relation to amplitude (am) function

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1: 22.16 Related Functions

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22.16.4 $$\mathrm{am}(x,0)=x,$$

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Symbols:

$\mathrm{am}(x,k)$: Jacobi’s amplitude function and $x$: real

Permalink:

http://dlmf.nist.gov/22.16.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §22.16(i), §22.16(i), §22.16 and Ch.22

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22.16.8 $$\mathrm{am}(x,k)=\mathrm{gd}x-\frac{1}{4}k_{}^{\prime }{}_{}{}^2(x-\sinh x\cosh x)\mathrm{sech}x+O\left(k_{}^{\prime }{}_{}{}^4\right).$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{gd}x$: Gudermannian function, $\mathrm{am}(x,k)$: Jacobi’s amplitude function, $\cosh z$: hyperbolic cosine function, $\mathrm{sech}z$: hyperbolic secant function, $\sinh z$: hyperbolic sine function, $x$: real, $k$: modulus and $k^{\prime }$: complementary modulus

A&S Ref:

16.15.4

Permalink:

http://dlmf.nist.gov/22.16.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §22.16(i), §22.16(i), §22.16 and Ch.22

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Relation to Elliptic Integrals

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2: 29.1 Special Notation

… ►(For other notation see Notation for the Special Functions.) … ►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). … ►The relation to the Lamé functions $L_{c\nu }^{(m)}$, $L_{s\nu }^{(m)}$of Jansen (1977) is given by …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 …

3: 29.6 Fourier Series

§29.6 Fourier Series

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§29.6(i) Function ${\mathrm{𝐸𝑐}}_{\nu }^{2m}(z,k^2)$

►With $\varphi =\frac{1}{2}\pi -\mathrm{am}(z,k)$, as in (29.2.5), we have … ►In addition, if $H$ satisfies (29.6.2), then (29.6.3) applies. … ►Consequently, ${\mathrm{𝐸𝑐}}_{\nu }^{2m}(z,k^2)$ reduces to a Lamé polynomial; compare §§29.12(i) and 29.15(i). …

4: 22.20 Methods of Computation

… ►A powerful way of computing the twelve Jacobian elliptic functions for real or complex values of both the argument $z$ and the modulus $k$ is to use the definitions in terms of theta functions given in §22.2, obtaining the theta functions via methods described in §20.14. … ►

§22.20(vi) Related Functions

► $\mathrm{am}(x,k)$ can be computed from its definition (22.16.1) or from its Fourier series (22.16.9). Alternatively, Sala (1989) shows how to apply the arithmetic-geometric mean to compute $\mathrm{am}(x,k)$. … ►For additional information on methods of computation for the Jacobi and related functions, see the introductory sections in the following books: Lawden (1989), Curtis (1964b), Milne-Thomson (1950), and Spenceley and Spenceley (1947). …

5: 19.25 Relations to Other Functions

§19.25 Relations to Other Functions

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§19.25(iv) Theta Functions

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§19.25(vii) Hypergeometric Function

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6: 29.2 Differential Equations

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§29.2(ii) Other Forms

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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

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