Lagrange inversion theorem

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31: 19.9 Inequalities

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19.9.11 $$\varphi \le F(\varphi ,k)\le \min (\varphi /\mathrm{\Delta },{\mathrm{gd}}^{-1}\left(\varphi \right)),$$

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Symbols:

$F(\varphi ,k)$: Legendre’s incomplete elliptic integral of the first kind, ${\mathrm{gd}}^{-1}\left(x\right)$: inverse Gudermannian function, $\varphi$: real or complex argument, $k$: real or complex modulus and $\mathrm{\Delta }$

Permalink:

http://dlmf.nist.gov/19.9.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §19.9(ii), §19.9 and Ch.19

►where ${\mathrm{gd}}^{-1}\left(\varphi \right)$ is given by (4.23.41) and (4.23.42). … ►Inequalities for both $F(\varphi ,k)$ and $E(\varphi ,k)$ involving inverse circular or inverse hyperbolic functions are given in Carlson (1961b, §4). … ►

$L=(1/\sigma )\mathrm{arctanh}\left(\sigma \sin \varphi \right),$ $\sigma =\sqrt{(1+k^2)/2}$,

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$U=\frac{1}{2}\mathrm{arctanh}\left(\sin \varphi \right)+\frac{1}{2}{k}^{-1}\mathrm{arctanh}\left(k\sin \varphi \right).$

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32: 18.18 Sums

… ►See Szegő (1975, Theorems 3.1.5 and 5.7.1). … ►

§18.18(ii) Addition Theorems

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Ultraspherical

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Legendre

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§18.18(iii) Multiplication Theorems

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33: 28.2 Definitions and Basic Properties

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§28.2(iii) Floquet’s Theorem and the Characteristic Exponents

… ►The solutions of (28.2.16) are given by $\nu ={\pi }^{-1}\arccos \left(w_{\text{I}}(\pi ;a,q)\right)$. If the inverse cosine takes its principal value (§4.23(ii)), then $\nu =\widehat{\nu }$, where $0\le \mathrm{\Re }\widehat{\nu }\le 1$. … ►If $q\ne 0$, then for a given value of $\nu$ the corresponding Floquet solution is unique, except for an arbitrary constant factor (Theorem of Ince; see also 28.5(i)). …

34: 4.24 Inverse Trigonometric Functions: Further Properties

§4.24 Inverse Trigonometric Functions: Further Properties

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§4.24(i) Power Series

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§4.24(ii) Derivatives

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§4.24(iii) Addition Formulas

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4.24.17 $$\mathrm{Arctan}u\pm \mathrm{Arccot}v=\mathrm{Arctan}\left(\frac{uv\pm 1}{v\mp u}\right)=\mathrm{Arccot}\left(\frac{v\mp u}{uv\pm 1}\right).$$

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Symbols:

$\mathrm{Arccot}z$: general arccotangent function and $\mathrm{Arctan}z$: general arctangent function

A&S Ref:

4.4.36

Permalink:

http://dlmf.nist.gov/4.24.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §4.24(iii), §4.24 and Ch.4

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35: 18.33 Polynomials Orthogonal on the Unit Circle

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18.33.23 $${\mathrm{\Phi }}_{n+1}(z)=z{\mathrm{\Phi }}_n(z)-\overline{{\alpha }_n}{\mathrm{\Phi }}_n^{\ast }(z),$$

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Symbols:

$\overline{z}$: complex conjugate, $z$: complex variable and $n$: nonnegative integer

Proved:

Simon (2005a, Theorem 1.5.2)(proved)

Referenced by:

§18.33(vi), §18.33(vi), §18.33(vi), §18.33(vi)

Permalink:

http://dlmf.nist.gov/18.33.E23

Encodings:

pMML, png, TeX

See also:

Annotations for §18.33(vi), §18.33(vi), §18.33 and Ch.18

… ►Equivalent to the recurrence relations (18.33.23), (18.33.24) are the inverse Szegő recurrence relations … ►

Verblunsky’s Theorem

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Szegő’s Theorem

►For $w(z)$ as in (18.33.19) (or more generally as the weight function of the absolutely continuous part of the measure $\mu$ in (18.33.17)) and with ${\alpha }_n$ the Verblunsky coefficients in (18.33.23), (18.33.24), Szegő’s theorem states that …

36: 7.17 Inverse Error Functions

§7.17 Inverse Error Functions

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§7.17(i) Notation

►The inverses of the functions $x=\mathrm{erf}y$, $x=\mathrm{erfc}y$, $y\in ℝ$, are denoted by … ►

§7.17(ii) Power Series

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§7.17(iii) Asymptotic Expansion of $\mathrm{inverfc}x$ for Small $x$

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37: 1.9 Calculus of a Complex Variable

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DeMoivre’s Theorem

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Jordan Curve Theorem

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Cauchy’s Theorem

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Liouville’s Theorem

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Dominated Convergence Theorem

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38: 4.15 Graphics

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§4.15(i) Real Arguments

… ►Figure 4.15.7 illustrates the conformal mapping of the strip onto the whole $w$-plane cut along the real axis from $-\mathrm{\infty }$ to $-1$ and $1$ to $\mathrm{\infty }$, where $w=\sin z$ and $z=\arcsin w$ (principal value). … ►

► … ►

Figure 4.15.7: Conformal mapping of sine and inverse sine. … Magnify

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§4.15(iii) Complex Arguments: Surfaces

… ►The corresponding surfaces for $\arccos \left(x+\mathrm{i}y\right)$, $\mathrm{arccot}\left(x+\mathrm{i}y\right)$, $\mathrm{arcsec}\left(x+\mathrm{i}y\right)$ can be visualized from Figures 4.15.9, 4.15.11, 4.15.13 with the aid of equations (4.23.16)–(4.23.18).

39: 18.15 Asymptotic Approximations

… ►The first term of this expansion also appears in Szegő (1975, Theorem 8.21.7). … ►For a bound on the error term in (18.15.10) see Szegő (1975, Theorem 8.21.11). … ►Another expansion follows from (18.15.10) by taking $\lambda =\frac{1}{2}$; see Szegő (1975, Theorem 8.21.5). … ►

18.15.18 $$\xi =\frac{1}{2}\left(\sqrt{x-x^2}+\arcsin (\sqrt{x})\right),$$ $0\le x\le 1$.

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Defines:

$\xi$ (locally)

Symbols:

$\arcsin z$: arcsine function and $x$: real variable

Permalink:

http://dlmf.nist.gov/18.15.E18

Encodings:

pMML, png, TeX

See also:

Annotations for §18.15(iv), §18.15(iv), §18.15 and Ch.18

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$$\zeta =-{\left(\frac{3}{4}\left(\arccos \left(\sqrt{x}\right)-\sqrt{x-x^2}\right)\right)}^{\frac{2}{3}},$$ $0\le x\le 1$,

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40: Bibliography C

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The Staudt-Clausen theorem. Math. Mag. 34, pp. 131–146.

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External Links:

MathReview (K. Goldberg), ZentralBlatt

Referenced by:

§24.6.

Encodings:

BibTeX

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New proof of the addition theorem for Gegenbauer polynomials. SIAM J. Math. Anal. 2, pp. 347–351.

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External Links:

ISSN 1095-7154, Document, MathReview (Mary L. Boas), ZentralBlatt

Referenced by:

§18.18(ii).

Encodings:

BibTeX

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Short proofs of three theorems on elliptic integrals. SIAM J. Math. Anal. 9 (3), pp. 524–528.

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External Links:

Document, MathReview (Y. L. Luke), ZentralBlatt

Referenced by:

§19.26(i).

Encodings:

BibTeX

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The universal von Staudt theorems. Trans. Amer. Math. Soc. 315 (2), pp. 591–603.

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External Links:

ISSN 0002-9947, FullText, MathReview (T. Metsänkylä), ZentralBlatt

Referenced by:

§24.16(iii).

Encodings:

BibTeX

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Modular Forms and Fermat’s Last Theorem. Springer-Verlag, New York.

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External Links:

ISBN 0-387-94609-8; 0-387-98998-6, MathReview (M. Ram Murty), ZentralBlatt

Referenced by:

§23.20(v).

Encodings:

BibTeX

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