## Saturday, February 15, 2020

### HP Prime and TI 84 Plus CE: Jacobi Elliptic Functions

HP Prime and TI 84 Plus CE:  Jacobi Elliptic Functions

Introduction

Jacobian Elliptic Functions are a set of twelve functions denoted by XY(U, K) where X and Y stands of letters c, s, n, and d.  Today's blog post will focus on three of the common Jacobi Elliptic Functions:

Sine Amplitude:  sn(u,k)
Cosine Amplitude:  cn(u,k)
Delta Amplitude:  dn(u,k)

Where u is a real number and k is a parameter between -1 and 1 inclusive

To determine any of the Jacobian Elliptic Functions, the integral has to be solved for X:

U = ∫( 1/√(1 - K^2 * sin^2(T)) dT from T = 0 to T = X)

Solving for X will represent the function am(U,K).

Then:
sn(U,K) = sin(X)
cn(U,K) = cos(X)
dn(U,K) = √(1 - K^2 * sin^2(X))

Radian angles are used.

HP Prime App:  Jacobi Elliptic Functions

In a different approach, I have created a custom app, which is based on the Solver App named Jacobi Elliptic Functions, which you can download on the link above.

Symb View:  The four equations that are used for this app.  Leave all four checked.

Num View:  This is where you enter U and K.  Leave these boxes unchecked.  Press or touch (Solve) to get the other values am (X), sn (S), cn (C), and dn (D).

If you want to program ths app yourself, please see the screen shots above.

TI-84 Plus CE Program:  ELLIPFX

"EWS 2020-01-22"
.5→X:1→F:1→N
ClrHome
Disp "JACOBIAN ELLIPTIC","­1≤K and K≤1"
Prompt U,K
Repeat abs(N/F)≤1E­10
fnInt((1-K^2*sin(T)^2)^(­-1/2),T,0,X)-U→N
(1-K^2*sin(X)^2)^(­-1/2)→F
X-N/F→X
End
sin(X)→S
cos(X)→C
√(1-K^2*sin(X)^2)→D
ClrHome
Disp "U="+toString(U)
Disp "K="+toString(K)
Disp "AM="+toString(X)
Disp "SN="+toString(S)
Disp "CN="+toString(C)
Disp "DN="+toString(D)

Examples

Example 1:
U = 3
K = 0.5

Results:
AM(U,K) = 2.772166899
SN(U,K) = 0.3610799872
CN(U,K) = -0.932534848
DN(U,K) = 0.9835676442

Example 2:
U = 1.5
K = 0

Results:
AM(U,K) = 1.5
SN(U,K) = 0.9974949866
CN(U,K) = 0.0707372017
DN(U,K) = 1

Sources

"Jacobi elliptic functions"  Wikipeida.  https://en.wikipedia.org/wiki/Jacobi_elliptic_functions  Retrieved December 23, 2019

"Jacobi elliptic function sn,cn,dn (chart) Calculator"  Ke!san Online Calculator https://keisan.casio.com/exec/system/1180573437  Retrieved January 22, 2020

Eddie

All original content copyright, © 2011-2020.  Edward Shore.   Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited.  This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author.