Jacobi 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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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: 22.21 Tables

§22.21 Tables

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4: 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:

pMML, png, TeX

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)). …

5: 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. … …

6: 31.16 Mathematical Applications

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31.16.1 $${\mathrm{𝐻𝑝}}_{n,m}\left(x\right){\mathrm{𝐻𝑝}}_{n,m}\left(y\right)=\sum _{j=0}^n{A}_j{\sin }^{2j}\theta P_{n-j}^{(\gamma +\delta +2j-1,ϵ-1)}\left(\cos \left(2\theta \right)\right)P_j^{(\delta -1,\gamma -1)}\left(\cos \left(2\varphi \right)\right),$$

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Symbols:

${\mathrm{𝐻𝑝}}_{n,m}(a,q_{n,m};-n,\beta ,\gamma ,\delta ;z)$: Heun polynomials, $P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, $\cos z$: cosine function, $\sin z$: sine function, $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $ϵ$: real or complex parameter, $j$: nonnegative integer, $m$: nonnegative integer, $n$: nonnegative integer, $a$: complex parameter, $q$: real or complex parameter, $\beta$: real or complex parameter, $\theta$: angle, $\varphi$: angle, $x$: variable, $y$: variable and $A_j$: coefficients

Referenced by:

§31.16(ii), §31.16(ii)

Permalink:

http://dlmf.nist.gov/31.16.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §31.16(ii), §31.16 and Ch.31

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7: 18.15 Asymptotic Approximations

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18.15.6 $${(\sin \frac{1}{2}\theta )}^{\alpha +\frac{1}{2}}{(\cos \frac{1}{2}\theta )}^{\beta +\frac{1}{2}}{P}_n^{(\alpha ,\beta )}\left(\cos \theta \right)=\frac{\mathrm{\Gamma }\left(n+\alpha +1\right)}{2^{\frac{1}{2}}{\rho }^{\alpha }n!}\left({\theta }^{\frac{1}{2}}{J}_{\alpha }\left(\rho \theta \right)\sum _{m=0}^M\frac{A_m(\theta )}{{\rho }^{2m}}+{\theta }^{\frac{3}{2}}{J}_{\alpha +1}\left(\rho \theta \right)\sum _{m=0}^{M-1}\frac{B_m(\theta )}{{\rho }^{2m+1}}+{\epsilon }_M(\rho ,\theta )\right),$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, $\cos z$: cosine function, $!$: factorial (as in $n!$), $\sin z$: sine function, $m$: nonnegative integer, $n$: nonnegative integer, $\rho$ , ${\epsilon }_M(\rho ,\theta )$ , $A_m(\theta )$: coefficient and $B_m(\theta )$: coefficient

Permalink:

http://dlmf.nist.gov/18.15.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §18.15(i), §18.15 and Ch.18

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8: 31.7 Relations to Other Functions

§31.7 Relations to Other Functions

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§31.7(i) Reductions to the Gauss Hypergeometric Function

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§31.7(ii) Relations to Lamé Functions

►With $z={\mathrm{sn}}^2(\zeta ,k)$ and … ►

$\gamma =\delta =ϵ=\frac{1}{2},$

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9: 31.2 Differential Equations

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31.2.8 $$\frac{d^{2}w}{{d\zeta }^2}+\left((2\gamma -1)\frac{\mathrm{cn}\zeta \mathrm{dn}\zeta }{\mathrm{sn}\zeta }-(2\delta -1)\frac{\mathrm{sn}\zeta \mathrm{dn}\zeta }{\mathrm{cn}\zeta }-(2ϵ-1)k^2\frac{\mathrm{sn}\zeta \mathrm{cn}\zeta }{\mathrm{dn}\zeta }\right)\frac{dw}{d\zeta }+4k^2(\alpha \beta {\mathrm{sn}}^2\zeta -q)w=0.$$

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Symbols:

$\mathrm{cn}(z,k)$: Jacobian elliptic function, $\mathrm{dn}(z,k)$: Jacobian elliptic function, $\mathrm{sn}(z,k)$: Jacobian elliptic function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $ϵ$: real or complex parameter, $q$: real or complex parameter, $\alpha$: real or complex parameter, $\beta$: real or complex parameter, $\zeta$: change of variable and $k$: modulus

Referenced by:

§31.7(ii)

Permalink:

http://dlmf.nist.gov/31.2.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §31.2(iv), §31.2(iv), §31.2 and Ch.31

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10: 20.15 Tables

… ►Tables of Neville’s theta functions ${\theta }_s(x,q)$, ${\theta }_c(x,q)$, ${\theta }_d(x,q)$, ${\theta }_n(x,q)$ (see §20.1) and their logarithmic $x$-derivatives are given in Abramowitz and Stegun (1964, pp. 582–585) to 9D for $\epsilon ,\alpha =0(5^{\circ }){90}^{\circ }$, where (in radian measure) $\epsilon =x/{\theta }_3^2(0,q)=\pi x/(2K\left(k\right))$, and $\alpha$ is defined by (20.15.1). …