epsilon function

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1: 22.16 Related Functions

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§22.16(ii) Jacobi’s Epsilon Function

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Integral Representations

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Quasi-Addition and Quasi-Periodic Formulas

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Relation to Theta Functions

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Figure 22.16.2: Jacobi’s epsilon function $ℰ(x,k)$ for $0\le x\le 10\pi$ and $k=0.4,0.7,0.99,0.999999$. … Magnify

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2: 22.1 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 $\mathrm{Z}\left(x|k\right)$. … ►

3: 33.14 Definitions and Basic Properties

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§33.14(ii) Regular Solution $f(ϵ,\mathrm{\ell };r)$

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§33.14(iii) Irregular Solution $h(ϵ,\mathrm{\ell };r)$

►For nonzero values of $ϵ$ and $r$ the function $h(ϵ,\mathrm{\ell };r)$ is defined by … ►

§33.14(iv) Solutions $s(ϵ,\mathrm{\ell };r)$ and $c(ϵ,\mathrm{\ell };r)$

… ►The function $s(ϵ,\mathrm{\ell };r)$ has the following properties: …

4: 22.21 Tables

§22.21 Tables

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5: 22.18 Mathematical Applications

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Ellipse

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22.18.3 $$l(u)=aℰ(u,k),$$

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Symbols:

$ℰ(x,k)$: Jacobi’s epsilon function, $k$: modulus, $a$: real, $u$: variable and $l(r)$: arc length

Permalink:

http://dlmf.nist.gov/22.18.E3

Encodings:

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See also:

Annotations for §22.18(i), §22.18(i), §22.18 and Ch.22

►where $ℰ(u,k)$ is Jacobi’s epsilon function (§22.16(ii)). …

6: 33.18 Limiting Forms for Large $\mathrm{\ell }$

§33.18 Limiting Forms for Large $\mathrm{\ell }$

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7: 33.15 Graphics

§33.15 Graphics

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§33.15(i) Line Graphs of the Coulomb Functions $f(ϵ,\mathrm{\ell };r)$ and $h(ϵ,\mathrm{\ell };r)$

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§33.15(ii) Surfaces of the Coulomb Functions $f(ϵ,\mathrm{\ell };r)$, $h(ϵ,\mathrm{\ell };r)$, $s(ϵ,\mathrm{\ell };r)$, and $c(ϵ,\mathrm{\ell };r)$

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Figure 33.15.9: $h(ϵ,\mathrm{\ell };r)$ with . Magnify 3D Help

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8: 33.20 Expansions for Small $|ϵ|$

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§33.20(i) Case $ϵ=0$

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§33.20(ii) Power-Series in $ϵ$ for the Regular Solution

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§33.20(iii) Asymptotic Expansion for the Irregular Solution

… ►where $A(ϵ,\mathrm{\ell })$ is given by (33.14.11), (33.14.12), and … ►

§33.20(iv) Uniform Asymptotic Expansions

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9: 33.17 Recurrence Relations and Derivatives

§33.17 Recurrence Relations and Derivatives

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33.17.1 $$(\mathrm{\ell }+1)rf(ϵ,\mathrm{\ell }-1;r)-(2\mathrm{\ell }+1)\left(\mathrm{\ell }(\mathrm{\ell }+1)-r\right)f(ϵ,\mathrm{\ell };r)+\mathrm{\ell }\left(1+{(\mathrm{\ell }+1)}^2ϵ\right)rf(ϵ,\mathrm{\ell }+1;r)=0,$$

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Symbols:

$f(ϵ,\mathrm{\ell };r)$: regular Coulomb function, $\mathrm{\ell }$: nonnegative integer, $r$: real variable and $ϵ$: real parameter

Permalink:

http://dlmf.nist.gov/33.17.E1

Encodings:

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See also:

Annotations for §33.17 and Ch.33

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33.17.2 $$(\mathrm{\ell }+1)\left(1+{\mathrm{\ell }}^2ϵ\right)rh(ϵ,\mathrm{\ell }-1;r)-(2\mathrm{\ell }+1)\left(\mathrm{\ell }(\mathrm{\ell }+1)-r\right)h(ϵ,\mathrm{\ell };r)+\mathrm{\ell }rh(ϵ,\mathrm{\ell }+1;r)=0,$$

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Symbols:

$h(ϵ,\mathrm{\ell };r)$: irregular Coulomb function, $\mathrm{\ell }$: nonnegative integer, $r$: real variable and $ϵ$: real parameter

Permalink:

http://dlmf.nist.gov/33.17.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §33.17 and Ch.33

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33.17.3 $$(\mathrm{\ell }+1)r{f}^{\prime }(ϵ,\mathrm{\ell };r)=\left({(\mathrm{\ell }+1)}^2-r\right)f(ϵ,\mathrm{\ell };r)-\left(1+{(\mathrm{\ell }+1)}^2ϵ\right)rf(ϵ,\mathrm{\ell }+1;r),$$

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Symbols:

$f(ϵ,\mathrm{\ell };r)$: regular Coulomb function, $\mathrm{\ell }$: nonnegative integer, $r$: real variable and $ϵ$: real parameter

Permalink:

http://dlmf.nist.gov/33.17.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §33.17 and Ch.33

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33.17.4 $$(\mathrm{\ell }+1)r{h}^{\prime }(ϵ,\mathrm{\ell };r)=\left({(\mathrm{\ell }+1)}^2-r\right)h(ϵ,\mathrm{\ell };r)-rh(ϵ,\mathrm{\ell }+1;r).$$

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Symbols:

$h(ϵ,\mathrm{\ell };r)$: irregular Coulomb function, $\mathrm{\ell }$: nonnegative integer, $r$: real variable and $ϵ$: real parameter

Permalink:

http://dlmf.nist.gov/33.17.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §33.17 and Ch.33

10: 22.20 Methods of Computation

… ►Jacobi’s epsilon function can be computed from its representation (22.16.30) in terms of theta functions and complete elliptic integrals; compare §20.14. … …