Jacobi's Imaginary Transformation

For Jacobi Elliptic Functions $\mathop{\rm sn}\nolimits u$, $\mathop{\rm cn}\nolimits u$, and $\mathop{\rm dn}\nolimits u$,

$$\mathop{\rm sn}\nolimits (iu,k) = i {\mathop{\rm sn}\nolimits (u,k')\over\mathop{\rm cn}\nolimits (u,k')}$$

$$\mathop{\rm cn}\nolimits (iu,k) = {1\over\mathop{\rm cn}\nolimits (u,k')}$$

$$\mathop{\rm dn}\nolimits (iu,k) = {\mathop{\rm dn}\nolimits (u,k')\over\mathop{\rm cn}\nolimits (u,k')}$$

(Abramowitz and Stegun 1972, Whittaker and Watson 1990).

See also Jacobi Elliptic Functions

References

Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 592 and 595, 1972.

Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge University Press, p. 505, 1990.