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symbolic operations

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1: 24.4 Basic Properties
§24.4(viii) Symbolic Operations
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.

  • Math Symbols
    4: 17.6 ϕ 1 2 Function
    17.6.26 𝒟 q n ( ( z ; q ) ( a b z / c ; q ) ϕ 1 2 ( a , b c ; q , z ) ) = ( c / a , c / b ; q ) n ( c ; q ) n ( 1 q ) n ( a b c ) n ( z q n ; q ) ( a b z / c ; q ) ϕ 1 2 ( a , b c q n ; q , z q n ) .
    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 ) = 𝒟 q n f ( z ) = { z n ( 1 q ) n j = 0 n q n j + ( j + 1 2 ) ( 1 ) j [ n j ] q f ( z q j ) , z 0 , f ( n ) ( 0 ) ( q ; q ) n n ! ( 1 q ) n , z = 0 .
    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
    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 [Uncaptioned image] indicating missing symbols, and thus insufficient 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 3 n - 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 𝐀 P is now being used in several places instead of 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

  • Linking

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

  • Usability

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

  • 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 α was more precisely identified as the Riemann-Liouville fractional integral operator of order α , 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).