sum rules

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1: 34.7 Basic Properties: $\mathit{9}j$ Symbol

… ►This equation is the sum rule. It constitutes an addition theorem for the $\mathit{9}j$ symbol. …

2: 34.5 Basic Properties: $\mathit{6}j$ Symbol

… ►Equations (34.5.15) and (34.5.16) are the sum rules. They constitute addition theorems for the $\mathit{6}j$ symbol. …

3: Bibliography R

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Finite-sum rules for Macdonald’s functions and Hankel’s symbols. Integral Transform. Spec. Funct. 10 (2), pp. 115–124.

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External Links:

ISSN 1065-2469, Document, MathReview, ZentralBlatt

Referenced by:

§10.60(iii).

Encodings:

BibTeX

…

4: 1.2 Elementary Algebra

… ►This is the row times column rule. … ►If $det(𝐀)=0$ then, depending on $𝐜$, there is either no solution or there are infinitely many solutions, being the sum of a particular solution of (1.2.61) and any solution of $𝐀𝐛=𝟎$. … ►Nonzero vectors ${𝐯}_1,\mathrm{\dots },{𝐯}_n$ are linearly independent if ${\sum }_{i=1}^n{c}_i{𝐯}_i=𝟎$ implies that all coefficients $c_i$ are zero. …The sum of all multiplicities is $n$. … ►Thus $\mathrm{tr}(𝐀)$ is the sum of the (counted according to their multiplicities) eigenvalues of $𝐀$. …

5: 29.20 Methods of Computation

… ►A second approach is to solve the continued-fraction equations typified by (29.3.10) by Newton’s rule or other iterative methods; see §3.8. …The Fourier series may be summed using Clenshaw’s algorithm; see §3.11(ii). …

6: 3.5 Quadrature

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§3.5(i) Trapezoidal Rules

… ►The composite trapezoidal rule is … ►

§3.5(ii) Simpson’s Rule

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§3.5(iv) Interpolatory Quadrature Rules

… ► …

7: 17.2 Calculus

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17.2.36 $$\sum _{j=0}^n\left(\genfrac{}{}{0.0pt}{}{n}{j}\right){(-z)}^j={(1-z)}^n.$$

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Symbols:

$\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $j$: nonnegative integer, $n$: nonnegative integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/17.2.E36

Encodings:

pMML, png, TeX

See also:

Annotations for §17.2(iii), §17.2 and Ch.17

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Product Rule

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Leibniz Rule

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17.2.45 $${\int }_0^{1}f(x)d_{q}x=(1-q)\sum _{j=0}^{\mathrm{\infty }}f(q^j)q^j,$$

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Symbols:

$\int$: integral, $d_{q}x$: $q$-differential, $q$: complex base, $j$: nonnegative integer and $x$: real variable

Permalink:

http://dlmf.nist.gov/17.2.E45

Encodings:

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See also:

Annotations for §17.2(v), §17.2 and Ch.17

… ►provided that ${\sum }_{j=-\mathrm{\infty }}^{\mathrm{\infty }}f(q^j)q^j$ converges. …

8: 1.4 Calculus of One Variable

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Chain Rule

… ►where the sum is over all nonnegative integers $m_1,m_2,\mathrm{\dots },m_n$ that satisfy $m_1+2m_2+\mathrm{\cdots }+n{m}_n=n$, and $k=m_1+m_2+\mathrm{\cdots }+m_n$. ►

L’Hôpital’s Rule

… ►Definite integrals over the Stieltjes measure $d\alpha (x)$ could represent a sum, an integral, or a combination of the two. Let $d\alpha (x)=w(x)dx+{\sum }_{n=1}^N{w}_n\delta \left(x-x_n\right)dx$, $x_n\in (a,b)$, $n=1,\mathrm{\dots }N$. …

9: 1.6 Vectors and Vector-Valued Functions

… ►where $𝐧$ is the unit vector normal to $𝐚$ and $𝐛$ whose direction is determined by the right-hand rule; see Figure 1.6.1. ►

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Figure 1.6.1: Vector notation. Right-hand rule for cross products. Magnify

… ►In almost all cases of repeated suffices, we can suppress the summation notation entirely, if it is understood that an implicit sum is to be taken over any repeated suffix. Thus pairs of indefinite suffices in an expression are resolved by being summed over (or “traced” over). …

10: 3.3 Interpolation

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3.3.1 $$P_n(z)=\sum _{k=0}^n{\mathrm{\ell }}_k(z)f_k={\omega }_{n+1}(z)\sum _{k=0}^n\frac{{\tilde{\omega }}_k}{z-z_k}{f}_k=\sum _{k=0}^n\frac{{\tilde{\omega }}_k{f}_k}{z-z_k}/\sum _{k=0}^n\frac{{\tilde{\omega }}_k}{z-z_k},$$

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Symbols:

$f_t$: function values, $z_k$: nodes of function, ${\mathrm{\ell }}_k(z)$: Lagrange interpolation polynomial and ${\omega }_{n+1}(z)$: nodal polynomial

A&S Ref:

25.2.1

Referenced by:

§3.3(i), Erratum (V1.1.3) for Subsection 3.3(i)

Permalink:

http://dlmf.nist.gov/3.3.E1

Encodings:

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See also:

Annotations for §3.3(i), §3.3 and Ch.3

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3.3.3_2 $$1=\sum _{k=0}^n{\mathrm{\ell }}_k(z)={\omega }_{n+1}(z)\sum _{k=0}^n\frac{{\tilde{\omega }}_k}{z-z_k},$$

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Symbols:

$z_k$: nodes of function, ${\mathrm{\ell }}_k(z)$: Lagrange interpolation polynomial and ${\omega }_{n+1}(z)$: nodal polynomial

Referenced by:

§3.3(i), Erratum (V1.1.3) for Additions

Permalink:

http://dlmf.nist.gov/3.3.E3_2

Encodings:

pMML, png, TeX

Addition (effective with 1.1.3):

This equation was added.

See also:

Annotations for §3.3(i), §3.3 and Ch.3

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3.3.4 $$f(z)=\sum _{k=0}^n{\mathrm{\ell }}_k(z)f_k+R_n(z).$$

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Symbols:

$f(z)$: function, $f_t$: function values, ${\mathrm{\ell }}_k(z)$: Lagrange interpolation polynomial and $R_n(x)$: remainder

A&S Ref:

25.2.1

Referenced by:

§3.3(ii), §3.4(i)

Permalink:

http://dlmf.nist.gov/3.3.E4

Encodings:

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See also:

Annotations for §3.3(i), §3.3 and Ch.3

… ►and compute an approximation to $a_1$ by using Newton’s rule (§3.8(ii)) with starting value $x=-2.5$. …Then by using $x_3$ in Newton’s interpolation formula, evaluating $\left[x_0,x_1,x_2,x_3\right]f=-\mathrm{0.26608\hspace{0.33em}28233}$ and recomputing $f^{\prime }(x)$, another application of Newton’s rule with starting value $x_3$ gives the approximation $x=\mathrm{2.33810\hspace{0.33em}7373}$, with 8 correct digits. …