About the Project

with imaginary periods

AdvancedHelp

(0.001 seconds)

1—10 of 39 matching pages

1: 29.10 Lamé Functions with Imaginary Periods
§29.10 Lamé Functions with Imaginary Periods
The first and the fourth functions have period 2 i K ; the second and the third have period 4 i K . …
2: 25.13 Periodic Zeta Function
The notation F ( x , s ) is used for the polylogarithm Li s ( e 2 π i x ) with x real:
25.13.1 F ( x , s ) n = 1 e 2 π i n x n s ,
25.13.2 F ( x , s ) = Γ ( 1 s ) ( 2 π ) 1 s ( e π i ( 1 s ) / 2 ζ ( 1 s , x ) + e π i ( s 1 ) / 2 ζ ( 1 s , 1 x ) ) , 0 < x < 1 , s > 1 ,
25.13.3 ζ ( 1 s , x ) = Γ ( s ) ( 2 π ) s ( e π i s / 2 F ( x , s ) + e π i s / 2 F ( x , s ) ) , s > 0 if 0 < x < 1 ; s > 1 if x = 1 .
3: 4.28 Definitions and Periodicity
The functions sinh z and cosh z have period 2 π i , and tanh z has period π i . …
4: 22.4 Periods, Poles, and Zeros
This half-period will be plus or minus a member of the triple K , i K , K + i K ; the other two members of this triple are quarter periods of p q ( z , k ) . …
5: 4.2 Definitions
It has period 2 π i : …
6: 28.31 Equations of Whittaker–Hill and Ince
Formal 2 π -periodic solutions can be constructed as Fourier series; compare §28.4: … where ( u 0 , u ) = ( 0 , i ) when ξ > 0 , and ( u 0 , u ) = ( 1 2 π , 1 2 π + i ) when ξ < 0 . … For ξ > 0 , the functions hc p m ( z , ξ ) , hs p m ( z , ξ ) behave asymptotically as multiples of exp ( 1 4 ξ cos ( 2 z ) ) ( cos z ) p as z ± i . All other periodic solutions behave as multiples of exp ( 1 4 ξ cos ( 2 z ) ) ( cos z ) p 2 . … All other periodic solutions behave as multiples of exp ( 1 4 ξ cos ( 2 z ) ) ( cos z ) p .
7: 27.10 Periodic Number-Theoretic Functions
8: 20.13 Physical Applications
The functions θ j ( z | τ ) , j = 1 , 2 , 3 , 4 , provide periodic solutions of the partial differential equation …with κ = i π / 4 . For τ = i t , with α , t , z real, (20.13.1) takes the form of a real-time t diffusion equation …Thus the classical theta functions are “periodized”, or “anti-periodized”, Gaussians; see Bellman (1961, pp. 18, 19). … In the singular limit τ 0 + , the functions θ j ( z | τ ) , j = 1 , 2 , 3 , 4 , become integral kernels of Feynman path integrals (distribution-valued Green’s functions); see Schulman (1981, pp. 194–195). …
9: 4.14 Definitions and Periodicity
§4.14 Definitions and Periodicity
4.14.1 sin z = e i z e i z 2 i ,
4.14.2 cos z = e i z + e i z 2 ,
4.14.3 cos z ± i sin z = e ± i z ,
10: 21.4 Graphics
21.4.1 Ω = [ 1.69098 3006 + 0.95105 6516 i 1.5 + 0.36327 1264 i 1.5 + 0.36327 1264 i 1.30901 6994 + 0.95105 6516 i ] .
Figure 21.4.1: θ ^ ( z | Ω ) parametrized by (21.4.1). …Shown are the real part (a), the imaginary part (b), and the modulus (c).
21.4.3 Ω 2 = [ 1 2 + i 1 2 1 2 i 1 2 1 2 i 1 2 1 2 i i 0 1 2 1 2 i 0 i ] .
See accompanying text
Figure 21.4.2: θ ^ ( x + i y , 0 | Ω 1 ) , 0 x 1 , 0 y 5 . (The imaginary part looks very similar.) Magnify 3D Help
See accompanying text
Figure 21.4.4: A real-valued scaled Riemann theta function: θ ^ ( i x , i y | Ω 1 ) , 0 x 4 , 0 y 4 . In this case, the quasi-periods are commensurable, resulting in a doubly-periodic configuration. Magnify 3D Help