with imaginary periods

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21: 21.2 Definitions

… ►With $z_1=x_1+\mathrm{i}{y}_1$, $z_2=x_2+\mathrm{i}{y}_2$, ►

21.2.4

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Symbols:

$\widehat{\theta }\left(𝐳\right|𝛀)$: scaled Riemann theta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm and $\mathrm{i}$: imaginary unit

Permalink:

http://dlmf.nist.gov/21.2.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §21.2(i), §21.2(i), §21.2 and Ch.21

… ►

21.2.7 $$\theta \left[\genfrac{}{}{0.0pt}{}{\mathrm{𝟎}}{\mathrm{𝟎}}\right]\left(𝐳\right|𝛀)=\theta \left(𝐳\right|𝛀).$$

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Symbols:

$\theta \left(𝐳\right|𝛀)$: Riemann theta function, $\theta \left[\genfrac{}{}{0.0pt}{}{𝜶}{𝜷}\right]\left(𝐳\right|𝛀)$: Riemann theta function with characteristics and $𝛀$: a Riemann matrix

Permalink:

http://dlmf.nist.gov/21.2.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §21.2(ii), §21.2 and Ch.21

►Characteristics whose elements are either $0$ or $\frac{1}{2}$ are called half-period characteristics. For given $𝛀$, there are $2^{2g}$ $g$-dimensional Riemann theta functions with half-period characteristics. …

22: 31.2 Differential Equations

… ►

§31.2(iv) Doubly-Periodic Forms

►

Jacobi’s Elliptic Form

… ►

Weierstrass’s Form

… ►

$\zeta =\mathrm{i}{K}^{\prime }+\xi {(e_1-e_3)}^{1/2},$

… ►where $2{\omega }_1$ and $2{\omega }_3$ with $\mathrm{\Im }\left({\omega }_3/{\omega }_1\right)>0$ are generators of the lattice $𝕃$ for $\mathrm{\wp }\left(z|𝕃\right)$. …

23: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►Case 3: Periodic Boundary Conditions: $\varphi (0)=\varphi (\pi )$ and ${\varphi }^{\prime }(0)={\varphi }^{\prime }(\pi )$. … ►More generally, continuous spectra may occur in sets of disjoint finite intervals $[{\lambda }_a,{\lambda }_b]\in (0,\mathrm{\infty })$, often called bands, when $q(x)$ is periodic, see Ashcroft and Mermin (1976, Ch 8) and Kittel (1996, Ch 7). … ►Then $dim{N}_z$ is constant for $\mathrm{\Im }z>0$ and also constant for . Put $n_{+}=dim{N}_z$ ($\mathrm{\Im }z>0$) and $n_{-}=dim{N}_z$ (), the deficiency indices for $T$. …Then any self-adjoint extension of $T$ is determined by a linear isometry $U:N_{\mathrm{i}}\to N_{-\mathrm{i}}$ and it is the restriction of $T^{\ast }$ to $\{v+w+Uw\mid v\in 𝒟(T^{\ast \ast }),w\in N_{\mathrm{i}}\}$. …

24: 23.2 Definitions and Periodic Properties

§23.2 Definitions and Periodic Properties

… ►

§23.2(iii) Periodicity

… ►Hence $\mathrm{\wp }\left(z\right)$ is an elliptic function, that is, $\mathrm{\wp }\left(z\right)$ is meromorphic and periodic on a lattice; equivalently, $\mathrm{\wp }\left(z\right)$ is meromorphic and has two periods whose ratio is not real. … ►The function $\zeta \left(z\right)$ is quasi-periodic: for $j=1,2,3$, … ►For further quasi-periodic properties of the $\sigma$-function see Lawden (1989, §6.2).

25: 2.10 Sums and Sequences

… ►As in §24.2, let $B_n$ and $B_n\left(x\right)$ denote the $n$th Bernoulli number and polynomial, respectively, and ${\tilde{B}}_n\left(x\right)$ the $n$th Bernoulli periodic function $B_n\left(x-\lfloor x\rfloor \right)$. … ►From §24.12(i), (24.2.2), and (24.4.27), ${\tilde{B}}_{2m}\left(x\right)-B_{2m}$ is of constant sign ${(-1)}^m$. … ►where $\alpha$ and $\beta$ are real constants with ${\mathrm{e}}^{\mathrm{i}\beta }\ne 1$. … ►where $𝒞$ comprises the two semicircles and two parts of the imaginary axis depicted in Figure 2.10.1. … ►Here the branch of ${\left({\mathrm{e}}^{-\mathrm{i}\alpha }-z\right)}^{-1/2}$ is continuous in the $z$-plane cut along the outward-drawn ray through $z={\mathrm{e}}^{-\mathrm{i}\alpha }$ and equals ${\mathrm{e}}^{\mathrm{i}\alpha /2}$ at $z=0$. …

26: 1.8 Fourier Series

… ►Formally, if $f(x)$ is a real- or complex-valued $2\pi$-periodic function, … ►If $f(x)$ is of period $2\pi$, and $f^{(m)}(x)$ is piecewise continuous, then … ►If $f(x)$ and $g(x)$ are continuous, have the same period and same Fourier coefficients, then $f(x)=g(x)$ for all $x$. … ►Let $f(x)$ be an absolutely integrable function of period $2\pi$, and continuous except at a finite number of points in any bounded interval. … ►If a function $f(x)\in C^2[0,2\pi ]$ is periodic, with period $2\pi$, then the series obtained by differentiating the Fourier series for $f(x)$ term by term converges at every point to $f^{\prime }(x)$. …

27: 23.7 Quarter Periods

§23.7 Quarter Periods

… ►

23.7.2 $$\mathrm{\wp }\left(\frac{1}{2}{\omega }_2\right)=e_2-\mathrm{i}\sqrt{(e_1-e_2)(e_2-e_3)}=e_2-\mathrm{i}{\omega }_1^{-2}{(K\left(k\right))}^{2}k{k}^{\prime },$$

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Symbols:

$\mathrm{\wp }\left(z\right)$ (= $\mathrm{\wp }\left(z|𝕃\right)$ = $\mathrm{\wp }(z;g_2,g_3)$): Weierstrass $\mathrm{\wp }$-function, $e_j$: Weierstrass lattice roots, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\mathrm{i}$: imaginary unit, $𝕃$: lattice, ${\omega }_1$, ${\omega }_3$, ${\omega }_2=-{\omega }_1-{\omega }_3$: lattice generators, $k$: modulus and $k^{\prime }$: complementary modulus

A&S Ref:

18.3.18

Permalink:

http://dlmf.nist.gov/23.7.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §23.7 and Ch.23

…

28: 1.15 Summability Methods

… ►If $f(\theta )$ is periodic and integrable on $[0,2\pi ]$, then as $n\to \mathrm{\infty }$ the Abel means $A(r,\theta )$ and the (C,1) means ${\sigma }_n(\theta )$ converge to … ►

1.15.25 $$\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}F(n){\mathrm{e}}^{\mathrm{i}n\theta }$$

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Symbols:

$\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $n$: nonnegative integer and $F(n)$

Permalink:

http://dlmf.nist.gov/1.15.E25

Encodings:

pMML, png, TeX

See also:

Annotations for §1.15(iii), §1.15(iii), §1.15 and Ch.1

… ►The imaginary part ►

1.15.40 $$\mathrm{\Im }\mathrm{\Phi }(x+\mathrm{i}y)=\frac{1}{\pi }{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}f(t)\frac{x-t}{{(x-t)}^2+y^2}dt$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{\Im }$: imaginary part, $\mathrm{i}$: imaginary unit, $\int$: integral and $\mathrm{\Phi }(z)$: function

Permalink:

http://dlmf.nist.gov/1.15.E40

Encodings:

pMML, png, TeX

See also:

Annotations for §1.15(v), §1.15(v), §1.15 and Ch.1

… ►Moreover, ${lim}_{y\to 0+}\mathrm{\Im }\mathrm{\Phi }(x+\mathrm{i}y)$ is the Hilbert transform of $f(x)$ (§1.14(v)). …

29: 21.9 Integrable Equations

… ►

21.9.2 $$\mathrm{i}{u}_t=-\frac{1}{2}{u}_{xx}\pm {|u|}^{2}u.$$

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Symbols:

$\mathrm{i}$: imaginary unit and $u(x,y,t)$: elevation

Permalink:

http://dlmf.nist.gov/21.9.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §21.9 and Ch.21

… ►The KP equation has a class of quasi-periodic solutions described by Riemann theta functions, given by … ►

►►

Figure 21.9.1: Two-dimensional periodic waves in a shallow water wave tank, taken from Hammack et al. (1995, p. 97) by permission of Cambridge University Press. … Magnify

…

30: 3.5 Quadrature

… ►If in addition $f$ is periodic, $f\in C^k(ℝ)$, and the integral is taken over a period, then … ►

3.5.37 $${\int }_{c-\mathrm{i}\mathrm{\infty }}^{c+\mathrm{i}\mathrm{\infty }}{\mathrm{e}}^{\zeta }{\zeta }^{-s}{p}_k(1/\zeta )p_{\mathrm{\ell }}(1/\zeta )d\zeta =0,$$ $k\ne \mathrm{\ell }$,

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Symbols:

$dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit and $\int$: integral

Permalink:

http://dlmf.nist.gov/3.5.E37

Encodings:

pMML, png, TeX

See also:

Annotations for §3.5(viii), §3.5 and Ch.3

… ►

3.5.39 $$g(t)=\frac{1}{2\pi \mathrm{i}}{\int }_{\sigma -\mathrm{i}\mathrm{\infty }}^{\sigma +\mathrm{i}\mathrm{\infty }}{\mathrm{e}}^{tp}G(p)dp,$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $g(t)$: function, $G(p)$: Laplace transform of $g(t)$ and $\sigma$: parameter

A&S Ref:

29.2.1 (in different form)

Referenced by:

§3.5(viii)

Permalink:

http://dlmf.nist.gov/3.5.E39

Encodings:

pMML, png, TeX

See also:

Annotations for §3.5(viii), §3.5(viii), §3.5 and Ch.3

… ►The steepest descent path is given by $\mathrm{\Im }\left(t-2\sqrt{t}\right)=0$, or in polar coordinates $t=r{\mathrm{e}}^{\mathrm{i}\theta }$ we have $r={\sec }^2\left(\frac{1}{2}\theta \right)$. …The integrand can be extended as a periodic $C^{\mathrm{\infty }}$ function on $ℝ$ with period $2\pi$ and as noted in §3.5(i), the trapezoidal rule is exceptionally efficient in this case. …