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11: 4.38 Inverse Hyperbolic Functions: Further Properties
§4.38(iii) Addition Formulas
12: 4.37 Inverse Hyperbolic Functions
§4.37(iii) Reflection Formulas
13: Errata
  • Changes


    • In Paragraph Inversion Formula in §35.2, the wording was changed to make the integration variable more apparent.

    • In many cases, the links from mathematical symbols to their definitions were corrected or improved. These links were also enhanced with ‘tooltip’ feedback, where supported by the user’s browser.

  • Other Changes


    • Equations (4.45.8) and (4.45.9) have been replaced with equations that are better for numerically computing arctan x .

    • A new Subsection 13.29(v) Continued Fractions, has been added to cover computation of confluent hypergeometric functions by continued fractions.

    • A new Subsection 14.5(vi) Addendum to §14.5(ii) μ = 0 , ν = 2 , containing the values of Legendre and Ferrers functions for degree ν = 2 has been added.

    • Subsection 14.18(iii) has been altered to identify Equations (14.18.6) and (14.18.7) as Christoffel’s Formulas.

    • A new Subsection 15.19(v) Continued Fractions, has been added to cover computation of the Gauss hypergeometric functions by continued fractions.

    • Special cases of normalization of Jacobi polynomials for which the general formula is undefined have been stated explicitly in Table 18.3.1.

    • Cross-references have been added in §§1.2(i), 10.19(iii), 10.23(ii), 17.2(iii), 18.15(iii), 19.2(iv), 19.16(i).

    • Several small revisions have been made. For details see §§5.11(ii), 10.12, 10.19(ii), 18.9(i), 18.16(iv), 19.7(ii), 22.2, 32.11(v), 32.13(ii).

    • Entries for the Sage computational system have been updated in the Software Index.

    • The default document format for DLMF is now HTML5 which includes MathML providing better accessibility and display of mathematics.

    • All interactive 3D graphics on the DLMF website have been recast using WebGL and X3DOM, improving portability and performance; WebGL it is now the default format.

  • 14: 4.23 Inverse Trigonometric Functions
    §4.23(iii) Reflection Formulas
    15: 13.10 Integrals
    Other formulas of this kind can be constructed by inversion of the differentiation formulas given in §13.3(ii). …
    16: 19.2 Definitions
    Formulas involving Π ( ϕ , α 2 , k ) that are customarily different for circular cases, ordinary hyperbolic cases, and (hyperbolic) Cauchy principal values, are united in a single formula by using R C ( x , y ) . In (19.2.18)–(19.2.22) the inverse trigonometric and hyperbolic functions assume their principal values (§§4.23(ii) and 4.37(ii)). When x and y are positive, R C ( x , y ) is an inverse circular function if x < y and an inverse hyperbolic function (or logarithm) if x > y :
    19.2.18 R C ( x , y ) = 1 y - x arctan y - x x = 1 y - x arccos x / y , 0 x < y ,
    19.2.19 R C ( x , y ) = 1 x - y arctanh x - y x = 1 x - y ln x + x - y y , 0 < y < x .
    17: Bibliography K
  • A. V. Kitaev and A. H. Vartanian (2004) Connection formulae for asymptotics of solutions of the degenerate third Painlevé equation. I. Inverse Problems 20 (4), pp. 1165–1206.
  • 18: 32.11 Asymptotic Approximations for Real Variables
    Connection formulas for d and θ 0 are given by … The connection formulas for k are … The connection formulas for σ , ρ , and θ are … The connection formulas relating (32.11.25) and (32.11.26) are … Connection formulas for d and θ 0 are given by …
    19: 6.14 Integrals
    6.14.3 0 e - a t si ( t ) d t = - 1 a arctan a , a > 0 .
    20: 6.18 Methods of Computation
    For large x and | z | , expansions in inverse factorial series (§6.10(i)) or asymptotic expansions (§6.12) are available. …Also, other ranges of ph z can be covered by use of the continuation formulas of §6.4. … For example, the Gauss-Laguerre formula3.5(v)) can be applied to (6.2.2); see Todd (1954) and Tseng and Lee (1998). For an application of the Gauss-Legendre formula3.5(v)) see Tooper and Mark (1968). …