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Symmetric Angle Doubling Formulas for JacobiSN | CN | DN .

Posted 5 months ago
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Doubly-Periodic function JacobiSN is a generalization of the Sin function. It fully determines the time parameterization of a genus one Riemann torus, the natural next step up from the genus zero hyperboloid. Harold Edwards discusses this fact in "A normal form for elliptic curves"; however, Edwards uses a slightly different notation, with $\psi(t)$, or more closely $\phi(t)$, as the standard, two-pole elliptic function. In Part III.21, page 419, Edwards gives angle doubling formula for $\phi(2t)$, which immediately allows derivation of a similar expression for $\text{sn}(2t)$. After slightly more analysis, we complete the following set of three: \begin{eqnarray} \text{sn}(2t) &=& \frac{2\;\text{cd}(t)\;\text{sn}(t)}{\text{cd}(t)^2+\text{sn}(t)^2} &=& \frac{2\;\psi(t')\;\phi(t')}{\psi(t')^2+\phi(t')^2}, \newline \text{cn}(2t) &=& \frac{\text{cd}(t)^2-\text{sn}(t)^2}{\text{cd}(t)^2+\text{sn}(t)^2} &=& \frac{\psi(t')^2-\phi(t')^2}{\psi(t')^2+\phi(t')^2}, \newline \text{dn}(2t) &=& \frac{1-m\;\text{cd}(t)^2 \;\text{sn}(t)^2}{\text{cd}(t)^2+\text{sn}(t)^2} &=& \frac{1-\psi(t')^2 \phi(t')^2}{1 + \psi(t')^2 \phi(t')^2}, \end{eqnarray}

with modulus $m=k^2=a^4$ and $t =2 K(\alpha) t'$. These formulae generalize the angle doubling formula of sine and cosine, and are well-known in some form or another. Comparing with Mathworld: Jacobi Elliptic Functions, Eq. 63-65, formulation in terms of $\text{sn}(t)$ and $\text{cd}(t)$ has the advantage of symmetry or antisymmetry whenever transposing $\text{sn}(t) \leftrightarrow \text{cd}(t)$. In Edwards normal form, this amounts to invariant transposition of $x$ and $y$ variables.

The derivation will soon be available, but for now, here are a few tests that should help to quell doubt if these expressions are indeed correct:

SNCNDN[m_] := {JacobiSN[t, m], JacobiCN[t, m], JacobiDN[t, m]}
AngleDoubleSNCNDN[m_] := {Divide[
   2 JacobiCD[t, m] JacobiSN[t, m],
   JacobiCD[t, m]^2 + JacobiSN[t, m]^2],
  Divide[JacobiCD[t, m]^2 - JacobiSN[t, m]^2,
   JacobiCD[t, m]^2 + JacobiSN[t, m]^2],
  Divide[1 - m JacobiCD[t, m]^2 JacobiSN[t, m]^2,
   JacobiCD[t, m]^2 + JacobiSN[t, m]^2]}

Test[m_] := With[{t0 = RandomComplex[{0, 4*EllipticK[m] + 2*I*EllipticK[1-m]}]},
  Subtract[AngleDoubleSNCNDN[m] /. t -> t0,
   SNCNDN[m] /. t -> 2*t0]]

FullSimplify[AngleDoubleSNCNDN[0] - (SNCNDN[0] /. t -> 2 t)]
FullSimplify[AngleDoubleSNCNDN[1] - (SNCNDN[1] /. t -> 2 t)]
{Mean[ReIm@#], StandardDeviation[ReIm@#]} &@
 Flatten[Table[Test[RandomReal[{0, 1}]], 100]]

With all outputs equal to zero, or approximately zero. If this is not enough, the rules above are essentially those found in Explicit Formulas Database, with $\text{dn}$ multiplied through numerator and denominator. Computer algebra proofs are available at EFD (though in sage language).

Posted 5 months ago

I will note that these duplication formulae for $\operatorname{sn}$, $\operatorname{cn}$, $\operatorname{dn}$ can also be respectively derived from the addition formulae 22.14, 22.16, and 22.18 in the DLMF, where Lawden's book is given as one of the references.

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