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→Fast computation: c_0 is used nowhere in the sequence; added jacobi zn result |
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Line 772:
Initialize
:<math>a_0=1,\, b_0=\sqrt{1-m
where <math>0<m<1</math>.
Define
Line 782:
:<math>\varphi_{n-1}=\frac{1}{2}\left(\varphi_n+\arcsin \left(\frac{c_n}{a_n}\sin \varphi_n\right)\right)</math>
for <math>n\ge 1</math>, then
:<math>\operatorname{am}(u,m)=\varphi_0,\quad \operatorname{zn}(u,m)=\sum_{n=1}^N c_n\sin\varphi_n</math>
as <math>N\to\infty</math>. This is notable for its rapid convergence. It is then trivial to compute all Jacobi elliptic functions from the Jacobi amplitude <math>\operatorname{am}</math> on the real line.<ref group="note">For the <math>\operatorname{dn}</math> function,
<math>\operatorname{dn}(u,m)=\frac{\operatorname{cn}(u,m)}{\operatorname{sn}(K(m)-u,m)}</math> can be used.</ref>
In conjunction with the addition theorems for elliptic functions (which hold for complex numbers in general) and the Jacobi transformations, the method of computation described above can be used to compute all Jacobi elliptic functions in the whole complex plane.
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