Table 2 Limiting forms of Jacobi elliptic functions for \(m \rightarrow 1\) and \(m \rightarrow 0\).

\(\textrm{sn}\,u\)

\(\tanh u\)

\(\sin u\)

\(\textrm{dc}\,u\)

1

\(\sec u\)

\(\textrm{cn}\,u\)

\(\textrm{sech}\,u\)

\(\cos u\)

\(\textrm{nc}\,u\)

\(\cosh u\)

\(\sec u\)

\(\textrm{dn}\,u\)

\(\textrm{sech}\,u\)

1

\(\textrm{sc}\,u\)

\(\sinh u\)

\(\tan u\)

\(\textrm{cd}\,u\)

\(\cosh u\)

1

\(\textrm{ns}\,u\)

\(\coth u\)

\(\csc u\)

\(\textrm{sd}\,u\)

\(\sinh u\)

\(\sin u\)

\(\textrm{ds}\,u\)

\(\textrm{csch}\,u\)

\(\cot u\)

\(\textrm{nd}\,u\)

\(\cosh u\)

1

\(\textrm{cs}\,u\)

\(\textrm{csch}\,u\)

\(\cot u\)