A summary of the definitions and properties of the Jacobian elliptic functions

May 13, 1999

We adopt the notations and definitions given in the books:
I.S. Grandshteyn and I.M. Ryzhik, it Table of Integrals, Series and Products, ed. A. Jeffrey (5^th edition), Academic Press, 1994.
J. Spanier and K.B. Oldham, An Atlas of Functions, Hemisphere Publishing Corporation, 1987

The 12 Jacobian elliptic functions are denoted by two letters taken from the quartet s, c, d, n and may be classified into four groups, each with three members, according to the second letter of the function's name. Thus, functions cd, ds and ns are said to be copolar: they all posess a pole of type "s".

The most popular Jacobian elliptic functions is a copolar trio of sine amplitude elliptic function - sn(x,k), cosine amplitude elliptic function - cn(x,k), and delta amplitude elliptic function - dn(x,k). These functions may be defined via the inverse of the incomplete elliptic integrals as follows:

$$F(sn(x,k),k)= x = \int_{0}^{sn(x,k)} \frac{dt}{\sqrt{(1-t^2)(1-k^2t^2)}}; \nonumber \$$

$$x = \int_{1}^{cn(x,k)} \frac{dt}{\sqrt{(1-t^2)(k'^2+k^2t^2)}}; \nonumber \$$

 

$$x = \int_{1}^{dn(x,k)} \frac{dt}{\sqrt{(1-t^2)(t^2-k'^2)}}. \eqnum{A0}$$ (1)

The second argument of the functions k - is a modulus of the elliptic function and $k'=\sqrt{1-k^2}$ is a complimentary modulus. The eight remaining Jacobian elliptic functions can be conveniently defined via the general identity relations

 

$$fg(x,k)=\frac{fe(k,x)}{ge(k,x)} \hspace{0.8in} e,f,g=s,c,d,n \eqnum{A1}$$ (2)

where ff(k,x) is interpreted as unity. The functions sn(x,k), cn(x,k), and dn(x,k) are real valued when their argument is real, and modulus k is either real or purely imaginary. They are also periodic:

$$sn(4n{\bf K}(k)+x,k) = sn(x,k) \nonumber \$$

$$cn(4n{\bf K}(k)+x,k) = cn(x,k) \nonumber \$$

 

$$dn(2n{\bf K}(k)+x,k) = dn(x,k) \eqnum{A2}$$ (3)

where n=0,1,2,..., and ${\bf K}(k)$ is a complete elliptic integral of the first kind

 

$${\bf K}(k) = \int_{0}^{1 } \frac{dt}{\sqrt{(1-t^2)(1-k^2t^2)}}. \eqnum{A3}$$ (4)

The following general functional relations hold for all (complex or real) types of arguments and moduli:

 

$$dn^2(x,k)+k^2sn^2(x,k)=1 \eqnum{A4}$$ (5)

The following expressions relating the functions of imaginary and real moduli hold:

$$sn(x,ik) = \frac{1}{\sqrt{1+k^2}} sd (x \sqrt{1+k^2},\frac{k}{\sqrt{1+k^2}}) \nonumber \$$

$$cn(x,ik) = cd (x \sqrt{1+k^2},\frac{k}{\sqrt{1+k^2}}) \nonumber \$$

 

$$dn(x,ik) = nd (x \sqrt{1+k^2},\frac{k}{\sqrt{1+k^2}}). \eqnum{A5}$$ (6)

Addition/substraction formulae for a copolar trio of sine, cosine and delta amplitude Jacobian functions are

$$sn (x \pm y;k) = \frac {sn (x;k) cn(y;k) dn(y;k) \pm cn(x;k) dn(x;k) sn(y;k)} {1-k^2 sn^2(x;k) sn^2(y;k)}; \nonumber \$$

$$cn (x \pm y;k) = \frac {cn (x;k) cn(y;k) \pm sn(x;k) dn(x;k) sn(y;k) dn(y;k)} {1-k^2 sn^2(x;k) sn^2(y;k)}; \nonumber \$$

 

$$dn (x \pm y;k) = \frac {dn (x;k) dn(y;k) \mp k^2 sn(x;k) cn(x;k) sn(y;k) cn(y;k)} {1-k^2 sn^2(x;k) sn^2(y;k)}.\eqnum{A6}$$ (7)

For the practical evaluation of Jacobian functions of modulus exceeding unity the so-called Jacobi's real transformations can be used:

$$sn(x,\frac{1}{k})=k sn(\frac{x}{k},k) \nonumber \$$

$$cn(x,\frac{1}{k})=dn(\frac{x}{k},k) \nonumber \$$

 

$$dn(x,\frac{1}{k})=cn(\frac{x}{k},k) \eqnum{A7}$$ (8)

There also exist the Jacobi's imaginary transformations for the transformation of an elliptic function of imaginary argument to one of real argument. Below we cite the one relevant for our purpose:

 

$$sn(ix,k)=i sc(x,k') \eqnum{A8}$$ (9)

It is also worth noting that all Jacobian elliptic functions reduce to a trigonometric function, or to unity, when k=0. Similarly, all Jacobian elliptic functions reduce to a hyperbolic function, or to unity, when k=1. For instance:

$$sn(x,0) = \sin{x}; \hspace{1.in} sn(x,1)=\tanh{x} \nonumber \$$

$$cn(x,0) = \cos(x); \hspace{1.in} cn(x,1)=\frac{1}{\cosh{x}} \nonumber \$$

 

$$dn(x,0) = 1; \hspace{1.in} dn(x,1)=\frac{1}{\cosh{x}} \eqnum{A9}$$ (10)