22 Jacobian Elliptic Functions Properties 22.15 Inverse Functions 22.17 Moduli Outside the Interval [0,1]

§22.16 Related Functions

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Permalink:: http://dlmf.nist.gov/22.16
See also:: Annotations for Ch.22

§22.16(i) Jacobi’s Amplitude () Function
§22.16(ii) Jacobi’s Epsilon Function
§22.16(iii) Jacobi’s Zeta Function
§22.16(iv) Graphs

§22.16(i) Jacobi’s Amplitude () Function

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Keywords:: amplitude () function
Notes:: See Walker (1996, pp. 138–141).
Referenced by:: §19.25(v), §22.14(i), §29.1, §29.2(ii)
Permalink:: http://dlmf.nist.gov/22.16.i
See also:: Annotations for §22.16 and Ch.22

Definition

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Keywords:: amplitude () function, definition
See also:: Annotations for §22.16(i), §22.16 and Ch.22

22.16.1
,
ⓘ Defines: : Jacobi’s amplitude function Symbols: : Jacobian elliptic function, : element of, : general arcsine function, : real line, : real and : modulus Referenced by: §22.20(vi) Permalink: http://dlmf.nist.gov/22.16.E1 Encodings: TeX, pMML, png See also: Annotations for §22.16(i), §22.16(i), §22.16 and Ch.22

where the inverse sine has its principal value when and is defined by continuity elsewhere. See Figure 22.16.1. is an infinitely differentiable function of .

Quasi-Periodicity

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Keywords:: amplitude () function, quasi-periodicity
See also:: Annotations for §22.16(i), §22.16 and Ch.22

22.16.2
ⓘ Symbols: : Jacobi’s amplitude function, : the ratio of the circumference of a circle to its diameter, : Legendre’s complete elliptic integral of the first kind, : real and : modulus Permalink: http://dlmf.nist.gov/22.16.E2 Encodings: TeX, pMML, png See also: Annotations for §22.16(i), §22.16(i), §22.16 and Ch.22

Integral Representation

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Keywords:: amplitude () function, integral representation
See also:: Annotations for §22.16(i), §22.16 and Ch.22

22.16.3
ⓘ Symbols: : Jacobi’s amplitude function, : Jacobian elliptic function, : differential of , : integral, : real and : modulus Referenced by: Figure 22.16.2, Figure 22.16.2 Permalink: http://dlmf.nist.gov/22.16.E3 Encodings: TeX, pMML, png See also: Annotations for §22.16(i), §22.16(i), §22.16 and Ch.22

Special Values

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Keywords:: Gudermannian function, amplitude () function, relation to amplitude () function, special values
See also:: Annotations for §22.16(i), §22.16 and Ch.22

22.16.4
ⓘ Symbols: : Jacobi’s amplitude function and : real Permalink: http://dlmf.nist.gov/22.16.E4 Encodings: TeX, pMML, png See also: Annotations for §22.16(i), §22.16(i), §22.16 and Ch.22
22.16.5
ⓘ Symbols: : Gudermannian function, : Jacobi’s amplitude function and : real Permalink: http://dlmf.nist.gov/22.16.E5 Encodings: TeX, pMML, png See also: Annotations for §22.16(i), §22.16(i), §22.16 and Ch.22

For the Gudermannian function see §4.23(viii).

Approximation for Small

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Keywords:: amplitude () function, approximations, small
See also:: Annotations for §22.16(i), §22.16 and Ch.22

22.16.6
ⓘ Symbols: : order not exceeding, : Jacobi’s amplitude function, : factorial (as in ), : real and : modulus Permalink: http://dlmf.nist.gov/22.16.E6 Encodings: TeX, pMML, png See also: Annotations for §22.16(i), §22.16(i), §22.16 and Ch.22

Approximations for Small ,

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Keywords:: amplitude () function, approximations, relation to Gudermannian function, small
See also:: Annotations for §22.16(i), §22.16 and Ch.22

22.16.7
ⓘ Symbols: : order not exceeding, : Jacobi’s amplitude function, : cosine function, : sine function, : real and : modulus A&S Ref: 16.13.4 Permalink: http://dlmf.nist.gov/22.16.E7 Encodings: TeX, pMML, png See also: Annotations for §22.16(i), §22.16(i), §22.16 and Ch.22

22.16.8
ⓘ Symbols: : order not exceeding, : Gudermannian function, : Jacobi’s amplitude function, : hyperbolic cosine function, : hyperbolic secant function, : hyperbolic sine function, : real, : modulus and : complementary modulus A&S Ref: 16.15.4 Permalink: http://dlmf.nist.gov/22.16.E8 Encodings: TeX, pMML, png See also: Annotations for §22.16(i), §22.16(i), §22.16 and Ch.22

Fourier Series

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Keywords:: Fourier series, amplitude () function
See also:: Annotations for §22.16(i), §22.16 and Ch.22

With as in (22.2.1) and ,

22.16.9
ⓘ Symbols: : Jacobi’s amplitude function, : the ratio of the circumference of a circle to its diameter, : Legendre’s complete elliptic integral of the first kind, : nome, : sine function, : real and : modulus Referenced by: §22.20(vi) Permalink: http://dlmf.nist.gov/22.16.E9 Encodings: TeX, pMML, png See also: Annotations for §22.16(i), §22.16(i), §22.16 and Ch.22

Relation to Elliptic Integrals

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Keywords:: Legendre’s elliptic integrals, function, amplitude () function, relation to elliptic integrals, relations to other functions
See also:: Annotations for §22.16(i), §22.16 and Ch.22

If , then the following four equations are equivalent:

22.16.10
ⓘ Symbols: : Legendre’s incomplete elliptic integral of the first kind, : real and : modulus Permalink: http://dlmf.nist.gov/22.16.E10 Encodings: TeX, pMML, png See also: Annotations for §22.16(i), §22.16(i), §22.16 and Ch.22

22.16.11
ⓘ Symbols: : Jacobi’s amplitude function, : real and : modulus Permalink: http://dlmf.nist.gov/22.16.E11 Encodings: TeX, pMML, png See also: Annotations for §22.16(i), §22.16(i), §22.16 and Ch.22

22.16.12
ⓘ Symbols: : Jacobi’s amplitude function, : Jacobian elliptic function, : sine function, : real and : modulus Permalink: http://dlmf.nist.gov/22.16.E12 Encodings: TeX, pMML, png See also: Annotations for §22.16(i), §22.16(i), §22.16 and Ch.22

22.16.13
ⓘ Symbols: : Jacobi’s amplitude function, : Jacobian elliptic function, : cosine function, : real and : modulus Permalink: http://dlmf.nist.gov/22.16.E13 Encodings: TeX, pMML, png See also: Annotations for §22.16(i), §22.16(i), §22.16 and Ch.22

For see §19.2(ii).

§22.16(ii) Jacobi’s Epsilon Function

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Keywords:: Jacobi’s epsilon function
Notes:: See Lawden (1989, pp. 60–65), Whittaker and Watson (1927, pp. 514–522).
Referenced by:: §22.14(ii), §22.18(i), Erratum (V1.0.1) for Clarifications
Permalink:: http://dlmf.nist.gov/22.16.ii
Clarification (effective with 1.0.1):: The presentation of some of the groups of integral representations in this subsection were rearranged for greater clarity.
See also:: Annotations for §22.16 and Ch.22

Integral Representations

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Keywords:: Jacobian elliptic functions, Jacobi’s epsilon function, definition, integral representations, integrals, of squares
See also:: Annotations for §22.16(ii), §22.16 and Ch.22

For ,

22.16.14

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Defines:

: Jacobi’s epsilon function

Symbols:

: Jacobian elliptic function, : differential of , : integral, : real and : modulus

Referenced by:

§22.16(ii), Erratum (V1.0.1) for Equation (22.16.14)

Permalink:

http://dlmf.nist.gov/22.16.E14

Encodings:

TeX, pMML, png

Errata (effective with 1.0.1):

Originally appeared with the upper limit as , rather than :

Reported 2010-07-08 by Charles Karney

See also:

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

compare (19.2.5). See Figure 22.16.2.

22.16.15
ⓘ Symbols: : Jacobi’s epsilon function, : Jacobian elliptic function, : differential of , : integral, : real and : modulus A&S Ref: 16.26.1 Permalink: http://dlmf.nist.gov/22.16.E15 Encodings: TeX, pMML, png See also: Annotations for §22.16(ii), §22.16(ii), §22.16 and Ch.22
22.16.16
ⓘ Symbols: : Jacobi’s epsilon function, : Jacobian elliptic function, : differential of , : integral, : real, : modulus and : complementary modulus A&S Ref: 16.26.2 Permalink: http://dlmf.nist.gov/22.16.E16 Encodings: TeX, pMML, png See also: Annotations for §22.16(ii), §22.16(ii), §22.16 and Ch.22
22.16.17
ⓘ Symbols: : Jacobi’s epsilon function, : Jacobian elliptic function, : differential of , : integral, : real and : modulus A&S Ref: 16.26.3 Referenced by: Figure 22.16.2, Figure 22.16.2 Permalink: http://dlmf.nist.gov/22.16.E17 Encodings: TeX, pMML, png See also: Annotations for §22.16(ii), §22.16(ii), §22.16 and Ch.22

22.16.18
ⓘ Symbols: : Jacobi’s epsilon function, : Jacobian elliptic function, : Jacobian elliptic function, : differential of , : integral, : real and : modulus A&S Ref: 16.26.4 Permalink: http://dlmf.nist.gov/22.16.E18 Encodings: TeX, pMML, png See also: Annotations for §22.16(ii), §22.16(ii), §22.16 and Ch.22
22.16.19
ⓘ Symbols: : Jacobi’s epsilon function, : Jacobian elliptic function, : Jacobian elliptic function, : Jacobian elliptic function, : differential of , : integral, : real, : modulus and : complementary modulus A&S Ref: 16.26.5 Permalink: http://dlmf.nist.gov/22.16.E19 Encodings: TeX, pMML, png See also: Annotations for §22.16(ii), §22.16(ii), §22.16 and Ch.22
22.16.20
ⓘ Symbols: : Jacobi’s epsilon function, : Jacobian elliptic function, : Jacobian elliptic function, : Jacobian elliptic function, : differential of , : integral, : real, : modulus and : complementary modulus A&S Ref: 16.26.6 Permalink: http://dlmf.nist.gov/22.16.E20 Encodings: TeX, pMML, png See also: Annotations for §22.16(ii), §22.16(ii), §22.16 and Ch.22

In Equations (22.16.21)–(22.16.23),

22.16.21
ⓘ Symbols: : Jacobi’s epsilon function, : Jacobian elliptic function, : Jacobian elliptic function, : differential of , : integral, : real and : modulus A&S Ref: 16.26.7 Referenced by: §22.16(ii) Permalink: http://dlmf.nist.gov/22.16.E21 Encodings: TeX, pMML, png See also: Annotations for §22.16(ii), §22.16(ii), §22.16 and Ch.22
22.16.22
ⓘ Symbols: : Jacobi’s epsilon function, : Jacobian elliptic function, : Jacobian elliptic function, : Jacobian elliptic function, : differential of , : integral, : real, : modulus and : complementary modulus A&S Ref: 16.26.8 Permalink: http://dlmf.nist.gov/22.16.E22 Encodings: TeX, pMML, png See also: Annotations for §22.16(ii), §22.16(ii), §22.16 and Ch.22
22.16.23
ⓘ Symbols: : Jacobi’s epsilon function, : Jacobian elliptic function, : Jacobian elliptic function, : Jacobian elliptic function, : differential of , : integral, : real, : modulus and : complementary modulus A&S Ref: 16.26.9 Referenced by: §22.16(ii) Permalink: http://dlmf.nist.gov/22.16.E23 Encodings: TeX, pMML, png See also: Annotations for §22.16(ii), §22.16(ii), §22.16 and Ch.22

In Equations (22.16.24)–(22.16.26), .

22.16.24
ⓘ Symbols: : Jacobi’s epsilon function, : Jacobian elliptic function, : Jacobian elliptic function, : Jacobian elliptic function, : differential of , : integral, : real and : modulus A&S Ref: 16.26.10 Referenced by: §22.16(ii) Permalink: http://dlmf.nist.gov/22.16.E24 Encodings: TeX, pMML, png See also: Annotations for §22.16(ii), §22.16(ii), §22.16 and Ch.22
22.16.25
ⓘ Symbols: : Jacobi’s epsilon function, : Jacobian elliptic function, : Jacobian elliptic function, : differential of , : integral, : real, : modulus and : complementary modulus A&S Ref: 16.26.11 Permalink: http://dlmf.nist.gov/22.16.E25 Encodings: TeX, pMML, png See also: Annotations for §22.16(ii), §22.16(ii), §22.16 and Ch.22
22.16.26
ⓘ Symbols: : Jacobi’s epsilon function, : Jacobian elliptic function, : Jacobian elliptic function, : Jacobian elliptic function, : differential of , : integral, : real and : modulus A&S Ref: 16.26.12 Referenced by: §22.16(ii) Permalink: http://dlmf.nist.gov/22.16.E26 Encodings: TeX, pMML, png See also: Annotations for §22.16(ii), §22.16(ii), §22.16 and Ch.22

Quasi-Addition and Quasi-Periodic Formulas

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Keywords:: Jacobi’s epsilon function, Legendre’s elliptic integrals, quasi-addition formula, quasi-periodicity, relations to other functions
See also:: Annotations for §22.16(ii), §22.16 and Ch.22

22.16.27
ⓘ Symbols: : Jacobi’s epsilon function, : Jacobian elliptic function, : real and : modulus Referenced by: §22.16(iii) Permalink: http://dlmf.nist.gov/22.16.E27 Encodings: TeX, pMML, png See also: Annotations for §22.16(ii), §22.16(ii), §22.16 and Ch.22

22.16.28
ⓘ Symbols: : Jacobi’s epsilon function, : Jacobian elliptic function, : Jacobian elliptic function, : Legendre’s complete elliptic integral of the first kind, : Legendre’s complete elliptic integral of the second kind, : real and : modulus Permalink: http://dlmf.nist.gov/22.16.E28 Encodings: TeX, pMML, png See also: Annotations for §22.16(ii), §22.16(ii), §22.16 and Ch.22

22.16.29
ⓘ Symbols: : Jacobi’s epsilon function, : Legendre’s complete elliptic integral of the first kind, : Legendre’s complete elliptic integral of the second kind, : real and : modulus Permalink: http://dlmf.nist.gov/22.16.E29 Encodings: TeX, pMML, png See also: Annotations for §22.16(ii), §22.16(ii), §22.16 and Ch.22

For see §19.2(ii).

Relation to Theta Functions

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Keywords:: Jacobi’s epsilon function, relation to theta functions, relations to other functions, theta functions
See also:: Annotations for §22.16(ii), §22.16 and Ch.22

22.16.30
ⓘ Symbols: : Jacobi’s epsilon function, : theta function, : Legendre’s complete elliptic integral of the first kind, : Legendre’s complete elliptic integral of the second kind, : derivative of with respect to , : nome, : real and : modulus Referenced by: §22.20(vi) Permalink: http://dlmf.nist.gov/22.16.E30 Encodings: TeX, pMML, png See also: Annotations for §22.16(ii), §22.16(ii), §22.16 and Ch.22

where . For see §20.2(i). For see §19.2(ii).

Relation to the Elliptic Integral

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Keywords:: Jacobi’s epsilon function, relation to Legendre’s elliptic integrals
See also:: Annotations for §22.16(ii), §22.16 and Ch.22

22.16.31
.
ⓘ Symbols: : Jacobi’s epsilon function, : Jacobi’s amplitude function, : Legendre’s complete elliptic integral of the first kind, : Legendre’s incomplete elliptic integral of the second kind, : real and : modulus Permalink: http://dlmf.nist.gov/22.16.E31 Encodings: TeX, pMML, png See also: Annotations for §22.16(ii), §22.16(ii), §22.16 and Ch.22

For see §19.2(ii). See also (22.16.14).

§22.16(iii) Jacobi’s Zeta Function

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Keywords:: Jacobi’s zeta function
Notes:: See Lawden (1989, pp. 65–67), Whittaker and Watson (1927, pp. 517–522).
Permalink:: http://dlmf.nist.gov/22.16.iii
See also:: Annotations for §22.16 and Ch.22

Definition

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Keywords:: Jacobi’s zeta function, Legendre’s elliptic integrals, definition, relations to other functions
See also:: Annotations for §22.16(iii), §22.16 and Ch.22

With and as in §19.2(ii) and ,

22.16.32
ⓘ Defines: : Jacobi’s zeta function Symbols: : Jacobi’s epsilon function, : Legendre’s complete elliptic integral of the first kind, : Legendre’s complete elliptic integral of the second kind, : real and : modulus Referenced by: §22.20(vi) Permalink: http://dlmf.nist.gov/22.16.E32 Encodings: TeX, pMML, png See also: Annotations for §22.16(iii), §22.16(iii), §22.16 and Ch.22

See Figure 22.16.3. (Sometimes in the literature is denoted by .)

Properties

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Keywords:: Jacobi’s zeta function, quasi-addition formula
See also:: Annotations for §22.16(iii), §22.16 and Ch.22

satisfies the same quasi-addition formula as the function , given by (22.16.27). Also,

22.16.33
ⓘ Symbols: : Jacobi’s zeta function, : Jacobian elliptic function, : Jacobian elliptic function, : Legendre’s complete elliptic integral of the first kind, : real and : modulus Permalink: http://dlmf.nist.gov/22.16.E33 Encodings: TeX, pMML, png See also: Annotations for §22.16(iii), §22.16(iii), §22.16 and Ch.22

22.16.34
ⓘ Symbols: : Jacobi’s zeta function, : Legendre’s complete elliptic integral of the first kind, : real and : modulus Permalink: http://dlmf.nist.gov/22.16.E34 Encodings: TeX, pMML, png See also: Annotations for §22.16(iii), §22.16(iii), §22.16 and Ch.22