jacobiAM

Jacobi amplitude function

Syntax

jacobiAM(u,m)

Description

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

Calculate Jacobi Amplitude Function for Numeric Inputs

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

Calculate Jacobi Amplitude Function for Symbolic Numbers

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)

Find Jacobi Amplitude Function for Symbolic Variables or Expressions

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 Jacobi Amplitude Function

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

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 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