ellipj

Jacobi elliptic functions

Syntax

[SN,CN,DN] = ellipj(u,m)

Description

[SN,CN,DN] = ellipj(u,m) returns the Jacobi sn, cn, and dn elliptic functions evaluated for the corresponding elements of u and m. Inputs u and m must be the same size, or either u or m must be scalar.

Examples

Compute Jacobi Elliptic Functions

Compute the Jacobi elliptic functions for u = 0.75 and m = 0.5. Because these numbers are not symbolic objects, you get floating-point results.

[SN,CN,DN] = ellipj(0.75,0.5)

SN = 
0.6585

CN = 
0.7526

DN = 
0.8850

Compute the Jacobi elliptic functions for the same numbers converted to symbolic objects. For most symbolic (exact) numbers, ellipj returns results using the jacobiSN, jacobiCN, and jacobiDN functions.

[SN,CN,DN] = ellipj(sym(3/4),sym(1/2))

SN = 
$$\mathrm{sn}\text{<a href="matlab:doc symbolic/jacobiSN">jacobiSN</a>}\left(\frac{3}{4}|\frac{1}{2}\right)$$

CN = 
$$\mathrm{cn}\text{<a href="matlab:doc symbolic/jacobiCN">jacobiCN</a>}\left(\frac{3}{4}|\frac{1}{2}\right)$$

DN = 
$$\mathrm{dn}\text{<a href="matlab:doc symbolic/jacobiDN">jacobiDN</a>}\left(\frac{3}{4}|\frac{1}{2}\right)$$

Use vpa to approximate the symbolic results with floating-point numbers.

vpa([SN,CN,DN],10)

ans = $$\left(\begin{array}{ccc}0.6585147441& 0.7525678254& 0.8849741046\end{array}\right)$$

Compute Jacobi Elliptic Functions When m Is Not Between 0 and 1

If the argument m is not in [0 1], then convert that argument to a symbolic object before using ellipj.

[SN,CN,DN] = ellipj(1,sym(pi/2))

SN = 
$$\mathrm{sn}\text{<a href="matlab:doc symbolic/jacobiSN">jacobiSN</a>}\left(1|\frac{\pi }{2}\right)$$

CN = 
$$\mathrm{cn}\text{<a href="matlab:doc symbolic/jacobiCN">jacobiCN</a>}\left(1|\frac{\pi }{2}\right)$$

DN = 
$$\mathrm{dn}\text{<a href="matlab:doc symbolic/jacobiDN">jacobiDN</a>}\left(1|\frac{\pi }{2}\right)$$

Alternatively, use jacobiSN, jacobiCN, and jacobiDN to compute the Jacobi elliptic functions separately.

SN = jacobiSN(1,sym(pi/2))

SN = 
$$\mathrm{sn}\text{<a href="matlab:doc symbolic/jacobiSN">jacobiSN</a>}\left(1|\frac{\pi }{2}\right)$$

CN = jacobiCN(1,sym(pi/2))

CN = 
$$\mathrm{cn}\text{<a href="matlab:doc symbolic/jacobiCN">jacobiCN</a>}\left(1|\frac{\pi }{2}\right)$$

DN = jacobiDN(1,sym(pi/2))

DN = 
$$\mathrm{dn}\text{<a href="matlab:doc symbolic/jacobiDN">jacobiDN</a>}\left(1|\frac{\pi }{2}\right)$$

Compute Jacobi Elliptic Functions for Matrix Input

Call ellipj for symbolic matrix input. When the input arguments are matrices with the same size, ellipj computes the Jacobi elliptic functions for each element.

[SN,CN,DN] = ellipj(sym([-1 0; 1 1/2]),sym([1 pi/2; -1 0]))

SN = 
$$\left(\begin{array}{cc}-\tanh \left(1\right)& 0\\ \mathrm{sn}\text{<a href="matlab:doc symbolic/jacobiSN">jacobiSN</a>}\left(1|-1\right)& \sin \left(\frac{1}{2}\right)\end{array}\right)$$

CN = 
$$\left(\begin{array}{cc}\frac{1}{\cosh \left(1\right)}& 1\\ \mathrm{cn}\text{<a href="matlab:doc symbolic/jacobiCN">jacobiCN</a>}\left(1|-1\right)& \cos \left(\frac{1}{2}\right)\end{array}\right)$$

DN = 
$$\left(\begin{array}{cc}\frac{1}{\cosh \left(1\right)}& 1\\ \mathrm{dn}\text{<a href="matlab:doc symbolic/jacobiDN">jacobiDN</a>}\left(1|-1\right)& 1\end{array}\right)$$

Input Arguments

u — Input

number | vector | matrix | array | symbolic number | symbolic variable | symbolic array | symbolic function | symbolic expression

Input, specified as a number, vector, matrix, or array, or a symbolic number, variable, array, function, or expression.

m — Input

number | vector | matrix | array | symbolic number | symbolic variable | symbolic array | symbolic function | symbolic expression

Input, specified as a number, vector, matrix, or array, or a symbolic number, variable, array, function, or expression.

Output Arguments

SN — Jacobi sn elliptic function

symbolic expression

Jacobi sn elliptic function, returned as a symbolic expression.

CN — Jacobi cn elliptic function

symbolic expression

Jacobi cn elliptic function, returned as a symbolic expression.

DN — Jacobi dn elliptic function

symbolic expression

Jacobi dn elliptic function, returned as a symbolic expression.

More About

Jacobi Elliptic Functions

The Jacobi elliptic functions are defined as

$$\begin{array}{l}\text{sn}(u,m)=\sin \varphi \\ \text{cn}(u,m)=\cos \varphi \\ \text{dn}(u,m)=\sqrt{1-m{\sin }^2\varphi }\end{array}$$

where ϕ satisfies the incomplete elliptic integral of the first kind

$$u={\displaystyle {\int }_0^{\varphi }\frac{d\theta }{\sqrt{1-m{\sin }^2\theta }}}.$$

Tips

• Calling ellipj for numbers that are not symbolic objects invokes the MATLAB^® ellipj function. This function accepts only 0 <= m <= 1. To compute the Jacobi elliptic functions for values out of this range, use sym or vpa to convert the numbers to symbolic objects, and then call ellipj for those symbolic objects. Alternatively, use the jacobiSN, jacobiCN, and jacobiDN functions to compute the elliptic functions separately.

• For most symbolic (exact) numbers, ellipj returns results using the jacobiSN, jacobiCN, and jacobiDN functions. You can approximate such results with floating-point numbers using vpa.

References

[1] Abramowitz, M. and I. A. Stegun, Handbook of Mathematical Functions, Dover Publications (1965), 17.6.

Version History

Introduced in R2017b