amplitude (am) function

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1: 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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Figure 22.16.1: Jacobi’s amplitude function $\mathrm{am}(x,k)$ for $0\le x\le 10\pi$ and $k=0.4,0.7,0.99,0.999999$. … Magnify

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2: 22.21 Tables

§22.21 Tables

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3: 22.19 Physical Applications

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§22.19(i) Classical Dynamics: The Pendulum

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

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Symbols:

$\mathrm{am}(x,k)$: Jacobi’s amplitude function, $\theta (t)$: angular displacement and $E$: energy

Referenced by:

§22.19(i), §22.19(i), Erratum (V1.0.8) for Equation (22.19.3)

Permalink:

http://dlmf.nist.gov/22.19.E3

Encodings:

pMML, png, TeX

Errata (effective with 1.0.8):

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

See also:

Annotations for §22.19(i), §22.19 and Ch.22

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Figure 22.19.1: Jacobi’s amplitude function $\mathrm{am}(x,k)$ for $0\le x\le 10\pi$ and $k=0.5,0.9999,1.0001,2$. … Magnify

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4: 22.1 Special Notation

… ►The functions treated in this chapter are the three principal Jacobian elliptic functions $\mathrm{sn}(z,k)$, $\mathrm{cn}(z,k)$, $\mathrm{dn}(z,k)$; the nine subsidiary Jacobian elliptic functions $\mathrm{cd}(z,k)$, $\mathrm{sd}(z,k)$, $\mathrm{nd}(z,k)$, $\mathrm{dc}(z,k)$, $\mathrm{nc}(z,k)$, $\mathrm{sc}(z,k)$, $\mathrm{ns}(z,k)$, $\mathrm{ds}(z,k)$, $\mathrm{cs}(z,k)$; the amplitude function $\mathrm{am}(x,k)$; Jacobi’s epsilon and zeta functions $ℰ(x,k)$ and $\mathrm{Z}\left(x|k\right)$. …

5: 22.20 Methods of Computation

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§22.20(vi) Related Functions

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6: 19.25 Relations to Other Functions

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19.25.30 $$\mathrm{am}(u,k)=R_C({\mathrm{cs}}^2(u,k),{\mathrm{ns}}^2(u,k)),$$

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Symbols:

$R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $\mathrm{am}(x,k)$: Jacobi’s amplitude function, $\mathrm{cs}(z,k)$: Jacobian elliptic function, $\mathrm{ns}(z,k)$: Jacobian elliptic function and $k$: real or complex modulus

Referenced by:

§19.25(v)

Permalink:

http://dlmf.nist.gov/19.25.E30

Encodings:

pMML, png, TeX

See also:

Annotations for §19.25(v), §19.25 and Ch.19

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7: Errata

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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: 22.14 Integrals

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22.14.3 $$\int \mathrm{dn}(x,k)dx=\mathrm{Arcsin}\left(\mathrm{sn}(x,k)\right)=\mathrm{am}(x,k).$$

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Symbols:

$\mathrm{am}(x,k)$: Jacobi’s amplitude function, $\mathrm{dn}(z,k)$: Jacobian elliptic function, $\mathrm{sn}(z,k)$: Jacobian elliptic function, $dx$: differential, $\int$: integral, $\mathrm{Arcsin}z$: general arcsine function, $x$: real and $k$: modulus

A&S Ref:

16.24.3

Permalink:

http://dlmf.nist.gov/22.14.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §22.14(i), §22.14 and Ch.22

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9: 29.2 Differential Equations

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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:

pMML, png, TeX

See also:

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

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10: 29.1 Special Notation

… ►(For other notation see Notation for the Special Functions.) … ►All derivatives are denoted by differentials, not by primes. … ►The notation for the eigenvalues and functions is due to Erdélyi et al. (1955, §15.5.1) and that for the polynomials is due to Arscott (1964b, §9.3.2). … ►where $\psi =\mathrm{am}(z,k)$; see §22.16(i). The relation to the Lamé functions ${\mathrm{Ec}}_{\nu }^m$, ${\mathrm{Es}}_{\nu }^m$ of Ince (1940b) is given by …