unity

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1: 29.21 Tables

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Arscott and Khabaza (1962) tabulates the coefficients of the polynomials $P$ in Table 29.12.1 (normalized so that the numerically largest coefficient is unity, i.e. monic polynomials), and the corresponding eigenvalues $h$ for $k^{2}=0.1(.1)0.9$ , $n=1(1)30$ . Equations from §29.6 can be used to transform to the normalization adopted in this chapter. Precision is 6S.

2: 21.5 Modular Transformations

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21.5.4

\theta\left(\left[[\mathbf{C}\boldsymbol{{\Omega}}+\mathbf{D}]^{-1}\right]^{% \mathrm{T}}\mathbf{z}\middle|[\mathbf{A}\boldsymbol{{\Omega}}+\mathbf{B}][% \mathbf{C}\boldsymbol{{\Omega}}+\mathbf{D}]^{-1}\right)=\xi(\boldsymbol{{% \Gamma}})\sqrt{\det[\mathbf{C}\boldsymbol{{\Omega}}+\mathbf{D}]}e^{\pi i% \mathbf{z}\cdot\left[[\mathbf{C}\boldsymbol{{\Omega}}+\mathbf{D}]^{-1}\mathbf{% C}\right]\cdot\mathbf{z}}\theta\left(\mathbf{z}\middle|\boldsymbol{{\Omega}}% \right).

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Symbols:: $\theta\left(\NVar{\mathbf{z}}\middle|\NVar{\boldsymbol{{\Omega}}}\right)$ : 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, $\NVar{\mathbf{A}}^{\mathrm{T}}$ : transpose of matrix, $\boldsymbol{{\Omega}}$ : a Riemann matrix, Matrix $\boldsymbol{{\Gamma}}$ and $\xi(\boldsymbol{{\Gamma}})$ : 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(\boldsymbol{{\Gamma}})

is an eighth root of unity, that is,

(\xi(\boldsymbol{{\Gamma}}))^{8}=1

. …

3: 1.11 Zeros of Polynomials

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

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4: 37.1 Notation

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$n$	nonnegative integer.
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$1_{V}$	characteristic (or indicator) function over a space $V$ ; unity on $V$ and zero elsewhere.

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5: 16.4 Argument Unity

§16.4 Argument Unity

… ►The function

{{}_{q+1}F_{q}}

with argument unity and general values of the parameters is discussed in Bühring (1992). … ►For continued fractions for ratios of

{{}_{3}F_{2}}

functions with argument unity, see Cuyt et al. (2008, pp. 315–317). …

6: 1.3 Determinants, Linear Operators, and Spectral Expansions

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1.3.15

\begin{vmatrix}a_{1}&a_{2}&\cdots&a_{n}\\ a_{n}&a_{1}&\cdots&a_{n-1}\\ \vdots&\vdots&\ddots&\vdots\\ a_{2}&a_{3}&\cdots&a_{1}\end{vmatrix}=\prod^{n}_{k=1}(a_{1}+a_{2}\omega_{k}+a_% {3}\omega_{k}^{2}+\dots+a_{n}\omega_{k}^{n-1}),

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Symbols:: $\det$ : determinant, $k$ : integer, $n$ : nonnegative integer and $\omega_{\NVar{j}}$ : 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},\dots,\omega_{n}

are the

n

th roots of unity (1.11.21). …

7: 27.10 Periodic Number-Theoretic Functions

… ►This is the sum of the

n

th powers of the primitive

k

th roots of unity. …

8: Bibliography O

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F. W. J. Olver (1997a) Asymptotic solutions of linear ordinary differential equations at an irregular singularity of rank unity. Methods Appl. Anal. 4 (4), pp. 375–403.

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External Links:: ISSN 1073-2772, MathReview (A. D. Wood), ZentralBlatt
Referenced by:: §2.7(ii).
Encodings:: BibTeX

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9: Mathematical Introduction

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$\mathbb{C}$	complex plane (excluding infinity).
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empty products	unity.
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10: 22.2 Definitions

… ►with the convention that functions with the same two letters are replaced by unity; e. …