symbolic operations

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1—10 of 33 matching pages

… ►The graphical method establishes a one-to-one correspondence between an analytic expression and a diagram by assigning a graphical symbol to each function and operation of the analytic expression. …

3: Guide to Searching the DLMF

… ►Search queries are made up of terms, textual phrases, and math expressions, combined with Boolean operators: ►

term:

a textual word, a number, or a math symbol.

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Boolean operator:

and and or

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proximity operator:

adj, prec/n, and near/n, where n is any positive natural number.

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Math Symbols

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4: 17.6 ${{}_{2}\phi_{1}}$ Function

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17.6.25

\mathcal{D}_{q}^{n}{{}_{2}\phi_{1}}\left({a,b\atop c};q,zd\right)=\frac{\left(% a,b;q\right)_{n}d^{n}}{\left(c;q\right)_{n}(1-q)^{n}}{{}_{2}\phi_{1}}\left({aq% ^{n},bq^{n}\atop cq^{n}};q,dz\right),

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Symbols:: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left({\NVar{a_{0},\dots,a_{r}}\atop\NVar{b_{1% },\dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : basic hypergeometric (or $q$ -hypergeometric) function, $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple $q$ -Pochhammer symbol, $q$ : complex base, $n$ : nonnegative integer, $z$ : complex variable and $\mathcal{D}_{q}$ : $q$ -differential operator
Permalink:: http://dlmf.nist.gov/17.6.E25
Encodings:: pMML, png, TeX
See also:: Annotations for §17.6(iv), §17.6(iv), §17.6 and Ch.17

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17.6.26

\mathcal{D}_{q}^{n}\left(\frac{\left(z;q\right)_{\infty}}{\left(abz/c;q\right)% _{\infty}}{{}_{2}\phi_{1}}\left({a,b\atop c};q,z\right)\right)=\frac{\left(c/a% ,c/b;q\right)_{n}}{\left(c;q\right)_{n}(1-q)^{n}}\left(\frac{ab}{c}\right)^{n}% \frac{\left(zq^{n};q\right)_{\infty}}{\left(abz/c;q\right)_{\infty}}{{}_{2}% \phi_{1}}\left({a,b\atop cq^{n}};q,zq^{n}\right).

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Symbols:: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left({\NVar{a_{0},\dots,a_{r}}\atop\NVar{b_{1% },\dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : basic hypergeometric (or $q$ -hypergeometric) function, $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple $q$ -Pochhammer symbol, $q$ : complex base, $n$ : nonnegative integer, $z$ : complex variable and $\mathcal{D}_{q}$ : $q$ -differential operator
Permalink:: http://dlmf.nist.gov/17.6.E26
Encodings:: pMML, png, TeX
See also:: Annotations for §17.6(iv), §17.6(iv), §17.6 and Ch.17

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5: 18.42 Software

… ►Also included is a website (CAOP) operated by a university department. … ►A more complete list of available software for computing these functions, and for generating formulas symbolically, is found in the Software Index. …

6: 17.2 Calculus

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17.2.42

f^{[n]}(z)=\mathcal{D}_{q}^{n}f(z)=\begin{cases}z^{-n}(1-q)^{-n}\sum_{j=0}^{n}% q^{-nj+\genfrac{(}{)}{0.0pt}{}{j+1}{2}}(-1)^{j}\genfrac{[}{]}{0.0pt}{}{n}{j}_{% q}f(zq^{j}),&z\neq 0,\\ \dfrac{f^{(n)}(0)\left(q;q\right)_{n}}{n!(1-q)^{n}},&z=0.\end{cases}

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Symbols:: $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $!$ : factorial (as in $n!$ ), $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), $\genfrac{[}{]}{0.0pt}{}{\NVar{n}}{\NVar{m}}_{\NVar{q}}$ : $q$ -binomial coefficient (or Gaussian polynomial), $q$ : complex base, $j$ : nonnegative integer, $n$ : nonnegative integer, $z$ : complex variable and $\mathcal{D}_{q}$ : $q$ -differential operator
Referenced by:: §17.2(iv)
Permalink:: http://dlmf.nist.gov/17.2.E42
Encodings:: pMML, png, TeX
See also:: Annotations for §17.2(iv), §17.2 and Ch.17

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7: 26.22 Software

… ►Also included are websites operated by university departments and consortia, research institutions, and peer-reviewed journals. … ►

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Inverse Symbolic Calculator (website).

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8: About MathML

… ►As a general rule, using the latest available version of your chosen browser, plugins and an updated operating system is helpful. … ►Since the display of mathematics involves many special symbols not often seen in plain text, a MathML renderer generally needs special fonts. … ►For other browsers, you may see a ? or a box like

indicating missing symbols, and thus insufficient fonts. …

9: Bibliography R

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J. Raynal (1979) On the definition and properties of generalized

6

j

symbols. J. Math. Phys. 20 (12), pp. 2398–2415.

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External Links:: ISSN 0022-2488, Document, MathReview (S. Saran), ZentralBlatt
Referenced by:: §16.4(iii), §16.4(iii).
Encodings:: BibTeX

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REDUCE (free interactive system)

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Notes:: Symbolic and extended-precision general purpose computer algebra system.
External Links:: Home Page
Referenced by:: § ‣ Software Cross Index.
Encodings:: BibTeX

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S. Ritter (1998) On the computation of Lamé functions, of eigenvalues and eigenfunctions of some potential operators. Z. Angew. Math. Mech. 78 (1), pp. 66–72.

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External Links:: ISSN 0044-2267, Document, MathReview Entry, ZentralBlatt
Referenced by:: §29.20(ii).
Encodings:: BibTeX

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C. C. J. Roothaan and S. Lai (1997) Calculation of

3n

j

symbols by Labarthe’s method. International Journal of Quantum Chemistry 63 (1), pp. 57–64.

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External Links:: Document
Referenced by:: §34.13.
Encodings:: BibTeX

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J. Rushchitsky and S. Rushchitska (2000) On Simple Waves with Profiles in the form of some Special Functions—Chebyshev-Hermite, Mathieu, Whittaker—in Two-phase Media. In Differential Operators and Related Topics, Vol. I (Odessa, 1997), Operator Theory: Advances and Applications, Vol. 117, pp. 313–322.

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External Links:: ISBN 3-7643-6287-1, MathReview Entry, ZentralBlatt
Referenced by:: 5th item.
Encodings:: BibTeX

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10: Errata

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Notation

In §3.7(iii), the symbol $\mathbf{A}_{P}$ is now being used in several places instead of $\mathbf{A}$ in order to disambiguate symbols.

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Equation (3.3.34)

In the online version, the leading divided difference operators were previously omitted from these formulas, due to programming error.

Reported by Nico Temme on 2021-06-01

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Linking

Pochhammer and $q$ -Pochhammer symbols in several equations now correctly link to their definitions.

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Usability

Linkage of mathematical symbols to their definitions were corrected or improved.

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Subsections 1.15(vi), 1.15(vii), 2.6(iii)

A number of changes were made with regard to fractional integrals and derivatives. In §1.15(vi) a reference to Miller and Ross (1993) was added, the fractional integral operator of order $\alpha$ was more precisely identified as the Riemann-Liouville fractional integral operator of order $\alpha$ , and a paragraph was added below (1.15.50) to generalize (1.15.47). In §1.15(vii) the sentence defining the fractional derivative was clarified. In §2.6(iii) the identification of the Riemann-Liouville fractional integral operator was made consistent with §1.15(vi).

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symbolic operations

1—10 of 33 matching pages

1: 24.4 Basic Properties

§24.4(viii) Symbolic Operations

2: 34.9 Graphical Method

3: Guide to Searching the DLMF

Math Symbols

4: 17.6 ${{}_{2}\phi_{1}}$ Function

5: 18.42 Software

6: 17.2 Calculus

7: 26.22 Software

8: About MathML

9: Bibliography R

10: Errata

symbolic operations

1—10 of 33 matching pages

1: 24.4 Basic Properties

§24.4(viii) Symbolic Operations

2: 34.9 Graphical Method

3: Guide to Searching the DLMF

Math Symbols

4: 17.6 ϕ 1 2 Function

5: 18.42 Software

6: 17.2 Calculus

7: 26.22 Software

8: About MathML

9: Bibliography R

10: Errata

4: 17.6 ${{}_{2}\phi_{1}}$ Function