for squares

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21—30 of 87 matching pages

21: 23.7 Quarter Periods

… ►where

k,k^{\prime}

and the square roots are real and positive when the lattice is rectangular; otherwise they are determined by continuity from the rectangular case.

22: 26.18 Counting Techniques

… ►With the notation of §26.15, the number of placements of

n

nonattacking rooks on an

n\times n

chessboard that avoid the squares in a specified subset

B

is …

23: 27.6 Divisor Sums

… ►

27.6.1

\sum_{d\mathbin{|}n}\lambda\left(d\right)=\begin{cases}1,&n\mbox{ is a square}% ,\\ 0,&\mbox{otherwise}.\end{cases}

ⓘ

Symbols:: $\lambda\left(\NVar{n}\right)$ : Liouville’s function, $d$ : positive integer and $n$ : positive integer
Permalink:: http://dlmf.nist.gov/27.6.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §27.6 and Ch.27

…

24: 1.8 Fourier Series

… ►

1.8.5

\frac{1}{\pi}\int^{\pi}_{-\pi}{\left|f(x)\right|}^{2}\,\mathrm{d}x=\tfrac{1}{2% }{\left|a_{0}\right|}^{2}+\sum^{\infty}_{n=1}({\left|a_{n}\right|}^{2}+{\left|% b_{n}\right|}^{2}),

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $n$ : nonnegative integer and $\left|\NVar{x}\right|$ : absolute value of $\NVar{x}$
Referenced by:: Erratum (V1.2.0) for Equations (1.8.5), (1.8.6)
Permalink:: http://dlmf.nist.gov/1.8.E5
Encodings:: pMML, png, TeX
Modification (effective with 1.2.0):: Previously this equation was given as an inequality. For square integrable functions this inequality $\geq$ can be sharpened to $=$ .
See also:: Annotations for §1.8(i), §1.8(i), §1.8 and Ch.1

►

1.8.6

\frac{1}{2\pi}\int^{\pi}_{-\pi}{\left|f(x)\right|}^{2}\,\mathrm{d}x=\sum^{% \infty}_{n=-\infty}{\left|c_{n}\right|}^{2},

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $n$ : nonnegative integer and $\left|\NVar{x}\right|$ : absolute value of $\NVar{x}$
Referenced by:: §1.8(i), Erratum (V1.2.0) for Equations (1.8.5), (1.8.6)
Permalink:: http://dlmf.nist.gov/1.8.E6
Encodings:: pMML, png, TeX
Modification (effective with 1.2.0):: Previously this equation was given as an inequality. For square integrable functions this inequality $\geq$ can be sharpened to $=$ .
See also:: Annotations for §1.8(i), §1.8(i), §1.8 and Ch.1

►where

f(x)

is square-integrable on

[-\pi,\pi]

and

a_{n},b_{n},c_{n}

are given by (1.8.2), (1.8.4). If

g(x)

is also square-integrable with Fourier coefficients

a_{n}^{\prime},b_{n}^{\prime}

c_{n}^{\prime}

then …

25: 10.49 Explicit Formulas

… ►

§10.49(iv) Sums or Differences of Squares

…

26: 10.67 Asymptotic Expansions for Large Argument

… ►

§10.67(ii) Cross-Products and Sums of Squares in the Case $\nu=0$

…

27: 14.28 Sums

… ►where the branches of the square roots have their principal values when

z_{1},z_{2}\in(1,\infty)

and are continuous when

z_{1},z_{2}\in\mathbb{C}\setminus(0,1]

. …

28: 22.6 Elementary Identities

… ►

§22.6(i) Sums of Squares

…

29: 22.17 Moduli Outside the Interval [0,1]

… ►In (22.17.5) either value of the square root can be chosen. …

30: 23.18 Modular Transformations

… ►where the square root has its principal value and …