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1: 21.4 Graphics

… ►Figure 21.4.1 provides surfaces of the scaled Riemann theta function $\widehat{\theta }\left(𝐳\right|𝛀)$, with … ►

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Figure 21.4.1: $\widehat{\theta }\left(𝐳\right|𝛀)$ parametrized by (21.4.1). The surface plots are of $\widehat{\theta }\left(x+\mathrm{i}y,0\right|𝛀)$, $0\le x\le 1$, $0\le y\le 5$ (suffix 1); $\widehat{\theta }\left(x,y\right|𝛀)$, $0\le x\le 1$, $0\le y\le 1$ (suffix 2); $\widehat{\theta }\left(\mathrm{i}x,\mathrm{i}y\right|𝛀)$, $0\le x\le 5$, $0\le y\le 5$ (suffix 3). …

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

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Figure 21.4.3: $\left|\widehat{\theta }\left(x+\mathrm{i}y,0\right|{𝛀}_1)\right|$, $0\le x\le 1$, $0\le y\le 2$. Magnify 3D Help

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2: 14.20 Conical (or Mehler) Functions

… ►Another real-valued solution ${\widehat{𝖰}}_{-\frac{1}{2}+\mathrm{i}\tau }^{-\mu }\left(x\right)$ of (14.20.1) was introduced in Dunster (1991). …${\widehat{𝖰}}_{-\frac{1}{2}+\mathrm{i}\tau }^{-\mu }\left(x\right)$ exists except when $\mu =\frac{1}{2},\frac{3}{2},\mathrm{\dots }$ and $\tau =0$; compare §14.3(i). …provided that ${\widehat{𝖰}}_{-\frac{1}{2}+\mathrm{i}\tau }^{-\mu }\left(x\right)$ exists. … ►

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Figure 14.20.2: ${\widehat{𝖰}}_{-\frac{1}{2}+\mathrm{i}\tau }^0\left(x\right)$, $\tau =0,\frac{1}{2},1,2,4$. Magnify

… ►Approximations for ${𝖯}_{-\frac{1}{2}+\mathrm{i}\tau }^{\mu }\left(x\right)$ and ${\widehat{𝖰}}_{-\frac{1}{2}+\mathrm{i}\tau }^{-\mu }\left(x\right)$ can then be achieved via (14.9.7) and (14.20.3). …

3: 21.2 Definitions

… ►For numerical purposes we use the scaled Riemann theta function $\widehat{\theta }\left(𝐳\right|𝛀)$, defined by (Deconinck et al. (2004)), ►

21.2.2 $$\widehat{\theta }\left(𝐳\right|𝛀)={\mathrm{e}}^{-\pi [\mathrm{\Im }𝐳]\cdot {[\mathrm{\Im }𝛀]}^{-1}\cdot [\mathrm{\Im }𝐳]}\theta \left(𝐳\right|𝛀).$$

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Defines:

$\widehat{\theta }\left(𝐳\right|𝛀)$: scaled Riemann theta function

Symbols:

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

Permalink:

http://dlmf.nist.gov/21.2.E2

Encodings:

pMML, png, TeX

See also:

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

► $\widehat{\theta }\left(𝐳\right|𝛀)$ is a bounded nonanalytic function of $𝐳$. … ►

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

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4: 14.34 Software

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5: 14.1 Special Notation

… ►The main functions treated in this chapter are the Legendre functions ${𝖯}_{\nu }\left(x\right)$, ${𝖰}_{\nu }\left(x\right)$, $P_{\nu }\left(z\right)$, $Q_{\nu }\left(z\right)$; Ferrers functions ${𝖯}_{\nu }^{\mu }\left(x\right)$, ${𝖰}_{\nu }^{\mu }\left(x\right)$ (also known as the Legendre functions on the cut); associated Legendre functions $P_{\nu }^{\mu }\left(z\right)$, $Q_{\nu }^{\mu }\left(z\right)$, ${𝑸}_{\nu }^{\mu }\left(z\right)$; conical functions ${𝖯}_{-\frac{1}{2}+\mathrm{i}\tau }^{\mu }\left(x\right)$, ${𝖰}_{-\frac{1}{2}+\mathrm{i}\tau }^{\mu }\left(x\right)$, ${\widehat{𝖰}}_{-\frac{1}{2}+\mathrm{i}\tau }^{\mu }\left(x\right)$, $P_{-\frac{1}{2}+\mathrm{i}\tau }^{\mu }\left(x\right)$, $Q_{-\frac{1}{2}+\mathrm{i}\tau }^{\mu }\left(x\right)$ (also known as Mehler functions). …

6: 14.23 Values on the Cut

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14.23.7 $${\widehat{𝖰}}_{-\frac{1}{2}+\mathrm{i}\tau }^{-\mu }\left(x\right)=\frac{1}{2}{\mathrm{e}}^{3\mu \pi \mathrm{i}/2}{Q}_{-\frac{1}{2}+\mathrm{i}\tau }^{-\mu }\left(x-\mathrm{i}0\right)+\frac{1}{2}{\mathrm{e}}^{-3\mu \pi \mathrm{i}/2}{Q}_{-\frac{1}{2}-\mathrm{i}\tau }^{-\mu }\left(x+\mathrm{i}0\right).$$

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Symbols:

${\widehat{𝖰}}_{-\frac{1}{2}+\mathrm{i}\tau }^{-\mu }\left(x\right)$: conical function, $Q_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the second kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $x$: real variable, $\tau$: real variable and $\mu$: general order

Referenced by:

§14.23

Permalink:

http://dlmf.nist.gov/14.23.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §14.23 and Ch.14

7: Guide to Searching the DLMF

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Table 4: Font and Accent Examples

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Query	Matches
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theta hat	$\widehat{\theta }$
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8: Software Index

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	Open Source													With Book						Commercial
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14.34(iv) ${𝖯}_{-\frac{1}{2}+\mathrm{i}\tau }\left(x\right)$, ${𝖰}_{-\frac{1}{2}+\mathrm{i}\tau }\left(x\right)$, ${\widehat{𝖰}}_{-\frac{1}{2}+\mathrm{i}\tau }^{\mu }\left(x\right)$, $P_{-\frac{1}{2}+\mathrm{i}\tau }\left(x\right)$, $Q_{-\frac{1}{2}+\mathrm{i}\tau }\left(x\right)$					✓									✓			✓
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9: Bibliography C

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Über die Fälle, wenn die Reihe von der Form $y=1+\frac{\alpha }{1}\cdot \frac{\beta }{\gamma }x+\frac{\alpha \cdot \alpha +1}{1\cdot 2}\cdot \frac{\beta \cdot \beta +1}{\gamma \cdot \gamma +1}{x}^2+$ etc. ein Quadrat von der Form $z=1+\frac{{\alpha }^{\prime }}{1}\cdot \frac{{\beta }^{\prime }}{{\gamma }^{\prime }}\cdot \frac{{\delta }^{\prime }}{{ϵ}^{\prime }}x+\frac{{\alpha }^{\prime }\cdot {\alpha }^{\prime }+1}{1\cdot 2}\cdot \frac{{\beta }^{\prime }\cdot {\beta }^{\prime }+1}{{\gamma }^{\prime }\cdot {\gamma }^{\prime }+1}\cdot \frac{{\delta }^{\prime }\cdot {\delta }^{\prime }+1}{{ϵ}^{\prime }\cdot {ϵ}^{\prime }+1}{x}^2+$ etc. hat. J. Reine Angew. Math. 3, pp. 89–91.

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External Links:

Link, MathReview Entry, ZentralBlatt

Referenced by:

§16.12.

Encodings:

BibTeX

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