# with respect to amplitude

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##### 1: 19.13 Integrals of Elliptic Integrals

###### §19.13(ii) Integration with Respect to the Amplitude

…##### 2: 22.19 Physical Applications

###### §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

##### 4: 2.7 Differential Equations

*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

##### 6: 22.16 Related Functions

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

►###### Definition

… ►###### Quasi-Periodicity

… ►###### Integral Representation

… ►###### Relation to Elliptic Integrals

…##### 7: Errata

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

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.*

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.

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.

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.*

##### 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

*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\mathrm{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

$l,m,n$ | nonnegative integers. |
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$\varphi $ | real or complex argument (or amplitude). |

… |