Lagrange inversion theorem

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21: 19.11 Addition Theorems

§19.11 Addition Theorems

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19.11.5 $$\mathrm{\Pi }(\theta ,{\alpha }^2,k)+\mathrm{\Pi }(\varphi ,{\alpha }^2,k)=\mathrm{\Pi }(\psi ,{\alpha }^2,k)-{\alpha }^2{R}_C(\gamma -\delta ,\gamma ),$$

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Symbols:

$R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$: Legendre’s incomplete elliptic integral of the third kind, $\varphi$: real or complex argument, $k$: real or complex modulus, ${\alpha }^2$: real or complex parameter, $\theta$: amplitude, $\psi$: angle, $\gamma$ and $\delta$

Referenced by:

§19.11(i), §19.11(i), Erratum (V1.0.24) for Subsection 19.11(i)

Permalink:

http://dlmf.nist.gov/19.11.E5

Encodings:

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See also:

Annotations for §19.11(i), §19.11 and Ch.19

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19.11.6_5 $$R_C(\gamma -\delta ,\gamma )=\frac{-1}{\sqrt{\delta }}\arctan \left(\frac{\sqrt{\delta }\sin \theta \sin \varphi \sin \psi }{{\alpha }^2-1-{\alpha }^2\cos \theta \cos \varphi \cos \psi }\right).$$

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Symbols:

$R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $\cos z$: cosine function, $\arctan z$: arctangent function, $\sin z$: sine function, $\varphi$: real or complex argument, ${\alpha }^2$: real or complex parameter, $\theta$: amplitude, $\psi$: angle, $\gamma$ and $\delta$

Referenced by:

§19.11(i), Erratum (V1.0.28) for Equations (15.2.3_5), (19.11.6_5)

Permalink:

http://dlmf.nist.gov/19.11.E6_5

Encodings:

pMML, png, TeX

Clarification (effective with 1.0.28):

This equation was given a number.

See also:

Annotations for §19.11(i), §19.11 and Ch.19

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19.11.10 $$\mathrm{\Pi }(\varphi ,{\alpha }^2,k)=\mathrm{\Pi }({\alpha }^2,k)-\mathrm{\Pi }(\theta ,{\alpha }^2,k)-{\alpha }^2{R}_C(\gamma -\delta ,\gamma ),$$

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Symbols:

$R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $\mathrm{\Pi }({\alpha }^2,k)$: Legendre’s complete elliptic integral of the third kind, $\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$: Legendre’s incomplete elliptic integral of the third kind, $\varphi$: real or complex argument, $k$: real or complex modulus, ${\alpha }^2$: real or complex parameter, $\theta$: amplitude, $\gamma$ and $\delta$

Permalink:

http://dlmf.nist.gov/19.11.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §19.11(ii), §19.11 and Ch.19

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19.11.16 $$\mathrm{\Pi }(\psi ,{\alpha }^2,k)=2\mathrm{\Pi }(\theta ,{\alpha }^2,k)+{\alpha }^2{R}_C(\gamma -\delta ,\gamma ),$$

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Symbols:

$R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$: Legendre’s incomplete elliptic integral of the third kind, $k$: real or complex modulus, ${\alpha }^2$: real or complex parameter, $\theta$: amplitude, $\delta$ , $\psi$: angle and $\gamma$

Permalink:

http://dlmf.nist.gov/19.11.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §19.11(iii), §19.11 and Ch.19

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22: 4.29 Graphics

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§4.29(i) Real Arguments

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Figure 4.29.2: Principal values of $\mathrm{arcsinh}x$ and $\mathrm{arccosh}x$. … Magnify

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Figure 4.29.4: Principal values of $\mathrm{arctanh}x$ and $\mathrm{arccoth}x$. … Magnify

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§4.29(ii) Complex Arguments

… ►The surfaces for the complex hyperbolic and inverse hyperbolic functions are similar to the surfaces depicted in §4.15(iii) for the trigonometric and inverse trigonometric functions. …

23: 4.1 Special Notation

… ►The main purpose of the present chapter is to extend these definitions and properties to complex arguments $z$. ►The main functions treated in this chapter are the logarithm $\ln z$, $\mathrm{Ln}z$; the exponential $\exp z$, ${\mathrm{e}}^z$; the circular trigonometric (or just trigonometric) functions $\sin z$, $\cos z$, $\tan z$, $\csc z$, $\sec z$, $\cot z$; the inverse trigonometric functions $\arcsin z$, $\mathrm{Arcsin}z$, etc. ; the hyperbolic trigonometric (or just hyperbolic) functions $\sinh z$, $\cosh z$, $\tanh z$, $\mathrm{csch}z$, $\mathrm{sech}z$, $\coth z$; the inverse hyperbolic functions $\mathrm{arcsinh}z$, $\mathrm{Arcsinh}z$, etc. ►Sometimes in the literature the meanings of $\ln$ and $\mathrm{Ln}$ are interchanged; similarly for $\arcsin z$ and $\mathrm{Arcsin}z$, etc. …${\sin }^{-1}z$ for $\arcsin z$ and ${\mathrm{Sin}}^{-1}z$ for $\mathrm{Arcsin}z$.

24: 30.10 Series and Integrals

… ►For an addition theorem, see Meixner and Schäfke (1954, p. 300) and King and Van Buren (1973). …

25: 10.44 Sums

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§10.44(i) Multiplication Theorem

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§10.44(ii) Addition Theorems

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Neumann’s Addition Theorem

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Graf’s and Gegenbauer’s Addition Theorems

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26: 3.4 Differentiation

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§3.4(i) Equally-Spaced Nodes

►The Lagrange $(n+1)$-point formula is … ► …

27: Bibliography R

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Normal limit theorems for symmetric random matrices. Probab. Theory Related Fields 112 (3), pp. 411–423.

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External Links:

ISSN 0178-8051, Document, MathReview (Alexei Khorunzhy), ZentralBlatt

Referenced by:

§35.9.

Encodings:

BibTeX

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Fourier analysis and signal processing by use of the Möbius inversion formula. IEEE Trans. Acoustics, Speech, Signal Processing 38, pp. 458–470.

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External Links:

ZentralBlatt

Referenced by:

§27.17.

Encodings:

BibTeX

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Universality properties of Gaussian quadrature, the derivative rule, and a novel approach to Stieltjes inversion.

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Notes:

arXiv:1812.02196

Referenced by:

§18.40(ii).

Encodings:

BibTeX

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13 Lectures on Fermat’s Last Theorem. Springer-Verlag, New York.

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External Links:

ISBN 0-387-90432-8, MathReview (T. Metsänkylä), ZentralBlatt

Referenced by:

§24.10(iv), §24.17(iii).

Encodings:

BibTeX

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Dawson’s integral and the sampling theorem. Computers in Physics 3 (2), pp. 85–87.

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Referenced by:

§7.25(vi).

Encodings:

BibTeX

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28: 9.8 Modulus and Phase

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9.8.4 $$\theta \left(x\right)=\arctan \left(\mathrm{Ai}\left(x\right)/\mathrm{Bi}\left(x\right)\right).$$

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Defines:

$\theta \left(z\right)$: Airy phase function

Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\mathrm{Bi}\left(z\right)$: Airy function, $\arctan z$: arctangent function, $x$: real variable and $z$: complex variable

Source:

Olver (1997b, (2.02), p. 395)

Referenced by:

(9.11.19), (9.8.11), (9.8.15), (9.8.17), (9.8.19)

Permalink:

http://dlmf.nist.gov/9.8.E4

Encodings:

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See also:

Annotations for §9.8(i), §9.8 and Ch.9

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9.8.8 $$\varphi \left(x\right)=\arctan \left({\mathrm{Ai}}^{\prime }\left(x\right)/{\mathrm{Bi}}^{\prime }\left(x\right)\right).$$

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Defines:

$\varphi \left(z\right)$: Airy phase function

Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\mathrm{Bi}\left(z\right)$: Airy function, $\arctan z$: arctangent function, $x$: real variable and $z$: complex variable

Source:

Olver (1997b, (2.09–2.10), p. 396); with different notation

Referenced by:

(9.11.19), (9.8.12), (9.8.16), (9.8.17)

Permalink:

http://dlmf.nist.gov/9.8.E8

Encodings:

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See also:

Annotations for §9.8(i), §9.8 and Ch.9

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9.8.11 $$\theta \left(x\right)=\frac{2}{3}\pi +\arctan \left(Y_{1/3}\left(\xi \right)/J_{1/3}\left(\xi \right)\right),$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $Y_{\nu }\left(z\right)$: Bessel function of the second kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\arctan z$: arctangent function, $\theta \left(z\right)$: Airy phase function, $x$: real variable and $\xi$: change of variable

Proof sketch:

Combine definition (9.8.4) with (9.6.17) and (9.6.19) using (10.4.3).

Permalink:

http://dlmf.nist.gov/9.8.E11

Encodings:

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See also:

Annotations for §9.8(i), §9.8 and Ch.9

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9.8.12 $$\varphi \left(x\right)=\frac{1}{3}\pi +\arctan \left(Y_{2/3}\left(\xi \right)/J_{2/3}\left(\xi \right)\right).$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $Y_{\nu }\left(z\right)$: Bessel function of the second kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\arctan z$: arctangent function, $\varphi \left(z\right)$: Airy phase function, $x$: real variable and $\xi$: change of variable

Proof sketch:

Combine definition (9.8.8) with (9.6.18) and (9.6.20) using (10.4.3).

Referenced by:

(9.8.20), (9.8.21), (9.8.22), (9.8.23)

Permalink:

http://dlmf.nist.gov/9.8.E12

Encodings:

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See also:

Annotations for §9.8(i), §9.8 and Ch.9

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9.8.22 $$\theta \left(x\right)\sim \frac{\pi }{4}+\frac{2}{3}{(-x)}^{3/2}\left(1+\frac{5}{32}\frac{1}{x^3}+\frac{1105}{6144}\frac{1}{x^6}+\frac{82825}{65536}\frac{1}{x^9}+\frac{\mathrm{12820\hspace{0.33em}31525}}{\mathrm{587\hspace{0.33em}20256}}\frac{1}{x^{12}}+\mathrm{\cdots }\right),$$

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Symbols:

$\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\theta \left(z\right)$: Airy phase function and $x$: real variable

Proof sketch:

Combine (9.8.9)–(9.8.12) with §10.18(iii). See also Nemes (2021, Theorem 2.1 and p. 4333).

Referenced by:

§9.8(iv), §9.8(iv), Erratum (V1.1.8) for Section 9.8(iv)

Permalink:

http://dlmf.nist.gov/9.8.E22

Encodings:

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See also:

Annotations for §9.8(iv), §9.8 and Ch.9

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29: 1.2 Elementary Algebra

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Binomial Theorem

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The Inverse

►If det($𝐀$) $\ne$ $0$, $𝐀$ has a unique inverse, ${𝐀}^{-1}$, such that …If $det(𝐀)=0$ then $𝐀𝐁=𝐀𝐂$ does not imply that $𝐁=𝐂$; if $det(𝐀)\ne 0$, then $𝐁=𝐂$, as both sides may be multiplied by ${𝐀}^{-1}$. … ►has a unique solution, $𝐛={𝐀}^{-1}𝐜$. …

30: 19.35 Other Applications

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§19.35(i) Mathematical

►Generalizations of elliptic integrals appear in analysis of modular theorems of Ramanujan (Anderson et al. (2000)); analysis of Selberg integrals (Van Diejen and Spiridonov (2001)); use of Legendre’s relation (19.7.1) to compute $\pi$ to high precision (Borwein and Borwein (1987, p. 26)). …