ Ν, Λ Ν ; z, z z ΛΝ C Λ Ν G, z Λ Λ Ν Ν, z Λ Ν, Λ Ν ; z, 0, Λ z ΛΝ C Ν Λ z z G,, z Λ Λ Ν Ν Λ Ν, Λ Ν ; z, 0 0, Λ z Λ Ν C Λ Ν z Ν, z Λ Ν G, Ν Λ, Λ Ν 0, Λ z Λ Ν z C Λ Ν z Ν, z Λ Ν G, Λ Ν, Ν 0, ; z, z Λ Ν z C Λ Ν z Λ Ν G,, z Λ, Λ Ν 0, Λ z Λ Ν C Ν Λ z z z Λ Ν G,, z Λ Ν, Λ Ν Ν, Λ Ν ; z, 0 Classical cases involving unit step Θ Λ Θ z z Λ Ν Λ C Λ Ν z Ν G,0, z Λ Ν, Λ Ν 0, Λ ; z, Λ Θz z Λ Ν Λ C Λ Ν z Ν G 0,, z Λ Ν, Λ Ν 0, Λ Θ z z Λ C Ν Λ Λ Ν Λ z Ν G,0, z Λ, Λ Λ Ν, Ν

34 Θz z Λ C Ν Λ Λ Ν Λ z Ν G 0,, z Λ, Λ Λ Ν, Ν ; z, Θ z z Λ C Ν Λ Λ Ν Λ z Ν G,0, z Ν Λ, Ν 0, ; z, Θz z Λ C Ν Λ Λ Λ Ν z G 0,, z Ν Ν Ν, Λ 0, Θ z z Λ C Ν Λ z Λ Λ Ν G,0, z Ν Λ, Λ Λ Ν, Ν ; z, Θz z Λ C Ν Λ z Λ Ν Λ Ν G 0,, z Λ, Λ Ν, Λ Ν Θ z z n Λ C n Λ z n Λ n G,0, z n Λ, n Λ ; n 0, Λ Θz z n Λ C n Λ z z n Λ G 0,, z n n Λ, n 0, ; n Θ z z n Λ z C Λ n z n Λ G,0, z n n Λ, Λ 0, Λ ; n Θz z n Λ C n Λ z z z n Λ n G 0,, z n Λ, Λ n n Λ, n ; n Θ z z Λ C Ν Λ z z Λ Ν Ν G,0, z Λ Ν, Λ Ν Λ Ν, Ν ; z, Θz z Λ C Ν Λ z z Λ Ν Ν G, 0, z Λ Ν, Λ Ν Λ Ν, Ν Generalized cases involving algebraic functions

35 z Λ Ν C Ν Λ z z Ν Λ Ν G,, z, Λ Ν, Ν 0, ; Rez z Λ Ν C Ν Λ z z z Λ Ν G,, z, Λ Ν, Λ Ν Ν, Λ Ν ; Rez 0 Generalized cases involving unit step Θ Θ z z Λ C Ν Λ z Λ Ν Ν Λ G,0, z, Ν Λ, Ν 0, Θz z Λ Λ Λ C Λ Ν Ν z G 0,, z, Ν Ν Ν, Λ 0, ; Rez Θ z z Λ C Ν Λ z Λ Λ Ν Ν G,0, z, Λ, Λ Λ Ν, Ν Θz z Λ C Ν Λ z Λ Ν Λ Ν G 0,, z, Λ, Λ Ν, Λ Ν ; Rez Θz z n Λ C n Λ z z n Λ n G 0,, z, n Λ, n 0, ; n Rez Θz z n Λ C n Λ z z z n Λ G 0,, z, n n Λ, Λ n n Λ, n ; n Rez Θ z z Λ C Ν Λ z z Λ Ν Ν G,0, z, Λ Ν, Λ Ν Λ Ν, Ν Θz z Λ C Ν Λ z z Λ Ν Ν G 0,, z, Λ Ν, Λ Ν Λ Ν, Ν ; Rez 0 Through other functions Involving some hypergeometric-type functions

36 C Ν Λ z Λ Ν Λ Ν Λ,Λ PΝ z C Ν Λ z C Ν Λ z Π Λ Ν Λ Ν P Ν Π z Λ Ν P Λ Ν Λ, z Ν Λ, z Representations through equivalent functions With related functions C Λ Ν z Λ C Λ Ν z Λ Π Ν Λ Ν Λ Π Λ Ν Ν Λ Λ z Λ z 4 P ΛΝ z Λ 4 z Λ Λ z 4 ΛΝ Π Λ C Λ Ν z Λ Ν Λ Λ Ν Λ Ν Ν Λ z 4 Λ Y ΛΝ cos z, 0 Theorems Expansions in generalized Fourier series f x 0 c Ψ x ; c f tψ tt, Ψ x Λ Λ Π Λ n n Λ n Λ x 4 C Λ x,. Eigenfunctions of the angular part of a d-dimensional Laplace operator The eigenfunctions of the angular part L d ij x i x j x j of a d-dimensional Laplace operator x i r d r rd L r, where u, r u are two unit vectors in d, has the representation L C n d u.u nn d C n d u.u. Removing Gibbs oscillations from Fourier series

37 37 Let f x be a doubly periodic function with f f. Let f be its Fourier components f f x Π x n x. Then the Fourier sum n f Π x exhibits Gibbs oscillations. It is possible to recover the original function f x as a sum without such Gibbs oscillations in the following n manner: 0 g n C Λ x, where g n 0 f 0 Λ n Λ l n 0 l J Λ Π l l Π Λ Π n f l /; Λ. 7 This sum converges pointwise to f x. Quantum mechanical eigenfunctions of the hydrogen atom The quantum mechanical eigenfunctions Ψ n m l p, Θ, Φ of the hydrogen atom in the momentum representation are: Ψ n m l p, Θ, Φ 6Π m Κ n l l n l n l Κ n p Κ n p Κ n p l l C nl Κ n p Κ n p Y l m Θ, Φ ; Κ n Τ ; Τ 0, n, l, l n, m, m l. n History L. Gegenbauer (893)

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