jacobiDN

Jacobi DN elliptic function

Syntax

``jacobiDN(u,m)``

Description

example

````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

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`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```

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)```

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 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

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

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

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Jacobi DN Elliptic Function

The Jacobi DN elliptic function is

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

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`.

Version History

Introduced in R2017b