jacobiSD

Jacobi SD elliptic function

Syntax

jacobiSD(u,m)

Description

jacobiSD(u,m) returns the Jacobi SD Elliptic Function of u and m. If u or m is an array, then jacobiSD acts element-wise.

Examples

Calculate Jacobi SD Elliptic Function for Numeric Inputs

jacobiSD(2,1)

ans =
    3.6269

Call jacobiSD on array inputs. jacobiSD acts element-wise when u or m is an array.

jacobiSD([2 1 -3],[1 2 3])

ans =
    3.6269    2.1629 -126.3078

Calculate Jacobi SD Elliptic Function for Symbolic Numbers

Convert numeric input to symbolic form using sym, and find the Jacobi SD elliptic function. For symbolic input where u = 0 or m = 0 or 1, jacobiSD returns exact symbolic output.

jacobiSD(sym(2),sym(1))

ans =
sinh(2)

Show that for other values of u or m, jacobiSD returns an unevaluated function call.

jacobiSD(sym(2),sym(3))

ans =
jacobiSD(2, 3)

Find Jacobi SD Elliptic Function for Symbolic Variables or Expressions

For symbolic variables or expressions, jacobiSD returns the unevaluated function call.

syms x y
f = jacobiSD(x,y)

f =
jacobiSD(x, y)

Substitute values for the variables by using subs, and convert values to double by using double.

f = subs(f, [x y], [3 5])

f =
jacobiSD(3, 5)

fVal = double(f)

fVal =
    0.0312

Calculate f to higher precision using vpa.

fVal = vpa(f)

fVal =
0.031220579864538785956650143970485

Plot Jacobi SD Elliptic Function

Plot the Jacobi SD elliptic function using fcontour. Set u on the x-axis and m on the y-axis by using the symbolic function f with the variable order (u,m). Fill plot contours by setting Fill to on.

syms f(u,m)
f(u,m) = jacobiSD(u,m);
fcontour(f,'Fill','on')
title('Jacobi SD Elliptic Function')
xlabel('u')
ylabel('m')

Input Arguments

u — Input

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

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

m — Input

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

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

More About

Jacobi SD Elliptic Function

The Jacobi SD elliptic function is

sd(u,m) = sn(u,m)/dn(u,m)

where sn and dn are the respective Jacobi elliptic functions.

The Jacobi elliptic functions are meromorphic and doubly periodic in their first argument with periods 4K(m) and 4iK'(m), where K is the complete elliptic integral of the first kind, implemented as ellipticK.

Version History

Introduced in R2017b