dot product

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1: 1.6 Vectors and Vector-Valued Functions

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Dot Product (or Scalar Product)

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1.6.2

\mathbf{a}\cdot\mathbf{b}=a_{1}b_{1}+a_{2}b_{2}+a_{3}b_{3}.

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Referenced by:: §1.2(v)
Permalink:: http://dlmf.nist.gov/1.6.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §1.6(i), §1.6(i), §1.6 and Ch.1

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2: 1.2 Elementary Algebra

… ►The dot product notation

\mathbf{u}\cdot\mathbf{v}

is reserved for the physical three-dimensional vectors of (1.6.2). …

3: 21.1 Special Notation

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$g, h$	positive integers.
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$\mathbf{a}\cdot\mathbf{b}$	scalar product of the vectors $\mathbf{a}$ and $\mathbf{b}$ .
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4: 1.3 Determinants, Linear Operators, and Spectral Expansions

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1.3.13

\begin{vmatrix}1&x_{1}&x^{2}_{1}&\cdots&x^{n-1}_{1}\\ 1&x_{2}&x^{2}_{2}&\cdots&x^{n-1}_{2}\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ 1&x_{n}&x^{2}_{n}&\cdots&x_{n}^{n-1}\end{vmatrix}=\prod_{1\leq j<k\leq n}(x_{k% }-x_{j}).

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Symbols:: $\det$ : determinant, $j$ : integer, $k$ : integer and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/1.3.E13
Encodings:: pMML, png, TeX
See also:: Annotations for §1.3(ii), §1.3(ii), §1.3 and Ch.1

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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

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5: 26.4 Lattice Paths: Multinomial Coefficients and Set Partitions

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26.4.2

\genfrac{(}{)}{0.0pt}{}{n_{1}+n_{2}+\cdots+n_{k}}{n_{1},n_{2},\ldots,n_{k}}=% \frac{(n_{1}+n_{2}+\cdots+n_{k})!}{n_{1}!\,n_{2}!\,\cdots\,n_{k}!}=\prod_{j=1}% ^{k-1}\genfrac{(}{)}{0.0pt}{}{n_{j}+n_{j+1}+\cdots+n_{k}}{n_{j}}.

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Symbols:: $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $!$ : factorial (as in $n!$ ), $\genfrac{(}{)}{0.0pt}{}{\NVar{n_{1}}+\NVar{n_{2}}+\dots+\NVar{n_{k}}}{\NVar{n_% {1}},\NVar{n_{2}},\ldots,\NVar{n_{k}}}$ : multinomial coefficient, $j$ : nonnegative integer, $k$ : nonnegative integer and $n$ : nonnegative integer
A&S Ref:: 24.1.2
Referenced by:: §26.4(i)
Permalink:: http://dlmf.nist.gov/26.4.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §26.4(i), §26.4 and Ch.26

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6: 21.6 Products

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21.6.4

\prod_{j=1}^{h}\theta\genfrac{[}{]}{0.0pt}{}{\sum_{k=1}^{h}T_{jk}\mathbf{c}_{k% }}{\sum_{k=1}^{h}T_{jk}\mathbf{d}_{k}}\left(\sum_{k=1}^{h}T_{jk}\mathbf{z}_{k}% \middle|\boldsymbol{{\Omega}}\right)=\frac{1}{\mathcal{D}^{g}}\sum_{\mathbf{A}% \in\mathcal{K}}\sum_{\mathbf{B}\in\mathcal{K}}e^{-2\pi i\sum_{j=1}^{h}\mathbf{% b}_{j}\cdot\mathbf{c}_{j}}\prod_{j=1}^{h}\theta\genfrac{[}{]}{0.0pt}{}{\mathbf% {a}_{j}+\mathbf{c}_{j}}{\mathbf{b}_{j}+\mathbf{d}_{j}}\left(\mathbf{z}_{j}% \middle|\boldsymbol{{\Omega}}\right),

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Symbols:: $\theta\genfrac{[}{]}{0.0pt}{}{\NVar{\boldsymbol{{\alpha}}}}{\NVar{\boldsymbol{% {\beta}}}}\left(\NVar{\mathbf{z}}\middle|\NVar{\boldsymbol{{\Omega}}}\right)$ : Riemann theta function with characteristics, $\pi$ : the ratio of the circumference of a circle to its diameter, $\in$ : element of, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $g$ : positive integer, $h$ : positive integer, $\boldsymbol{{\Omega}}$ : a Riemann matrix, $T_{jk}$ : element, $\mathcal{K}$ : set of matrices and $\mathcal{D}$ : number of elements
Permalink:: http://dlmf.nist.gov/21.6.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §21.6(i), §21.6 and Ch.21

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7: 17.14 Constant Term Identities

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17.14.1

\frac{\left(q;q\right)_{a_{1}+a_{2}+\cdots+a_{n}}}{\left(q;q\right)_{a_{1}}% \left(q;q\right)_{a_{2}}\cdots\left(q;q\right)_{a_{n}}}=\mbox{ coeff. of }x_{1% }^{0}x_{2}^{0}\cdots x_{n}^{0}\mbox{ in }\prod_{1\leq j<k\leq n}\left(\frac{x_% {j}}{x_{k}};q\right)_{a_{j}}\left(\frac{qx_{k}}{x_{j}};q\right)_{a_{k}}.

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Symbols:: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), $k$ : nonnegative integer, $q$ : complex base, $j$ : nonnegative integer, $n$ : nonnegative integer and $x$ : real variable
Referenced by:: §17.14
Permalink:: http://dlmf.nist.gov/17.14.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §17.14, §17.14 and Ch.17

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8: 17.2 Calculus

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17.2.49

1+\sum_{n=1}^{\infty}\frac{q^{n^{2}}}{(1-q)(1-q^{2})\cdots(1-q^{n})}=\prod_{n=% 0}^{\infty}\frac{1}{(1-q^{5n+1})(1-q^{5n+4})},

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Symbols:: $q$ : complex base and $n$ : nonnegative integer
Referenced by:: §17.14
Permalink:: http://dlmf.nist.gov/17.2.E49
Encodings:: pMML, png, TeX
See also:: Annotations for §17.2(vi), §17.2 and Ch.17

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17.2.50

1+\sum_{n=1}^{\infty}\frac{q^{n^{2}+n}}{(1-q)(1-q^{2})\cdots(1-q^{n})}=\prod_{% n=0}^{\infty}\frac{1}{(1-q^{5n+2})(1-q^{5n+3})}.

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Symbols:: $q$ : complex base and $n$ : nonnegative integer
Referenced by:: §17.14
Permalink:: http://dlmf.nist.gov/17.2.E50
Encodings:: pMML, png, TeX
See also:: Annotations for §17.2(vi), §17.2 and Ch.17

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9: 26.13 Permutations: Cycle Notation

… ►A permutation with cycle type

{\left(a_{1},a_{2},\ldots,a_{n}\right)}

can be written as a product of

a_{2}+2a_{3}+\cdots+(n-1)a_{n}=n-(a_{1}+a_{2}+\cdots+a_{n})

transpositions, and no fewer. …

10: 5.17 Barnes’ $G$ -Function (Double Gamma Function)

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5.17.2

G\left(n\right)=(n-2)!(n-3)!\cdots 1!,

n=2,3,\dots

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Symbols:: $G\left(\NVar{z}\right)$ : Barnes’ $G$ -function (or double gamma function), $!$ : factorial (as in $n!$ ) and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/5.17.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §5.17 and Ch.5

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