Jacobi Elliptic Functions

The Jacobi elliptic functions are standard forms of Elliptic Functions. The three basic functions are denoted $\mathop{\rm cn}\nolimits (u,k)$, $\mathop{\rm dn}\nolimits (u,k)$, and $\mathop{\rm sn}\nolimits (u,k)$, where $k$ is known as the Modulus. In terms of Theta Functions,

$$\mathop{\rm sn}\nolimits (u,k) = {\vartheta_3\over\vartheta_4} {\vartheta_1({u\vartheta_3}^{-2})\over\vartheta_4({u\vartheta_3}^{-2})}$$ (1)

$$\mathop{\rm cn}\nolimits (u,k) = {\vartheta_4\over\vartheta_2} {\vartheta_2({u\vartheta_3}^{-2})\over\vartheta_4({u\vartheta_3}^{-2})}$$ (2)

$$\mathop{\rm dn}\nolimits (u,k) = {\vartheta_4\over\vartheta_3} {\vartheta_3({u\vartheta_3}^{-2})\over\vartheta_4({u\vartheta_3}^{-2})}$$ (3)

(Whittaker and Watson 1990, p. 492), where $\vartheta_i\equiv\vartheta_i(0)$ (Whittaker and Watson 1990, p. 464). Ratios of Jacobi elliptic functions are denoted by combining the first letter of the Numerator elliptic function with the first of the Denominator elliptic function. The multiplicative inverses of the elliptic functions are denoted by reversing the order of the two letters. These combinations give a total of 12 functions: cd, cn, cs, dc, dn, ds, nc, nd, ns, sc, sd, and sn. The Amplitude $\phi$ is defined in terms of $\mathop{\rm sn}\nolimits u$ by

$$y=\sin\phi=\mathop{\rm sn}\nolimits (u,k).$$ (4)

The $k$ argument is often suppressed for brevity so, for example, $\mathop{\rm sn}\nolimits (u,k)$ can be written $\mathop{\rm sn}\nolimits u$.

The Jacobi elliptic functions are periodic in $K(k)$ and $K'(k)$ as

$$\mathop{\rm sn}\nolimits (u+2mK+2niK',k)=(-1)^m\mathop{\rm sn}\nolimits (u,k)$$ (5)

$$\mathop{\rm cn}\nolimits (u+2mK+2niK',k)=(-1)^{m+n}\mathop{\rm cn}\nolimits (u,k)$$ (6)

$$\mathop{\rm dn}\nolimits (u+2mK+2niK',k)=(-1)^n\mathop{\rm dn}\nolimits (u,k),$$ (7)

where $K(k)$ is the complete Elliptic Integral of the First Kind, $K'(k)\equiv K(k')$, and $k'\equiv\sqrt{1-k^2}$ (Whittaker and Watson 1990, p. 503).

The $\mathop{\rm cn}\nolimits x$, $\mathop{\rm dn}\nolimits x$, and $\mathop{\rm sn}\nolimits x$ functions may also be defined as solutions to the differential equations

$${d^2y\over dx^2}=-(1+k^2)y+2k^2y^3$$ (8)

$${d^2y\over dx^2}=-(1-2k^2)y-2k^2y^3$$ (9)

$${d^2y\over dx^2}=(2-k^2)y-2y^3.$$ (10)

The standard Jacobi elliptic functions satisfy the identities

$$\mathop{\rm sn}\nolimits ^2 u+\mathop{\rm cn}\nolimits ^2 u = 1$$ (11)

$$k^2\mathop{\rm sn}\nolimits ^2 u+\mathop{\rm dn}\nolimits ^2 u = 1$$ (12)

$$k^2\mathop{\rm cn}\nolimits ^2 u+k'^2 = \mathop{\rm dn}\nolimits ^2 u$$ (13)

$$\mathop{\rm cn}\nolimits ^2 u+k'^2\mathop{\rm sn}\nolimits ^2 u = \mathop{\rm dn}\nolimits ^2 u.$$ (14)

Special values are

$$\mathop{\rm cn}\nolimits (0) = 1$$ (15)

$$\mathop{\rm dn}\nolimits (0) = 1$$ (16)

$$\mathop{\rm cn}\nolimits (K) = 0$$ (17)

$$\mathop{\rm dn}\nolimits (K) = k'\equiv\sqrt{1-k^2},$$ (18)

$$\mathop{\rm sn}\nolimits (K) = 1,$$ (19)

where $K=K(k)$ is a complete Elliptic Integral of the First Kind and $k'$ is the complementary Modulus (Whittaker and Watson 1990, pp. 498-499).

In terms of integrals,

$$u = \int_0^{\mathop{\rm sn}\nolimits u} (1-t^2)^{1-/2}(1-k^2t^2)^{-1/2}\,dt$$ (20)

$$= \int_{\mathop{\rm ns}\nolimits u}^\infty (t^2-1)^{-1/2}(t^2-l^2)^{-1/2}\,dt$$ (21)

$$= \int_{\mathop{\rm cn}\nolimits u}^1 (1-t^2)^{-1/2}(k'^2+k^2t^2)^{-1/2}\,dt$$ (22)

$$= \int_1^{\mathop{\rm nc}\nolimits u} (t^2-1)^{-1/2}(k'^2t^2+k^2)^{-1/2}\,dt$$ (23)

$$= \int_{\mathop{\rm dn}\nolimits u}^1 (1-t^2)^{-1/2}(t^2-k'^2)^{-1/2}\,dt$$ (24)

$$= \int_1^{\mathop{\rm nd}\nolimits u}(t^2-1)^{-1/2}(1-k'^2t^2)^{-1/2}\,dt$$ (25)

$$= \int_0^{\mathop{\rm sc}\nolimits u} (1+t^2)^{-1/2}(1+k'^2t^2)^{-1/2}\,dt$$ (26)

$$= \int_{\mathop{\rm cs}\nolimits u}^\infty (t^2+1)^{-1/2}(t^2+k'^2)^{-1/2}\,dt$$ (27)

$$= \int_0^{\mathop{\rm sd}\nolimits u}(1-k'^2t^2)^{-1/2}(1+k^2t^2)^{-1/2}\,dt$$ (28)

$$= \int_{\mathop{\rm ds}\nolimits u}^\infty (t^2-k'^2)^{-1/2}(t^2+k^2)^{-1/2}\,dt$$ (29)

$$= \int_1^{\mathop{\rm cd}\nolimits u} (1-t^2)^{-1/2}(1-k^2t^2)^{-1/2}\,dt$$ (30)

$$= \int_{\mathop{\rm dc}\nolimits u}^1 (t^2-1)^{-1/2}(t^2-k^2)^{-1/2}\,dt$$ (31)

(Whittaker and Watson 1990, p. 494).

Jacobi elliptic functions addition formulas include

$$\mathop{\rm sn}\nolimits (u+v) = {\mathop{\rm sn}\nolimits u\mathop{\rm cn}\nolimits v\mathop{\rm ... ...olimits u\over 1-k^2\mathop{\rm sn}\nolimits ^2 u\mathop{\rm sn}\nolimits ^2 v}$$ (32)

$$\mathop{\rm cn}\nolimits (u+v) = {\mathop{\rm cn}\nolimits u\mathop{\rm cn}\nolimits v-\mathop{\rm... ...olimits v\over 1-k^2\mathop{\rm sn}\nolimits ^2 u\mathop{\rm sn}\nolimits ^2 v}$$ (33)

$$\mathop{\rm dn}\nolimits (u+v) = {\mathop{\rm dn}\nolimits u\mathop{\rm dn}\nolimits v-k^2\mathop{... ...limits v\over 1-k^2\mathop{\rm sn}\nolimits ^2 u\mathop{\rm sn}\nolimits ^2 v}.$$ (34)

Extended to integral periods,

$$\mathop{\rm sn}\nolimits (u+K) = {\mathop{\rm cn}\nolimits u\over\mathop{\rm dn}\nolimits u}$$ (35)

$$\mathop{\rm cn}\nolimits (u+K) = -{k'\mathop{\rm sn}\nolimits u\over\mathop{\rm dn}\nolimits u}$$ (36)

$$\mathop{\rm dn}\nolimits (u+K) = {k'\over\mathop{\rm dn}\nolimits u}$$ (37)

$$\mathop{\rm sn}\nolimits (u+2K) = -\mathop{\rm sn}\nolimits u$$ (38)

$$\mathop{\rm cn}\nolimits (u+2K) = -\mathop{\rm cn}\nolimits u$$ (39)

$$\mathop{\rm dn}\nolimits (u+2K) = \mathop{\rm dn}\nolimits u$$ (40)

For Complex arguments,

$$\mathop{\rm sn}\nolimits (u+iv)={\mathop{\rm sn}\nolimits (u... ...hop{\rm dn}\nolimits ^2(u,k)\mathop{\rm sn}\nolimits ^2(v,k')}$$ (41)

$$\mathop{\rm cn}\nolimits (u+iv)={\mathop{\rm cn}\nolimits (u... ...hop{\rm dn}\nolimits ^2(u,k)\mathop{\rm sn}\nolimits ^2(v,k')}$$ (42)

$$\mathop{\rm dn}\nolimits (u+iv)={\mathop{\rm dn}\nolimits (u... ...op{\rm dn}\nolimits ^2(u,k)\mathop{\rm sn}\nolimits ^2(v,k')}.$$ (43)

Derivatives of the Jacobi elliptic functions include

$${d\mathop{\rm sn}\nolimits u\over du} = \mathop{\rm cn}\nolimits u\mathop{\rm dn}\nolimits u$$ (44)

$${d\mathop{\rm cn}\nolimits u\over du} = \mathop{\rm sn}\nolimits u\mathop{\rm dn}\nolimits u$$ (45)

$${d\mathop{\rm dn}\nolimits u\over du} = -k^2\mathop{\rm sn}\nolimits u\mathop{\rm cn}\nolimits u.$$ (46)

Double-period formulas involving the Jacobi elliptic functions include

$$\mathop{\rm sn}\nolimits (2u) = {2\mathop{\rm sn}\nolimits u\mathop{\rm cn}\nolimits u\mathop{\rm dn}\nolimits u\over 1-k^2 \mathop{\rm sn}\nolimits ^4 u}$$ (47)

$$\mathop{\rm cn}\nolimits (2u) = {1-2\mathop{\rm sn}\nolimits ^2 u+k^2\mathop{\rm sn}\nolimits ^4 u\over 1-k^2\mathop{\rm sn}\nolimits ^4 u}$$ (48)

$$\mathop{\rm dn}\nolimits (2u) = {1-2k^2\mathop{\rm sn}\nolimits ^2 u+k^2\mathop{\rm sn}\nolimits ^4 u\over 1-k^2\mathop{\rm sn}\nolimits ^4 u}.$$ (49)

Half-period formulas involving the Jacobi elliptic functions include

$$\mathop{\rm sn}\nolimits ({\textstyle{1\over 2}}K) = {1\over\sqrt{1+k'}}$$ (50)

$$\mathop{\rm cn}\nolimits ({\textstyle{1\over 2}}K) = \sqrt{k'\over 1+k'}$$ (51)

$$\mathop{\rm dn}\nolimits ({\textstyle{1\over 2}}K) = \sqrt{k'}.$$ (52)

Squared formulas include

$$\mathop{\rm sn}\nolimits ^2 u = {1-\mathop{\rm cn}\nolimits (2u)\over 1+\mathop{\rm dn}\nolimits (2u)}$$ (53)

$$\mathop{\rm cn}\nolimits ^2 u = {\mathop{\rm dn}\nolimits (2u)+\mathop{\rm cn}\nolimits (2u)\over 1+\mathop{\rm dn}\nolimits (2u)}$$ (54)

$$\mathop{\rm dn}\nolimits ^2 u = {\mathop{\rm dn}\nolimits (2u)+\mathop{\rm cn}\nolimits (2u)\over 1+\mathop{\rm cn}\nolimits (2u)}.$$ (55)

See also Amplitude, Elliptic Function, Jacobi's Imaginary Transformation, Theta Function, Weierstraß Elliptic Function

References

Abramowitz, M. and Stegun, C. A. (Eds.). ``Jacobian Elliptic Functions and Theta Functions.'' Ch. 16 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 567-581, 1972.

Bellman, R. E. A Brief Introduction to Theta Functions. New York: Holt, Rinehart and Winston, 1961.

Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, p. 433, 1953.

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. ``Elliptic Integrals and Jacobi Elliptic Functions.'' §6.11 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 254-263, 1992.

Spanier, J. and Oldham, K. B. ``The Jacobian Elliptic Functions.'' Ch. 63 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 635-652, 1987.

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