with respect to amplitude

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1: 19.13 Integrals of Elliptic Integrals

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§19.13(ii) Integration with Respect to the Amplitude

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2: 22.19 Physical Applications

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§22.19(i) Classical Dynamics: The Pendulum

… ►This formulation gives the bounded and unbounded solutions from the same formula (22.19.3), for $k\ge 1$ and $k\le 1$, respectively. …Figure 22.19.1 shows the nature of the solutions $\theta (t)$ of (22.19.3) by graphing $\mathrm{am}(x,k)$ for both $0\le k\le 1$, as in Figure 22.16.1, and $k\ge 1$, where it is periodic. … ►Its dynamics for purely imaginary time is connected to the theory of instantons (Itzykson and Zuber (1980, p. 572), Schäfer and Shuryak (1998)), to WKB theory, and to large-order perturbation theory (Bender and Wu (1973), Simon (1982)). ►For $\beta$ real and positive, three of the four possible combinations of signs give rise to bounded oscillatory motions. …

3: 29.2 Differential Equations

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29.2.1 $$\frac{d^{2}w}{{dz}^2}+(h-\nu (\nu +1)k^2{\mathrm{sn}}^2(z,k))w=0,$$

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Symbols:

$\mathrm{sn}(z,k)$: Jacobian elliptic function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex variable, $h$: real parameter, $k$: real parameter, $\nu$: real parameter and $w(z)$: solution

Referenced by:

§29.10, §29.11, §29.12(i), §29.17(i), §29.17(ii), §29.17(iii), §29.18(i), §29.2(i), §29.20(i), §29.3(i), §29.7(ii), §29.8, §29.9

Permalink:

http://dlmf.nist.gov/29.2.E1

Encodings:

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See also:

Annotations for §29.2(i), §29.2 and Ch.29

… ►This equation has regular singularities at the points $2pK+(2q+1)\mathrm{i}{K}^{\prime }$, where $p,q\in \mathbb{Z}$, and $K$, $K^{\prime }$ are the complete elliptic integrals of the first kind with moduli $k$, $k^{\prime }(={(1-k^2)}^{1/2})$, respectively; see §19.2(ii). … ►

29.2.5 $$\varphi =\frac{1}{2}\pi -\mathrm{am}(z,k).$$

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Defines:

$\varphi$: change of variable (locally)

Symbols:

$\mathrm{am}(x,k)$: Jacobi’s amplitude function, $\pi$: the ratio of the circumference of a circle to its diameter, $z$: complex variable and $k$: real parameter

Referenced by:

§29.15(i), §29.15(ii), §29.6(i)

Permalink:

http://dlmf.nist.gov/29.2.E5

Encodings:

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See also:

Annotations for §29.2(ii), §29.2 and Ch.29

►For $\mathrm{am}(z,k)$ see §22.16(i). … ►For the Weierstrass function $\mathrm{\wp }$ see §23.2(ii). …

4: 2.7 Differential Equations

… ►as $z\to \mathrm{\infty }$ in the sectors … ►This phenomenon is an example of resurgence, a classification due to Écalle (1981a, b). … ►For extensions to singularities of higher rank see Olver and Stenger (1965). … ►The solutions $w_1(z)$ and $w_2(z)$ are respectively recessive and dominant as $\mathrm{\Re }z\to -\mathrm{\infty }$, and vice versa as $\mathrm{\Re }z\to +\mathrm{\infty }$. … ►In oscillatory intervals, and again following Miller (1950), we call a pair of solutions numerically satisfactory if asymptotically they have the same amplitude and are $\frac{1}{2}\pi$ out of phase.

5: 36.12 Uniform Approximation of Integrals

… ►The function $g$ has a smooth amplitude. … ►Define a mapping $u(t;𝐲)$ by relating $f(u;𝐲)$ to the normal form (36.2.1) of ${\mathrm{\Phi }}_K(t;𝐱)$ in the following way: … ►This technique can be applied to generate a hierarchy of approximations for the diffraction catastrophes ${\mathrm{\Psi }}_K(𝐱;k)$ in (36.2.10) away from $𝐱=\mathrm{𝟎}$, in terms of canonical integrals ${\mathrm{\Psi }}_J\left(\xi (𝐱;k)\right)$ for . For example, the diffraction catastrophe ${\mathrm{\Psi }}_2(x,y;k)$ defined by (36.2.10), and corresponding to the Pearcey integral (36.2.14), can be approximated by the Airy function ${\mathrm{\Psi }}_1\left(\xi (x,y;k)\right)$ when $k$ is large, provided that $x$ and $y$ are not small. … ►Also, ${\mathrm{\Delta }}^{1/4}/\sqrt{f_{+}^{\prime \prime }}$ and ${\mathrm{\Delta }}^{1/4}/\sqrt{-f_{-}^{\prime \prime }}$ are chosen to be positive real when $y$ is such that both critical points are real, and by analytic continuation otherwise. …

6: 22.16 Related Functions

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§22.16(i) Jacobi’s Amplitude ($\mathrm{am}$) Function

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Definition

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Quasi-Periodicity

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Integral Representation

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Relation to Elliptic Integrals

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7: Errata

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These equations, originally added in Other Changes and Other Changes, respectively, have been assigned interpolated numbers.

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The Weierstrass lattice roots $e_j,$ were linked inadvertently as the base of the natural logarithm. In order to resolve this inconsistency, the lattice roots $e_j$, and lattice invariants $g_2$, $g_3$, now link to their respective definitions (see §§23.2(i), 23.3(i)).

Reported by Felix Ospald.

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These equations have been generalized to include the additional cases of $\partial J_{-\nu }\left(z\right)/\partial \nu$, $\partial I_{-\nu }\left(z\right)/\partial \nu$, respectively.

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The titles have been changed to $\mu =0$ , $\nu =0,1$ , and Addendum to §14.5(ii): $\mu =0$, $\nu =2$ , respectively, in order to be more descriptive of their contents.

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22.19.3 $$\theta (t)=2\mathrm{am}(t\sqrt{E/2},\sqrt{2/E})$$

Originally the first argument to the function $\mathrm{am}$ was given incorrectly as $t$. The correct argument is $t\sqrt{E/2}$.

Reported 2014-03-05 by Svante Janson.

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8: 19.25 Relations to Other Functions

§19.25 Relations to Other Functions

… ►All terms on the right-hand sides are nonnegative when $k^2\le 0$, $0\le k^2\le 1$, or $1\le k^2\le c$, respectively. … ► … ►($F_1$ and $F_D$ are equivalent to the $R$-function of 3 and $n$ variables, respectively, but lack full symmetry.)

9: 18.39 Applications in the Physical Sciences

… ►For further details about the Schrödinger equation, including applications in physics and chemistry, see Gottfried and Yan (2004) and Pauling and Wilson (1985), respectively, among many others. … ►Then $\omega =2\pi \nu =\sqrt{k/m}$ is the circular frequency of oscillation (with $\nu$ the ordinary frequency), independent of the amplitude of the oscillations. … ►allows anharmonic, or amplitude dependent, frequencies of oscillation about $x_e$, and also escape of the particle to $x=+\mathrm{\infty }$ with dissociation energy $D$. … ►Orthogonality and normalization of eigenfunctions of this form is respect to the measure $r^{2}dr\sin \theta d\theta d\varphi$. … ►For applications and an extension of the Szegő–Szász inequality (18.14.20) for Legendre polynomials ($\alpha =\beta =0$) to obtain global bounds on the variation of the phase of an elastic scattering amplitude, see Cornille and Martin (1972, 1974). …

10: 19.1 Special Notation

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$l,m,n$	nonnegative integers.
$\varphi$	real or complex argument (or amplitude).
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… ►of the first, second, and third kinds, respectively, and Legendre’s incomplete integrals …of the first, second, and third kinds, respectively. …The functions (19.1.1) and (19.1.2) are used in Erdélyi et al. (1953b, Chapter 13), except that $\mathrm{\Pi }({\alpha }^2,k)$ and $\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$ are denoted by ${\mathrm{\Pi }}_1(\nu ,k)$ and $\mathrm{\Pi }(\varphi ,\nu ,k)$, respectively, where $\nu =-{\alpha }^2$. ►In Abramowitz and Stegun (1964, Chapter 17) the functions (19.1.1) and (19.1.2) are denoted, in order, by $K(\alpha )$, $E(\alpha )$, $\mathrm{\Pi }(n\backslash \alpha )$, $F(\varphi \backslash \alpha )$, $E(\varphi \backslash \alpha )$, and $\mathrm{\Pi }(n;\varphi \backslash \alpha )$, where $\alpha =\arcsin k$ and $n$ is the ${\alpha }^2$ (not related to $k$) in (19.1.1) and (19.1.2). …