Using Mathematica and scipy to clarify the definition of Jacobi elliptic function

This, this, that, that Jacobi, the Oval function SN and CN resemble trigonometric functions sine and cosine. They appear in applications such as nonlinear vibration and conformal mapping. Unfortunately, there are many conventions for defining these functions. The purpose of this article is to clarify the confusion surrounding these different conventions.

The image above is a graph of function sn[1].

modulus, parameters and modulus angle
The Jacobi function has two inputs. We generally think that the Jacobi function is the first input function, and the second input is fixed. This second input is a "dial-up" and you can turn it to change their behavior.

There are several ways to specify this dialing. I started with the word "dialing" instead of "parameters" because in this context the parameters have technical significance, a way of describing the dials. In addition to the parameters, you can also describe the Jacobi function as a modulus or a modulo angle. This article will be a Rosetta stone that shows how each of the ways of describing the Jacobian elliptic function is related.

This, this, that, that parameter m is the real number in [0,1]. This, this, that, that complementary parameter is m ' = 1-m. Abramovitz and Stogon For example, the Jacobi function Sn and CN are written as SN (u | m) and CN (U | m) They also use m1= instead of M ' to represent complementary parameters.

This, this, that, the square root m of the modulus K. If M represents modulus, but this is not tradition. Instead, m indicates that the parameter and K are modulus. This, this, that, that complementary modulus K ' is the square root of the complementary parameter.

This, this, that, the modulo angle α is by M=sin 2α.

Note that as the parameter M equals zero, modulus is also zero. K and module angle α. When any of these three functions becomes zero, the Jacobi function SN and CN converge to their corresponding sine and cosine. Therefore, whether your dial is a parameter, modulus, or die angle, when you turn the dial to zero, the SN converges to sine, and the CN converges to the cosine.

As the parameter M equals 1, the modulus is also 1. K, but the modular angle α enters the Π/2. So if your dial is a parameter or a modulus, it will turn 1. However, if you think your dialing is a modular angle, it will go into Π/2. In both cases, when you turn the dial to the right, SN converges to the hyperbolic secant, and the CN converges to the constant function 1.

Quarterly Period
In addition to parameters, modulus, and die angles, you can also see the Jacobi function K and K "" These are called quarterly periods for good reason. The function SN and CN have a period of 4. K When you move along a real axis, or horizontally anywhere in a complex plane. They also have the fourth period IK "" that is, the function repeats when the moving distance is 4 o'clock. K ' vertical [2].

The One-fourth period is a function of modulus. Quarter-period k along the true axis

function K (m) is called "The first class of complete elliptic integrals".

Amplitude
So far, we've focused on the second input to the Jacobi function and the three conventions that specify it.

There are two conventions that specify the first parameter, either as φ or as U. These are all related to.

Angular φ is called amplitude. (yes, this is an angle, but it is called amplitude.)

When we say that the Jacobi function has a 4-cycle time K, this is from a variable point of view. U Please note that when φ=π/2,u = K.

The Jacobi elliptic function in mathematics
Mathematica uses the conventions of the first parameter of U and the parameter conventions of the second parameter.

The mathematical function jacobisn[u, M] calculates the function snu the parameter m with the parameter. In a&s notation, sn (u | m).

Similarly, Jacobicn[u, M] calculates the function CNU parameter m with a parameter. In a&s notation, cn (U | m).

So far, we have not discussed the Jacobi function DN, but it is implemented in Mathematica in Jacobidn[u, M].

The amplitude φ as U is jacobiamplitude[um m].

Calculates the function for the quarter period. K from parameter M is elliptick[m].

Jacobi elliptic functions in Python
The SciPy library has a Python function that calculates four math functions at the same time. The function SCIPY.SPECIAL.ELLIPJ has two arguments, U and M, just like Mathematica, and returns SN (U | m), CN (U | m), DN (U | m), and amplitude φ (U, m).

The function K (m) is implemented as SCIPY.SPECIAL.ELLIPK in Python.

Related Posts
[1] The episode is described here with jacobisn[0.5, z] and functional Complexplot.

[2] Strictly speaking, 4IK "is" a period. This is the minimum vertical period for the CN, but the 2IK is the minimum vertical period of the SN.

Using Mathematica and scipy to clarify the definition of Jacobi elliptic function