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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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Relation to Elliptic Integrals

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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: William P. Reinhardt

… ►Older work on the scattering theory of the atomic Coulomb problem led to the discovery of new classes of orthogonal polynomials relating to the spectral theory of Schrödinger operators, and new uses of old ones: this work was strongly motivated by his original ownership of a 1964 hard copy printing of the original AMS 55 NBS Handbook of Mathematical Functions. …

8: 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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9: Mathematical Introduction

… ►The NIST Handbook has essentially the same objective as the Handbook of Mathematical Functions that was issued in 1964 by the National Bureau of Standards as Number 55 in the NBS Applied Mathematics Series (AMS). … ►As a consequence, in addition to providing more information about the special functions that were covered in AMS 55, the NIST Handbook includes several special functions that have appeared in the interim in applied mathematics, the physical sciences, and engineering, as well as in other areas. …

10: 8.21 Generalized Sine and Cosine Integrals

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8.21.4 $$\mathrm{si}(a,z)={\int }_z^{\mathrm{\infty }}{t}^{a-1}\sin tdt,$$ ,

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Symbols:

$dx$: differential, $\mathrm{si}(a,z)$: generalized sine integral, $\int$: integral, $\mathrm{\Re }$: real part, $\sin z$: sine function, $z$: complex variable and $a$: parameter

A&S Ref:

6.5.8 (This function is called $S(z,a)$ in AMS 55.)

Referenced by:

§8.21(iii), §8.21(iii), §8.21(v), §8.21(vii)

Permalink:

http://dlmf.nist.gov/8.21.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §8.21(iii), §8.21 and Ch.8

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8.21.5 $$\mathrm{ci}(a,z)={\int }_z^{\mathrm{\infty }}{t}^{a-1}\cos tdt,$$ ,

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Symbols:

$\cos z$: cosine function, $dx$: differential, $\mathrm{ci}(a,z)$: generalized cosine integral, $\int$: integral, $\mathrm{\Re }$: real part, $z$: complex variable and $a$: parameter

A&S Ref:

6.5.7 (This function is called $C(z,a)$ in AMS 55.)

Referenced by:

§8.21(iii), §8.21(iii), §8.21(v), §8.21(vii)

Permalink:

http://dlmf.nist.gov/8.21.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §8.21(iii), §8.21 and Ch.8

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8.21.18 $$f(a,z)=\mathrm{si}(a,z)\cos z-\mathrm{ci}(a,z)\sin z,$$

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Defines:

$f(a,z)$: auxiliary function (locally)

Symbols:

$\cos z$: cosine function, $\mathrm{ci}(a,z)$: generalized cosine integral, $\mathrm{si}(a,z)$: generalized sine integral, $\sin z$: sine function, $z$: complex variable and $a$: parameter

A&S Ref:

5.2.6 (This generalizes the form in AMS 55.)

Referenced by:

§8.21(vii)

Permalink:

http://dlmf.nist.gov/8.21.E18

Encodings:

pMML, png, TeX

See also:

Annotations for §8.21(vii), §8.21 and Ch.8

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8.21.19 $$g(a,z)=\mathrm{si}(a,z)\sin z+\mathrm{ci}(a,z)\cos z.$$

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Defines:

$g(a,z)$: auxiliary function (locally)

Symbols:

$\cos z$: cosine function, $\mathrm{ci}(a,z)$: generalized cosine integral, $\mathrm{si}(a,z)$: generalized sine integral, $\sin z$: sine function, $z$: complex variable and $a$: parameter

A&S Ref:

5.2.7 (This generalizes the form in AMS 55.)

Referenced by:

§8.21(vii)

Permalink:

http://dlmf.nist.gov/8.21.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §8.21(vii), §8.21 and Ch.8

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8.21.23 $$g(a,z)={\int }_0^{\mathrm{\infty }}\frac{\cos t}{{(t+z)}^{1-a}}dt.$$

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Symbols:

$\cos z$: cosine function, $dx$: differential, $\int$: integral, $z$: complex variable, $a$: parameter and $g(a,z)$: auxiliary function

A&S Ref:

5.2.13 (This generalizes the form in AMS 55.)

Referenced by:

§8.21(vii)

Permalink:

http://dlmf.nist.gov/8.21.E23

Encodings:

pMML, png, TeX

See also:

Annotations for §8.21(vii), §8.21 and Ch.8

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