relation to inverse phase functions

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1: 10.21 Zeros

… ►The functions ${\rho }_{\nu }(t)$ and ${\sigma }_{\nu }(t)$ are related to the inverses of the phase functions ${\theta }_{\nu }\left(x\right)$ and ${\varphi }_{\nu }\left(x\right)$ defined in §10.18(i): if $\nu \ge 0$, then … ►

${\varphi }_{\nu }\left(y_{\nu ,m}^{\prime }\right)=m\pi ,$ $m=1,2,\mathrm{\dots }$.

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2: 9.8 Modulus and Phase

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§9.8(i) Definitions

… ►(These definitions of $\theta \left(x\right)$ and $\varphi \left(x\right)$ differ from Abramowitz and Stegun (1964, Chapter 10), and agree more closely with those used in Miller (1946) and Olver (1997b, Chapter 11).) … ►

§9.8(ii) Identities

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§9.8(iii) Monotonicity

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3: Errata

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The Gegenbauer function $C_{\alpha }^{(\lambda )}\left(z\right)$, was labeled inadvertently as the ultraspherical (Gegenbauer) polynomial $C_n^{(\lambda )}\left(z\right)$. In order to resolve this inconsistency, this function now links correctly to its definition. This change affects Gegenbauer functions which appear in §§14.3(iv), 15.9(iii).

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The wording was changed to make the integration variable more apparent.

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The validity constraint was added. Additionally, specific source citations are now given in the metadata for all equations in Chapter 9 Airy and Related Functions.

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A number of additions and changes have been made to the metadata to reflect new and changed references as well as to how some equations have been derived.

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This figure was rescaled, with symmetry lines added, to make evident the symmetry due to the inverse relationship between the two functions.

Reported 2015-11-12 by James W. Pitman.

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4: 32.11 Asymptotic Approximations for Real Variables

… ►Any nontrivial real solution of (32.11.4) that satisfies (32.11.5) is asymptotic to $k\mathrm{Ai}\left(x\right)$, for some nonzero real $k$, where $\mathrm{Ai}$ denotes the Airy function (§9.2). … ►where $\mathrm{\Gamma }$ is the gamma function (§5.2(i)), and the branch of the $\mathrm{ph}$ function is immaterial. … ►The connection formulas relating (32.11.25) and (32.11.26) are … ►Now suppose $x\to -\mathrm{\infty }$. …and the branch of the $\mathrm{ph}$ function is immaterial. …

5: 10.68 Modulus and Phase Functions

§10.68 Modulus and Phase Functions

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§10.68(i) Definitions

… ►where $M_{\nu }\left(x\right)(>0)$, $N_{\nu }\left(x\right)(>0)$, ${\theta }_{\nu }\left(x\right)$, and ${\varphi }_{\nu }\left(x\right)$ are continuous real functions of $x$ and $\nu$, with the branches of ${\theta }_{\nu }\left(x\right)$ and ${\varphi }_{\nu }\left(x\right)$ chosen to satisfy (10.68.18) and (10.68.21) as $x\to \mathrm{\infty }$. … ►

§10.68(ii) Basic Properties

… ►However, care needs to be exercised with the branches of the phases. …

6: 15.9 Relations to Other Functions

§15.9 Relations to Other Functions

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§15.9(i) Orthogonal Polynomials

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Jacobi

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Legendre

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Meixner

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7: 13.8 Asymptotic Approximations for Large Parameters

… ►as $b\to \mathrm{\infty }$ in $|\mathrm{ph}b|\le \pi -\delta$, where $q_0(z,a)=1$ and … ►For the parabolic cylinder function $U$ see §12.2, and for an extension to an asymptotic expansion see Temme (1978). … ►where $w=\mathrm{arccosh}\left(1+{(2a)}^{-1}x\right)$, and $\beta =(w+\sinh w)/2$. … ►For an extension to an asymptotic expansion complete with error bounds see Temme (1990b), and for related results see §13.21(i). … ►When $a\to \mathrm{\infty }$ in $\left|\mathrm{ph}a\right|\le \pi -\delta$ and $b$ and $z$ fixed, …

8: 36.7 Zeros

… ►The zeros in Table 36.7.1 are points in the $𝐱=(x,y)$ plane, where $\mathrm{ph}{\mathrm{\Psi }}_2\left(𝐱\right)$ is undetermined. … ►The zeros are lines in $𝐱=(x,y,z)$ space where $\mathrm{ph}{\mathrm{\Psi }}^{(\mathrm{E})}\left(𝐱\right)$ is undetermined. …Near $z=z_n$, and for small $x$ and $y$, the modulus $|{\mathrm{\Psi }}^{(\mathrm{E})}\left(𝐱\right)|$ has the symmetry of a lattice with a rhombohedral unit cell that has a mirror plane and an inverse threefold axis whose $z$ and $x$ repeat distances are given by …Outside the bifurcation set (36.4.10), each rib is flanked by a series of zero lines in the form of curly “antelope horns” related to the “outside” zeros (36.7.2) of the cusp canonical integral. There are also three sets of zero lines in the plane $z=0$ related by $2\pi /3$ rotation; these are zeros of (36.2.20), whose asymptotic form in polar coordinates $(x=r\cos \theta ,y=r\sin \theta )$ is given by …

9: 3.5 Quadrature

… ►For the classical orthogonal polynomials related to the following Gauss rules, see §18.3. … ►The monic and orthonormal recursion relations of this section are both closely related to the Lanczos recursion relation in §3.2(vi). … ►are related to Bessel polynomials (§§10.49(ii) and 18.34). … … ►

Example. Laplace Transform Inversion

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10: 19.6 Special Cases

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19.6.8 $$F(\varphi ,1)=(\sin \varphi )R_C(1,{\cos }^2\varphi )={\mathrm{gd}}^{-1}\left(\varphi \right).$$

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Symbols:

$R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $\cos z$: cosine function, $F(\varphi ,k)$: Legendre’s incomplete elliptic integral of the first kind, ${\mathrm{gd}}^{-1}\left(x\right)$: inverse Gudermannian function, $\sin z$: sine function and $\varphi$: real or complex argument

Referenced by:

§19.10(ii), §19.10(ii)

Permalink:

http://dlmf.nist.gov/19.6.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §19.6(ii), §19.6 and Ch.19

►For the inverse Gudermannian function ${\mathrm{gd}}^{-1}\left(\varphi \right)$ see §4.23(viii). … ►

§19.6(v) $R_C(x,y)$

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$R_C(x,y)\to +\mathrm{\infty },$ $y\to 0+$ or $y\to 0-$, $x>0$,

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$R_C(0,y)=\frac{1}{2}\pi y^{-1/2},$ ,

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