# with imaginary periods

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##### 1: 29.10 Lamé Functions with Imaginary Periods
###### §29.10 Lamé Functions with ImaginaryPeriods
The first and the fourth functions have period $2\mathrm{i}{K^{\prime}}$; the second and the third have period $4\mathrm{i}{K^{\prime}}$. …
##### 2: 25.13 Periodic Zeta Function
The notation $F\left(x,s\right)$ is used for the polylogarithm $\operatorname{Li}_{s}\left(e^{2\pi ix}\right)$ with $x$ real:
25.13.2 $F\left(x,s\right)=\frac{\Gamma\left(1-s\right)}{(2\pi)^{1-s}}\*\left(e^{\pi i(% 1-s)/2}\zeta\left(1-s,x\right)+e^{\pi i(s-1)/2}\zeta\left(1-s,1-x\right)\right),$ $0, $\Re s>1$,
25.13.3 $\zeta\left(1-s,x\right)=\frac{\Gamma\left(s\right)}{(2\pi)^{s}}\left(e^{-\pi is% /2}F\left(x,s\right)+e^{\pi is/2}F\left(-x,s\right)\right),$ $\Re s>0$ if $0; $\Re s>1$ if $x=1$.
##### 3: 4.28 Definitions and Periodicity
The functions $\sinh z$ and $\cosh z$ have period $2\pi i$, and $\tanh z$ has period $\pi i$. …
##### 4: 22.4 Periods, Poles, and Zeros
This half-period will be plus or minus a member of the triple ${K,iK^{\prime},K+iK^{\prime}}$; the other two members of this triple are quarter periods of $\operatorname{pq}\left(z,k\right)$. …
##### 5: 4.2 Definitions
It has period $2\pi i$: …
##### 6: 28.31 Equations of Whittaker–Hill and Ince
Formal $2\pi$-periodic solutions can be constructed as Fourier series; compare §28.4: … where $(u_{0},u_{\infty})=(0,\mathrm{i}\infty)$ when $\xi>0$, and $(u_{0},u_{\infty})=(\tfrac{1}{2}\pi,\tfrac{1}{2}\pi+\mathrm{i}\infty)$ when $\xi<0$. … For $\xi>0$, the functions $\mathit{hc}_{p}^{m}(z,\xi)$, $\mathit{hs}_{p}^{m}(z,\xi)$ behave asymptotically as multiples of $\exp\left(-\tfrac{1}{4}\xi\cos\left(2z\right)\right)\left(\cos z\right)^{p}$ as $z\to\pm\mathrm{i}\infty$. All other periodic solutions behave as multiples of $\exp\left(\tfrac{1}{4}\xi\cos\left(2z\right)\right)(\cos z)^{-p-2}$. … All other periodic solutions behave as multiples of $\exp\left(-\tfrac{1}{4}\xi\cos\left(2z\right)\right)\left(\cos z\right)^{p}$.
##### 7: 27.10 Periodic Number-Theoretic Functions
27.10.2 $f(n)=\sum_{m=1}^{k}g(m)e^{2\pi\mathrm{i}mn/k},$
27.10.3 $g(m)=\dfrac{1}{k}\sum_{n=1}^{k}f(n)e^{-2\pi\mathrm{i}mn/k}.$
##### 8: 20.13 Physical Applications
The functions $\theta_{j}\left(z\middle|\tau\right)$, $j=1,2,3,4$, provide periodic solutions of the partial differential equation …with $\kappa=-i\pi/4$. For $\tau=it$, with $\alpha,t,z$ 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 $\Im\tau\rightarrow 0+$, the functions $\theta_{j}\left(z\middle|\tau\right)$, $j=1,2,3,4$, 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 $\sin z=\frac{e^{\mathrm{i}z}-e^{-\mathrm{i}z}}{2\mathrm{i}},$
4.14.2 $\cos z=\frac{e^{\mathrm{i}z}+e^{-\mathrm{i}z}}{2},$
##### 10: 21.4 Graphics
21.4.1 $\boldsymbol{{\Omega}}=\begin{bmatrix}1.69098\;3006+0.95105\;6516\,i&1.5+0.3632% 7\;1264\,i\\ 1.5+0.36327\;1264\,i&1.30901\;6994+0.95105\;6516\,i\end{bmatrix}.$
21.4.2 $\boldsymbol{{\Omega}}_{1}=\begin{bmatrix}i&-\tfrac{1}{2}\\ -\tfrac{1}{2}&i\end{bmatrix},$ Figure 21.4.2: ℜ ⁡ θ ^ ⁡ ( x + i ⁢ y , 0 | 𝛀 1 ) , 0 ≤ x ≤ 1 , 0 ≤ y ≤ 5 . (The imaginary part looks very similar.) Magnify 3D Help 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