roots of unity

SearchAdvancedHelp

(0.001 seconds)

4 matching pages

1: 1.11 Zeros of Polynomials

… ►

§1.11(iv) Roots of Unity and of Other Constants

…

2: 21.5 Modular Transformations

… ►

21.5.4 $$\theta \left({\left[{[𝐂𝛀+𝐃]}^{-1}\right]}^{\mathrm{T}}𝐳\right|[𝐀𝛀+𝐁]{[𝐂𝛀+𝐃]}^{-1})=\xi (𝚪)\sqrt{det[𝐂𝛀+𝐃]}{\mathrm{e}}^{\pi \mathrm{i}𝐳\cdot \left[{[𝐂𝛀+𝐃]}^{-1}𝐂\right]\cdot 𝐳}\theta \left(𝐳\right|𝛀).$$

ⓘ

Symbols:

$\theta \left(𝐳\right|𝛀)$: Riemann theta function, $\pi$: the ratio of the circumference of a circle to its diameter, $det$: determinant, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, ${𝐀}^{\mathrm{T}}$: transpose of matrix, $𝛀$: a Riemann matrix, Matrix $𝚪$ and $\xi (𝚪)$: eighth root of unity

Referenced by:

§21.5(i)

Permalink:

http://dlmf.nist.gov/21.5.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §21.5(i), §21.5 and Ch.21

►Here $\xi (𝚪)$ is an eighth root of unity, that is, ${(\xi (𝚪))}^8=1$. …

3: 1.3 Determinants, Linear Operators, and Spectral Expansions

… ►

1.3.15

ⓘ

Symbols:

$det$: determinant, $k$: integer, $n$: nonnegative integer and ${\omega }_j$: roots of unity

Permalink:

http://dlmf.nist.gov/1.3.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §1.3(ii), §1.3(ii), §1.3 and Ch.1

►where ${\omega }_1,{\omega }_2,\mathrm{\dots },{\omega }_n$ are the $n$th roots of unity (1.11.21). …

4: 27.10 Periodic Number-Theoretic Functions

… ►This is the sum of the $n$th powers of the primitive $k$th roots of unity. …