Jacobi Elliptic Functions - Relations Between Squares of The Functions

Relations Between Squares of The Functions

-\operatorname{dn}^2(u)+m_1= -m\;\operatorname{cn}^2(u) = m\;\operatorname{sn}^2(u)-m
-m_1\;\operatorname{nd}^2(u)+m_1= -mm_1\;\operatorname{sd}^2(u) = m\;\operatorname{cd}^2(u)-m
m_1\;\operatorname{sc}^2(u)+m_1= m_1\;\operatorname{nc}^2(u) = \operatorname{dc}^2(u)-m
\operatorname{cs}^2(u)+m_1=\operatorname{ds}^2(u)=\operatorname{ns}^2(u)-m

where m + m1 = 1 and m = k2.

Additional relations between squares can be obtained by noting that pq2 ยท qp2 = 1 and that pq = pr / qr where p, q, r are any of the letters s, c, d, n and ss = cc = dd = nn = 1.

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