How can I compute Jacobi Elliptic Functions in Matlab where the modulus is > 1?

I'm interested in computing the elliptic function cd(u|m) where cd(u|m) = cn(u|m) / dn(u|m). I can compute both cn and dn using the built in function "ellipj" in Matlab, however it doesn't work when m > 1.

Suggestions?

Thanks,

Mark

Answer 1

[Karan Gill]

Starting R2017b, all Jacobi Elliptic functions in MATLAB with Symbolic Math Toolbox. Example:

>> jacobiCD(1,3)
ans =
  535.8838

• jacobiAM for Jacobi amplitude function
• jacobiCD for Jacobi CD elliptic function
• jacobiCN for Jacobi CN elliptic function
• jacobiCS for Jacobi CS elliptic function
• jacobiDC for Jacobi DC elliptic function
• jacobiDN for Jacobi DN elliptic function
• jacobiDS for Jacobi DS elliptic function
• jacobiNC for Jacobi NC elliptic function
• jacobiND for Jacobi ND elliptic function
• jacobiNS for Jacobi NS elliptic function
• jacobiSC for Jacobi SC elliptic function
• jacobiSD for Jacobi SD elliptic function
• jacobiSN for Jacobi SN elliptic function
• jacobiZeta for Jacobi zeta function

See all Symbolic special functions here: https://www.mathworks.com/help/symbolic/mathematical-functions.html

Answer 2

[Matt Fig]

Use (for m>1):

sn(u,m) = sqrt(1/m)*sn(u*sqrt(m),1/m)  
cn(u,m) = dn(u*sqrt(m),1/m)           
dn(u,m) = cn(u*sqrt(m),1/m)

[Mark]

Fantastic! I'm relieved to see that it is this simple! Thanks for the help Matt.

Mark