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11: 4.38 Inverse Hyperbolic Functions: Further Properties

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§4.38(iii) Addition Formulas

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12: 4.37 Inverse Hyperbolic Functions

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§4.37(iii) Reflection Formulas

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13: Errata

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• 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.

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• 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) $\mu \mathrm{=}\mathrm{0}$, $\nu \mathrm{=}\mathrm{2}$ , containing the values of Legendre and Ferrers functions for degree $\nu =2$ 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.

• Bibliographic citations have been added in §§4.13, 5.6(i), 5.11(iii), 7.25(iii), 8.13(i), 10.37, 12.18, 14.11, 15.12(ii), 16.6, 18.16(ii), 18.16(iv), 18.24, 18.27(iv), 18.27(v),18.28(i), 24.13(i), 28.36(iii).

• 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.

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14: 4.23 Inverse Trigonometric Functions

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§4.23(iii) Reflection Formulas

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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 $\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$ 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 and an inverse hyperbolic function (or logarithm) if $x>y$: ►

19.2.18 $$R_C(x,y)=\frac{1}{\sqrt{y-x}}\arctan \sqrt{\frac{y-x}{x}}=\frac{1}{\sqrt{y-x}}\arccos \sqrt{x/y},$$ ,

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Symbols:

$R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $\arccos z$: arccosine function and $\arctan z$: arctangent function

Referenced by:

§19.2(iv)

Permalink:

http://dlmf.nist.gov/19.2.E18

Encodings:

pMML, png, TeX

See also:

Annotations for §19.2(iv), §19.2 and Ch.19

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19.2.19 $$R_C(x,y)=\frac{1}{\sqrt{x-y}}\mathrm{arctanh}\sqrt{\frac{x-y}{x}}=\frac{1}{\sqrt{x-y}}\ln \frac{\sqrt{x}+\sqrt{x-y}}{\sqrt{y}},$$ .

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Symbols:

$R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $\mathrm{arctanh}z$: inverse hyperbolic tangent function and $\ln z$: principal branch of logarithm function

Referenced by:

§19.26(i), Figure 19.3.2, Figure 19.3.2, §19.36(ii)

Permalink:

http://dlmf.nist.gov/19.2.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §19.2(iv), §19.2 and Ch.19

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17: Bibliography K

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Connection formulae for asymptotics of solutions of the degenerate third Painlevé equation. I. Inverse Problems 20 (4), pp. 1165–1206.

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External Links:

ISSN 0266-5611, Document, MathReview (Shun Shimomura), ZentralBlatt

Referenced by:

§32.11(iv).

Encodings:

BibTeX

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18: 32.11 Asymptotic Approximations for Real Variables

… ►Connection formulas for $d$ and ${\theta }_0$ are given by … ►The connection formulas for $k$ are … ►The connection formulas for $\sigma$, $\rho$, and $\theta$ are … ►The connection formulas relating (32.11.25) and (32.11.26) are … ►Connection formulas for $d$ and ${\theta }_0$ are given by …

19: 6.14 Integrals

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6.14.3 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-at}\mathrm{si}\left(t\right)dt=-\frac{1}{a}\arctan a,$$ $\mathrm{\Re }a>0$.

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Symbols:

$dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\arctan z$: arctangent function, $\mathrm{\Re }$: real part and $\mathrm{si}\left(z\right)$: sine integral

A&S Ref:

5.2.29

Permalink:

http://dlmf.nist.gov/6.14.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §6.14(i), §6.14 and Ch.6

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20: 6.18 Methods of Computation

… ►For large $x$ and $\left|z\right|$, expansions in inverse factorial series (§6.10(i)) or asymptotic expansions (§6.12) are available. …Also, other ranges of $\mathrm{ph}z$ can be covered by use of the continuation formulas of §6.4. … ►For example, the Gauss-Laguerre formula (§3.5(v)) can be applied to (6.2.2); see Todd (1954) and Tseng and Lee (1998). For an application of the Gauss-Legendre formula (§3.5(v)) see Tooper and Mark (1968). …