# symbolic operations

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## 1—10 of 33 matching pages

##### 2: 34.9 Graphical Method
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.

• Boolean operator:

and and or

• proximity operator:

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

##### 4: 17.6 ${{}_{2}\phi_{1}}$ Function
17.6.25 $\mathcal{D}_{q}^{n}{{}_{2}\phi_{1}}\left({a,b\atop c};q,zd\right)=\frac{\left(% a,b;q\right)_{n}d^{n}}{\left(c;q\right)_{n}(1-q)^{n}}{{}_{2}\phi_{1}}\left({aq% ^{n},bq^{n}\atop cq^{n}};q,dz\right),$
17.6.26 $\mathcal{D}_{q}^{n}\left(\frac{\left(z;q\right)_{\infty}}{\left(abz/c;q\right)% _{\infty}}{{}_{2}\phi_{1}}\left({a,b\atop c};q,z\right)\right)=\frac{\left(c/a% ,c/b;q\right)_{n}}{\left(c;q\right)_{n}(1-q)^{n}}\left(\frac{ab}{c}\right)^{n}% \frac{\left(zq^{n};q\right)_{\infty}}{\left(abz/c;q\right)_{\infty}}{{}_{2}% \phi_{1}}\left({a,b\atop cq^{n}};q,zq^{n}\right).$
##### 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
17.2.42 $f^{[n]}(z)=\mathcal{D}_{q}^{n}f(z)=\begin{cases}z^{-n}(1-q)^{-n}\sum_{j=0}^{n}% q^{-nj+\genfrac{(}{)}{0.0pt}{}{j+1}{2}}(-1)^{j}\genfrac{[}{]}{0.0pt}{}{n}{j}_{% q}f(zq^{j}),&z\neq 0,\\ \dfrac{f^{(n)}(0)\left(q;q\right)_{n}}{n!(1-q)^{n}},&z=0.\end{cases}$
##### 7: 26.22 Software
Also included are websites operated by university departments and consortia, research institutions, and peer-reviewed journals. …
• Inverse Symbolic Calculator (website).

• ##### 8: About MathML
Since the display of mathematics involves many unusual and special symbols, a MathML renderer generally needs a set of special fonts. … As a general rule, using the latest available version of your chosen browser, plugins and an updated operating system is helpful. Check the websites for your browser, plugin and operating system for more information about installing fonts. …
##### 9: Bibliography R
• J. Raynal (1979) On the definition and properties of generalized $6$-$j$ symbols. J. Math. Phys. 20 (12), pp. 2398–2415.
• REDUCE (free interactive system)
• 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.
• 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.
• 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.
• ##### 10: Errata
• 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.

• 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

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