with imaginary periods

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1: 29.10 Lamé Functions with Imaginary Periods

§29.10 Lamé Functions with Imaginary Periods

… ►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 $F(x,s)$ is used for the polylogarithm ${\mathrm{Li}}_s\left({\mathrm{e}}^{2\pi \mathrm{i}x}\right)$ with $x$ real: ►

25.13.1 $$F(x,s)\equiv \sum _{n=1}^{\mathrm{\infty }}\frac{{\mathrm{e}}^{2\pi \mathrm{i}nx}}{n^s},$$

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Defines:

$F(x,s)$: periodic zeta function

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\equiv$: equals by definition, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $n$: nonnegative integer, $x$: real variable and $s$: complex variable

Keywords:

definition, infinite series, series representation

Source:

Apostol (1976, (9), p. 257)

Permalink:

http://dlmf.nist.gov/25.13.E1

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Annotations for §25.13 and Ch.25

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25.13.2 $$F(x,s)=\frac{\mathrm{\Gamma }\left(1-s\right)}{{(2\pi )}^{1-s}}\left({\mathrm{e}}^{\pi \mathrm{i}(1-s)/2}\zeta (1-s,x)+{\mathrm{e}}^{\pi \mathrm{i}(s-1)/2}\zeta (1-s,1-x)\right),$$ , $\mathrm{\Re }s>1$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\zeta (s,a)$: Hurwitz zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $F(x,s)$: periodic zeta function, $\mathrm{\Re }$: real part, $x$: real variable and $s$: complex variable

Keywords:

connection formula

Source:

Apostol (1976, Exercise 2, p. 273)

Permalink:

http://dlmf.nist.gov/25.13.E2

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Annotations for §25.13 and Ch.25

►

25.13.3 $$\zeta (1-s,x)=\frac{\mathrm{\Gamma }\left(s\right)}{{(2\pi )}^s}\left({\mathrm{e}}^{-\pi \mathrm{i}s/2}F(x,s)+{\mathrm{e}}^{\pi \mathrm{i}s/2}F(-x,s)\right),$$ $\mathrm{\Re }s>0$ if ; $\mathrm{\Re }s>1$ if $x=1$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\zeta (s,a)$: Hurwitz zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $F(x,s)$: periodic zeta function, $\mathrm{\Re }$: real part, $x$: real variable and $s$: complex variable

Keywords:

connection formula

Source:

Apostol (1976, (10), p. 257)

Notes:

This formula is equivalent to (25.11.9).

Referenced by:

Erratum (V1.1.4) for Equation (25.13.3)

Permalink:

http://dlmf.nist.gov/25.13.E3

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Annotations for §25.13 and Ch.25

3: 4.28 Definitions and Periodicity

… ►The functions $\sinh z$ and $\cosh z$ have period $2\pi \mathrm{i}$, and $\tanh z$ has period $\pi \mathrm{i}$. …

4: 22.4 Periods, Poles, and Zeros

… ►This half-period will be plus or minus a member of the triple $K,\mathrm{i}{K}^{\prime },K+\mathrm{i}{K}^{\prime }$; the other two members of this triple are quarter periods of $\mathrm{p}\mathrm{q}(z,k)$. …

5: 4.2 Definitions

… ►It has period $2\pi \mathrm{i}$: …

6: 28.31 Equations of Whittaker–Hill and Ince

… ►Formal $2\pi$-periodic solutions can be constructed as Fourier series; compare §28.4: … ►where $(u_0,u_{\mathrm{\infty }})=(0,\mathrm{i}\mathrm{\infty })$ when $\xi >0$, and $(u_0,u_{\mathrm{\infty }})=(\frac{1}{2}\pi ,\frac{1}{2}\pi +\mathrm{i}\mathrm{\infty })$ when . … ►For $\xi >0$, the functions ${\mathrm{ℎ𝑐}}_p^m(z,\xi )$, ${\mathrm{ℎ𝑠}}_p^m(z,\xi )$ behave asymptotically as multiples of $\exp \left(-\frac{1}{4}\xi \cos \left(2z\right)\right){\left(\cos z\right)}^p$ as $z\to \pm \mathrm{i}\mathrm{\infty }$. All other periodic solutions behave as multiples of $\exp \left(\frac{1}{4}\xi \cos \left(2z\right)\right){(\cos z)}^{-p-2}$. … ►All other periodic solutions behave as multiples of $\exp \left(-\frac{1}{4}\xi \cos \left(2z\right)\right){\left(\cos z\right)}^p$.

7: 27.10 Periodic Number-Theoretic Functions

… ►

27.10.2 $$f(n)=\sum _{m=1}^{k}g(m){\mathrm{e}}^{2\pi \mathrm{i}mn/k},$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $k$: positive integer, $m$: positive integer, $n$: positive integer, $f(n)$: function and $g(m)$: periodic function

Permalink:

http://dlmf.nist.gov/27.10.E2

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Annotations for §27.10 and Ch.27

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27.10.3 $$g(m)=\frac{1}{k}\sum _{n=1}^{k}f(n){\mathrm{e}}^{-2\pi \mathrm{i}mn/k}.$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $k$: positive integer, $m$: positive integer, $n$: positive integer, $f(n)$: function and $g(m)$: periodic function

Permalink:

http://dlmf.nist.gov/27.10.E3

Encodings:

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See also:

Annotations for §27.10 and Ch.27

…

8: 20.13 Physical Applications

… ►The functions ${\theta }_j\left(z\right|\tau )$, $j=1,2,3,4$, provide periodic solutions of the partial differential equation …with $\kappa =-\mathrm{i}\pi /4$. ►For $\tau =\mathrm{i}t$, 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 $\mathrm{\Im }\tau \to 0+$, the functions ${\theta }_j\left(z\right|\tau )$, $j=1,2,3,4$, 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 $$\sin z=\frac{{\mathrm{e}}^{\mathrm{i}z}-{\mathrm{e}}^{-\mathrm{i}z}}{2\mathrm{i}},$$

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Defines:

$\sin z$: sine function

Symbols:

$\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit and $z$: complex variable

A&S Ref:

4.3.1

Referenced by:

§4.45(ii), (9.5.3)

Permalink:

http://dlmf.nist.gov/4.14.E1

Encodings:

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Annotations for §4.14 and Ch.4

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4.14.2 $$\cos z=\frac{{\mathrm{e}}^{\mathrm{i}z}+{\mathrm{e}}^{-\mathrm{i}z}}{2},$$

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Defines:

$\cos z$: cosine function

Symbols:

$\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit and $z$: complex variable

A&S Ref:

4.3.2

Referenced by:

(9.5.3)

Permalink:

http://dlmf.nist.gov/4.14.E2

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Annotations for §4.14 and Ch.4

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4.14.3 $$\cos z\pm \mathrm{i}\sin z={\mathrm{e}}^{\pm \mathrm{i}z},$$

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Symbols:

$\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\sin z$: sine function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/4.14.E3

Encodings:

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See also:

Annotations for §4.14 and Ch.4

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10: 21.4 Graphics

… ►

…

Figure 21.4.1: $\widehat{\theta }\left(𝐳\right|𝛀)$ parametrized by (21.4.1). …Shown are the real part (a), the imaginary part (b), and the modulus (c).

… ►

21.4.2

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Symbols:

$\mathrm{i}$: imaginary unit and $𝛀$: a Riemann matrix

Permalink:

http://dlmf.nist.gov/21.4.E2

Encodings:

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See also:

Annotations for §21.4 and Ch.21

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21.4.3

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Symbols:

$\mathrm{i}$: imaginary unit and $𝛀$: a Riemann matrix

Permalink:

http://dlmf.nist.gov/21.4.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §21.4 and Ch.21

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►

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Figure 21.4.2: $\mathrm{\Re }\widehat{\theta }\left(x+\mathrm{i}y,0\right|{𝛀}_1)$, $0\le x\le 1$, $0\le y\le 5$. (The imaginary part looks very similar.) Magnify 3D Help

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Figure 21.4.4: A real-valued scaled Riemann theta function: $\widehat{\theta }\left(\mathrm{i}x,\mathrm{i}y\right|{𝛀}_1)$, $0\le x\le 4$, $0\le y\le 4$. In this case, the quasi-periods are commensurable, resulting in a doubly-periodic configuration. Magnify 3D Help

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