Lagrange inversion theorem

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1: 4.37 Inverse Hyperbolic Functions

§4.37 Inverse Hyperbolic Functions

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Inverse Hyperbolic Sine

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Inverse Hyperbolic Cosine

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Inverse Hyperbolic Tangent

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Other Inverse Functions

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2: 4.23 Inverse Trigonometric Functions

§4.23 Inverse Trigonometric Functions

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Inverse Sine

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Inverse Cosine

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Inverse Tangent

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Other Inverse Functions

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3: 22.15 Inverse Functions

§22.15 Inverse Functions

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§22.15(i) Definitions

… ►Each of these inverse functions is multivalued. The principal values satisfy … ►

4: 3.3 Interpolation

§3.3 Interpolation

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§3.3(i) Lagrange Interpolation

… ► ►With an error term the Lagrange interpolation formula for $f$ is given by … ►

§3.3(ii) Lagrange Interpolation with Equally-Spaced Nodes

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5: 1.10 Functions of a Complex Variable

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Picard’s Theorem

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§1.10(iv) Residue Theorem

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Rouché’s Theorem

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Lagrange Inversion Theorem

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Extended Inversion Theorem

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6: 18.40 Methods of Computation

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§18.40(ii) The Classical Moment Problem

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Stieltjes Inversion via (approximate) Analytic Continuation

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Histogram Approach

… ►In what follows this is accomplished in two ways: i) via the Lagrange interpolation of §3.3(i) ; and ii) by constructing a pointwise continued fraction, or PWCF, as follows: … ►Comparisons of the precisions of Lagrange and PWCF interpolations to obtain the derivatives, are shown in Figure 18.40.2. …

7: 2.2 Transcendental Equations

… ►where $F_0=f_0$ and $s{F}_s$ ($s\ge 1$) is the coefficient of $x^{-1}$ in the asymptotic expansion of ${(f(x))}^s$ (Lagrange’s formula for the reversion of series). …

8: 28.27 Addition Theorems

§28.27 Addition Theorems

►Addition theorems provide important connections between Mathieu functions with different parameters and in different coordinate systems. They are analogous to the addition theorems for Bessel functions (§10.23(ii)) and modified Bessel functions (§10.44(ii)). …

9: Bibliography D

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Note on the addition theorem of parabolic cylinder functions. J. Indian Math. Soc. (N. S.) 4, pp. 29–30.

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External Links:

MathReview (M. C. Gray), ZentralBlatt

Referenced by:

§12.13(ii).

Encodings:

BibTeX

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Computation of the incomplete gamma function ratios and their inverses. ACM Trans. Math. Software 12 (4), pp. 377–393.

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External Links:

ISSN 0098-3500, Document, ZentralBlatt

Referenced by:

§8.12, §8.25(iii).

Encodings:

BibTeX

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Ramanujan’s master theorem for symmetric cones. Pacific J. Math. 175 (2), pp. 447–490.

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External Links:

ISSN 0030-8730, MathReview (Jacques Faraut), ZentralBlatt

Referenced by:

§35.7(ii), §35.8(iv), §35.8(v).

Encodings:

BibTeX

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On Vandermonde’s theorem, and some more general expansions. Proc. Edinburgh Math. Soc. 25, pp. 114–132.

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External Links:

ZentralBlatt

Referenced by:

§15.4(ii).

Encodings:

BibTeX

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Oscillatory integrals, Lagrange immersions and unfolding of singularities. Comm. Pure Appl. Math. 27, pp. 207–281.

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External Links:

MathReview (Alan Weinstein), ZentralBlatt

Referenced by:

§36.12(iii).

Encodings:

BibTeX

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10: 4.38 Inverse Hyperbolic Functions: Further Properties

§4.38 Inverse Hyperbolic Functions: Further Properties

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§4.38(i) Power Series

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§4.38(ii) Derivatives

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§4.38(iii) Addition Formulas

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4.38.19 $$\mathrm{Arctanh}u\pm \mathrm{Arccoth}v=\mathrm{Arctanh}\left(\frac{uv\pm 1}{v\pm u}\right)=\mathrm{Arccoth}\left(\frac{v\pm u}{uv\pm 1}\right).$$

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Symbols:

$\mathrm{Arccoth}z$: general inverse hyperbolic cotangent function and $\mathrm{Arctanh}z$: general inverse hyperbolic tangent function

A&S Ref:

4.6.30

Permalink:

http://dlmf.nist.gov/4.38.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §4.38(iii), §4.38 and Ch.4

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