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# jacobiAM

Jacobi amplitude function

## Syntax

``jacobiAM(u,m)``

## Description

example

````jacobiAM(u,m)` returns the Jacobi Amplitude Function of `u` and `m`. If `u` or `m` is an array, then `jacobiAM` acts element-wise.```

## Examples

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`jacobiAM(2,1)`
```ans = 1.3018```

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

`jacobiAM([2 1 -3],[1 2 3])`
```ans = 1.3018 0.7370 0.6155```

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

`jacobiAM(sym(2),sym(1))`
```ans = 2*atan(exp(2)) - pi/2```

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

`jacobiAM(sym(2),sym(3))`
```ans = jacobiAM(2, 3)```

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

```syms x y f = jacobiAM(x,y)```
```f = jacobiAM(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 = jacobiAM(3, 5)```
`fVal = double(f)`
```fVal = 0.0311```

Calculate `f` to higher precision using `vpa`.

`fVal = vpa(f)`
```fVal = 0.031149815412430844987208470634926```

Plot the Jacobi amplitude function using `fcontours`. 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) = jacobiAM(u,m); fcontour(f,'Fill','on') title('Jacobi Amplitude 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.

## More About

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### Jacobi Amplitude Function

The Jacobi amplitude function am(u,m) is defined by am(u,m) = φ where F(φ,m) = u and F represents the incomplete elliptic integral of the first kind. F is implemented as `ellipticF`.

## Version History

Introduced in R2017b