Weierstrass elliptic functions

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31: 31.2 Differential Equations

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Jacobi’s Elliptic Form

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Weierstrass’s Form

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$\zeta =\mathrm{i}{K}^{\prime }+\xi {(e_1-e_3)}^{1/2},$

… ►where $2{\omega }_1$ and $2{\omega }_3$ with $\mathrm{\Im }\left({\omega }_3/{\omega }_1\right)>0$ are generators of the lattice $𝕃$ for $\mathrm{\wp }\left(z|𝕃\right)$. … ►

31.2.10 $$w(\xi )={\left(\mathrm{\wp }\left(\xi \right)-e_3\right)}^{(1-2\gamma )/4}{\left(\mathrm{\wp }\left(\xi \right)-e_2\right)}^{(1-2\delta )/4}{\left(\mathrm{\wp }\left(\xi \right)-e_1\right)}^{(1-2ϵ)/4}W(\xi ),$$

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Symbols:

$\mathrm{\wp }\left(z\right)$ (= $\mathrm{\wp }\left(z|𝕃\right)$ = $\mathrm{\wp }(z;g_2,g_3)$): Weierstrass $\mathrm{\wp }$-function, $e_j$: Weierstrass lattice roots, $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $ϵ$: real or complex parameter, $𝕃$: lattice and $W(\xi )$: solution

Permalink:

http://dlmf.nist.gov/31.2.E10

Encodings:

pMML, png, TeX

See also:

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

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32: 23.19 Interrelations

§23.19 Interrelations

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23.19.1 $$\lambda \left(\tau \right)=16{\left(\frac{{\eta }^2\left(2\tau \right)\eta \left(\frac{1}{2}\tau \right)}{{\eta }^3\left(\tau \right)}\right)}^8,$$

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Symbols:

$\eta \left(\tau \right)$: Dedekind’s eta function (or Dedekind modular function), $\lambda \left(\tau \right)$: elliptic modular function and $\tau$: complex variable

Permalink:

http://dlmf.nist.gov/23.19.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §23.19 and Ch.23

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23.19.2 $$J\left(\tau \right)=\frac{4}{27}\frac{{\left(1-\lambda \left(\tau \right)+{\lambda }^2\left(\tau \right)\right)}^3}{{\left(\lambda \left(\tau \right)\left(1-\lambda \left(\tau \right)\right)\right)}^2},$$

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Symbols:

$J\left(\tau \right)$: Klein’s complete invariant, $\lambda \left(\tau \right)$: elliptic modular function and $\tau$: complex variable

Permalink:

http://dlmf.nist.gov/23.19.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §23.19 and Ch.23

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23.19.3 $$J\left(\tau \right)=\frac{g_2^3}{g_2^3-27g_3^2},$$

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Symbols:

$J\left(\tau \right)$: Klein’s complete invariant, $g_j$: Weierstrass lattice invariants $g_2$, $g_3$ , $𝕃$: lattice and $\tau$: complex variable

Permalink:

http://dlmf.nist.gov/23.19.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §23.19 and Ch.23

►where $g_2,g_3$ are the invariants of the lattice $𝕃$ with generators $1$ and $\tau$; see §23.3(i). …

33: Errata

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This subsection has been significantly updated. In particular, the following formulae have been corrected. Equation (19.25.35) has been replaced by

19.25.35 $$z+2\omega =\pm R_F(\mathrm{\wp }\left(z\right)-e_1,\mathrm{\wp }\left(z\right)-e_2,\mathrm{\wp }\left(z\right)-e_3),$$

in which the left-hand side $z$ has been replaced by $z+2\omega$ for some $2\omega \in 𝕃$, and the right-hand side has been multiplied by $\pm 1$. Equation (19.25.37) has been replaced by

19.25.37 $$\zeta \left(z+2\omega \right)+(z+2\omega )\mathrm{\wp }\left(z\right)=\pm 2R_G(\mathrm{\wp }\left(z\right)-e_1,\mathrm{\wp }\left(z\right)-e_2,\mathrm{\wp }\left(z\right)-e_3),$$

in which the left-hand side $\zeta \left(z\right)+z\mathrm{\wp }\left(z\right)$ has been replaced by $\zeta \left(z+2\omega \right)+(z+2\omega )\mathrm{\wp }\left(z\right)$ and the right-hand side has been multiplied by $\pm 1$. Equation (19.25.39) has been replaced by

19.25.39 $$\zeta \left({\omega }_j\right)+{\omega }_j{e}_j=2R_G(0,e_j-e_k,e_j-e_{\mathrm{\ell }}),$$

in which the left-hand side ${\eta }_j$ was replaced by $\zeta \left({\omega }_j\right)$, for some $2{\omega }_j\in 𝕃$ and $\mathrm{\wp }\left({\omega }_j\right)=e_j$. Equation (19.25.40) has been replaced by

19.25.40 $$z+2\omega =\pm \sigma \left(z\right)R_F({\sigma }_1^2(z),{\sigma }_2^2(z),{\sigma }_3^2(z)),$$

in which the left-hand side $z$ has been replaced by $z+2\omega$, and the right-hand side was multiplied by $\pm 1$. For more details see §19.25(vi).

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The Weierstrass lattice roots $e_j,$ were linked inadvertently as the base of the natural logarithm. In order to resolve this inconsistency, the lattice roots $e_j$, and lattice invariants $g_2$, $g_3$, now link to their respective definitions (see §§23.2(i), 23.3(i)).

Reported by Felix Ospald.

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The Weierstrass zeta function was incorrectly linked to the definition of the Riemann zeta function. However, to the eye, the function appeared correct. The link was corrected.

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These equations were rewritten with the modulus (second argument) of the Jacobian elliptic function defined explicitly in the preceding line of text.

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Originally a minus sign was missing in the entries for $\mathrm{cd}u$ and $\mathrm{dc}u$ in the second column (headed $z+K+i{K}^{\prime }$). The correct entries are $-k^{-1}\mathrm{ns}z$ and $-k\mathrm{sn}z$. Note: These entries appear online but not in the published print edition. More specifically, Table 22.4.3 in the published print edition is restricted to the three Jacobian elliptic functions $\mathrm{sn},\mathrm{cn},\mathrm{dn}$, whereas Table 22.4.3 covers all 12 Jacobian elliptic functions.

	$u$
	$z+K$	$z+K+\mathrm{i}{K}^{\prime }$	$z+\mathrm{i}{K}^{\prime }$	$z+2K$	$z+2K+2\mathrm{i}{K}^{\prime }$	$z+2\mathrm{i}{K}^{\prime }$
$\mathrm{⋮}$
$\mathrm{cd}u$	$-\mathrm{sn}z$	$-k^{-1}\mathrm{ns}z$	$k^{-1}\mathrm{dc}z$	$-\mathrm{cd}z$	$-\mathrm{cd}z$	$\mathrm{cd}z$
$\mathrm{⋮}$
$\mathrm{dc}u$	$-\mathrm{ns}z$	$-k\mathrm{sn}z$	$k\mathrm{cd}z$	$-\mathrm{dc}z$	$-\mathrm{dc}z$	$\mathrm{dc}z$
$\mathrm{⋮}$

Reported 2014-02-28 by Svante Janson.

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