jacobiDN

Jacobi DN elliptic function

Syntax

jacobiDN(u,m)

Description

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

Examples

Calculate Jacobi DN Elliptic Function for Numeric Inputs

jacobiDN(2,1)

ans =
    0.2658

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

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

ans =
    0.2658    0.3107   -0.0046

Calculate Jacobi DN Elliptic Function for Symbolic Numbers

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

jacobiDN(sym(2),sym(1))

ans =
1/cosh(2)

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

jacobiDN(sym(2),sym(3))

ans =
jacobiDN(2, 3)

Find Jacobi DN Elliptic Function for Symbolic Variables or Expressions

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

syms x y
f = jacobiDN(x,y)

f =
jacobiDN(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 =
jacobiDN(3, 5)

fVal = double(f)

fVal =
    0.9976

Calculate f to higher precision using vpa.

fVal = vpa(f)

fVal =
0.99757205953668099307853539907267

Plot Jacobi DN Elliptic Function

Plot the Jacobi DN 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) = jacobiDN(u,m);
fcontour(f,'Fill','on')
title('Jacobi DN 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 DN Elliptic Function

The Jacobi DN elliptic function is

$$\text{dn}\left(u,m\right)=\sqrt{1-m\sin {(\varphi )}^2}$$

where ϕ is such that F(ϕ,m) = u and F represents the incomplete elliptic integral of the first kind. F is implemented as ellipticF.

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.