inversion formula
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11: Errata
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Paragraph Inversion Formula (in §35.2)
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The wording was changed to make the integration variable more apparent.
12: 3.5 Quadrature
13: 4.37 Inverse Hyperbolic Functions
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§4.37(iii) Reflection Formulas
…14: 4.23 Inverse Trigonometric Functions
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§4.23(iii) Reflection Formulas
…15: 13.10 Integrals
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►Other formulas of this kind can be constructed by inversion of the differentiation formulas given in §13.3(ii).
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16: 19.2 Definitions
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►Formulas involving that are customarily different for circular cases, ordinary hyperbolic cases, and (hyperbolic) Cauchy principal values, are united in a single formula by using .
►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 and are positive, is an inverse circular function if and an inverse hyperbolic function (or logarithm) if :
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19.2.18
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19.2.19
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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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18: 32.11 Asymptotic Approximations for Real Variables
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►Connection formulas for and are given by
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►The connection formulas for are
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►The connection formulas for , , and are
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►The connection formulas relating (32.11.25) and (32.11.26) are
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►Connection formulas for and are given by
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19: 6.14 Integrals
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6.14.3
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20: 6.18 Methods of Computation
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►For large and , expansions in inverse factorial series (§6.10(i)) or asymptotic expansions (§6.12) are available.
…Also, other ranges of can be covered by use of the continuation formulas of §6.4.
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►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).
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