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Legendre’s equation,

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Legendre's equation An equation of the form (1-x2)⋅(d2y/dx2)-2⋅x⋅ (dy/dx)+n⋅ (n+1) ⋅y = 0, where n is a real number, is known as the Legendre's differential equation. Any solution for this equation is known as a Legendre's function. When n is a nonnegative integer, the solutions are called Legendre's polynomials. Legendre's polynomial of order n is given by ∑ Pn (x) = M (−1) m m=0 ⋅ 2n ⋅ (2n − 2m)! m!⋅(n − m)!⋅(n − 2m)! ⋅x n−2m = (2n)! 2n ⋅ (n!)2 ⋅ xn − 2n (2n − 2)! ⋅1!⋅(n −1)!(n − ⋅ 2)! x n−2 + ... − .. where M = n/2 or (n-1)/2, whichever is an integer. Legendre's polynomials are pre-programmed in the calculator and can be recalled by using the function LEGENDRE given the order of the polynomial, n. The function LEGENDRE can be obtained from the command catalog (,N) or through the menu ARITHMETIC/POLYNOMIAL menu (see Chapter 5). In RPN mode, the first six Legendre polynomials are obtained as follows: 0 LEGENDRE, result: 1, i.e., 1 LEGENDRE, result: 'X', i.e., 2 LEGENDRE, result: '(3*X^2-1)/2', i.e., 3 LEGENDRE, result: '(5*X^3-3*X)/2', i.e., 4 LEGENDRE, result: '(35*X^4-30*X^2+3)/8', i.e., P0(x) = 1.0. P1(x) = x. P2(x) = (3x2-1)/2. P3(x) =(5x3-3x)/2. P4(x) =(35x4-30x2+3)/8. 5 LEGENDRE, result: '(63*X^5-70*X^3+15*X)/8', i.e., P5(x) =(63x5-70x3+15x)/8. The ODE (1-x2)⋅(d2y/dx2)-2⋅x⋅ (dy/dx)+[n⋅ (n+1)-m2/(1-x2)] ⋅y = 0, has for solution the function y(x) = Pnm(x)= (1-x2)m/2⋅(dmPn/dxm). This function is referred to as an associated Legendre function. Page 16-54

Section	Page
Table of contents	5
A note about screenshots in this guide	27
Chapter 1 Getting started	30
Basic Operations	30
Batteries	30
Turning the calculator on and off	31
Adjusting the display contrast	31
Contents of the calculator’s display	31
Menus	32
SOFT menus vs. CHOOSE boxes	33
Selecting SOFT menus or CHOOSE boxes	34
The TOOL menu	35
Setting time and date	36
Introducing the calculator’s keyboard	39
Selecting calculator modes	41
Operating Mode	42
Number Format and decimal dot or comma	46
Angle Measure	51
Coordinate System	51
Beep, Key Click, and Last Stack	53
Selecting CAS settings	54
Selecting Display modes	54
Selecting the display font	55
Selecting properties of the line editor	56
Selecting properties of the Stack	56
Selecting properties of the equation writer (EQW)	57
Selecting the size of the header	58
Selecting the clock display	58
Chapter 2	59
Calculator objects	59
Editing expressions in the screen	61
Creating arithmetic expressions	61
Editing arithmetic expressions	64
Creating algebraic expressions	65
Editing algebraic expressions	66
Using the Equation Writer (EQW) to create expressions	68
Creating arithmetic expressions	69
Editing arithmetic expressions	74
Creating algebraic expressions	77
Editing algebraic expressions	78
Creating and editing summations, derivatives, and integrals	86
Organizing data in the calculator	90
Functions for manipulation of variables	91
The HOME directory	92
The CASDIR sub-directory	93
Typing directory and variable names	95
Creating subdirectories	96
Moving among subdirectories	100
Deleting subdirectories	100
Variables	104
Creating variables	104
Checking variables contents	109
Replacing the contents of variables	111
Copying variables	112
Reordering variables in a directory	115
Moving variables using the FILES menu	116
Deleting variables	117
UNDO and CMD functions	119
Flags	120
Example of flag setting: general solutions vs. principal value	121
Other flags of interest	122
CHOOSE boxes vs. Soft MENU	123
Selected CHOOSE boxes	124
Chapter 3 Calculation with real numbers	126
Checking calculators settings	126
Checking calculator mode	127
Real number calculations	127
Changing sign of a number, variable, or expression	128
The inverse function	128
Addition, subtraction, multiplication, division	128
Using parentheses	129
Absolute value function	129
Squares and square roots	129
Powers and roots	130
Base-10 logarithms and powers of 10	130
Using powers of 10 in entering data	130
Natural logarithms and exponential function	131
Trigonometric functions	131
Inverse trigonometric functions	131
Differences between functions and operators	132
Real number functions in the MTH menu	132
Hyperbolic functions and their inverses	134
Real number functions	136
Special functions	139
Calculator constants	141
Operations with units	142
The UNITS menu	142
Available units	143
Converting to base units	146
Attaching units to numbers	147
Operations with units	150
Units manipulation tools	152
Physical constants in the calculator	154
Special physical functions	156
Function ZFACTOR	157
Function F0λ	157
Function SIDENS	158
Function TDELTA	158
Function TINC	158
Defining and using functions	159
Functions defined by more than one expression	160
The IFTE function	161
Combined IFTE functions	161
Chapter 4 Calculations with complex numbers	163
Definitions	163
Setting the calculator to COMPLEX mode	163
Entering complex numbers	164
Polar representation of a complex number	165
Simple operations with complex numbers	166
Changing sign of a complex number	166
Entering the unit imaginary number	167
The CMPLX menus	167
CMPLX menu through the MTH menu	167
CMPLX menu in keyboard	169
Functions applied to complex numbers	170
Functions from the MTH menu	170
Function DROITE: equation of a straight line	171
Chapter 5 Algebraic and arithmetic operations	172
Entering algebraic objects	172
Simple operations with algebraic objects	173
Functions in the ALG menu	174
COLLECT	176
EXPAND	176
FACTOR	176
LNCOLLECT	176
LIN	176
PARTFRAC	176
SOLVE	176
SUBST	176
TEXPAND	176
Other forms of substitution in algebraic expressions	177
Operations with transcendental functions	178
Expansion and factoring using log-exp functions	179
Expansion and factoring using trigonometric functions	179
Functions in the ARITHMETIC menu	180
DIVIS	181
FACTORS	181
LGCD	181
PROPFRAC	181
SIMP2	181
INTEGER menu	181
POLYNOMIAL menu	182
MODULO menu	183
Applications of the ARITHMETIC menu	183
Modular arithmetic	183
Finite arithmetic rings in the calculator	186
Polynomials	189
Modular arithmetic with polynomials	189
The CHINREM function	190
The EGCD function	190
The GCD function	190
The HERMITE function	191
The HORNER function	191
The variable VX	191
The LAGRANGE function	192
The LCM function	192
The LEGENDRE function	193
The PCOEF function	193
The PROOT function	193
The PTAYL function	193
The QUOT and REMAINDER functions	194
The EPSX0 function and the CAS variable EPS	194
The PEVAL function	194
The TCHEBYCHEFF function	195
Fractions	195
The SIMP2 function	195
The PROPFRAC function	195
The PARTFRAC function	196
The FCOEF function	196
The FROOTS function	197
Step-by-step operations with polynomials and fractions	197
The CONVERT Menu and algebraic operations	198
UNITS convert menu	198
BASE convert menu	199
TRIGONOMETRIC convert menu	199
MATRICES convert menu	199
REWRITE convert menu	199
Chapter 6 Solution to single equations	202
Symbolic solution of algebraic equations	202
Function ISOL	202
Function SOLVE	203
Function SOLVEVX	205
Function ZEROS	205
Numerical solver menu	206
Polynomial Equations	207
Financial calculations	210
Solving equations with one unknown through NUM.SLV	215
The SOLVE soft menu	228
The ROOT sub-menu	228
Function ROOT	228
Variable EQ	228
The SOLVR sub-menu	229
The DIFFE sub-menu	231
The POLY sub-menu	231
The SYS sub-menu	232
The TVM sub-menu	232
Chapter 7 Solving multiple equations	235
Rational equation systems	235
Example 1 – Projectile motion	235
Example 2 – Stresses in a thick wall cylinder	236
Example 3 - System of polynomial equations	238
Solution to simultaneous equations with MSLV	238
Example 1 – Example from the help facility	239
Example 2 - Entrance from a lake into an open channel	240
Using the Multiple Equation Solver (MES)	244
Application 1 - Solution of triangles	244
Application 2 - Velocity and acceleration in polar coordinates	252
Chapter 8 Operations with lists	256
Definitions	256
Creating and storing lists	258
Changing sign	258
Addition, subtraction, multiplication, division	258
Real number functions from the keyboard	260
Real number functions from the MTH menu	260
Examples of functions that use two arguments	261
Lists of complex numbers	262
Lists of algebraic objects	263
The MTH/LIST menu	263
Manipulating elements of a list	265
List size	265
Extracting and inserting elements in a list	265
Element position in the list	266
HEAD and TAIL functions	266
The SEQ function	266
The MAP function	267
Defining functions that use lists	268
Applications of lists	269
Harmonic mean of a list	270
Geometric mean of a list	271
Weighted average	272
Statistics of grouped data	273
Chapter 9 Vectors	276
Definitions	276
Entering vectors	277
Typing vectors in the stack	277
Storing vectors into variables	278
Using the Matrix Writer (MTRW) to enter vectors	278
Building a vector with → ARRY	281
Identifying, extracting, and inserting vector elements	282
Simple operations with vectors	284
Changing sign	284
Addition, subtraction	284
Multiplication by a scalar, and division by a scalar	285
Absolute value function	285
The MTH/VECTOR menu	285
Magnitude	286
Dot product	286
Cross product	286
Decomposing a vector	287
Building a two-dimensional vector	287
Building a three-dimensional vector	288
Changing coordinate system	288
Application of vector operations	291
Resultant of forces	291
Angle between vectors	291
Moment of a force	292
Equation of a plane in space	293
Row vectors, column vectors, and lists	294
Function OBJ →	295
Function → LIST	295
Function → ARRY	296
Function DROP	296
Transforming a row vector into a column vector	296
Transforming a column vector into a row vector	297
Transforming a list into a vector	299
Transforming a vector (or matrix) into a list	300
Chapter 10 Creating and manipulating matrices	301
Definitions	301
Entering matrices in the stack	302
Using the matrix editor	302
Typing in the matrix directly into the stack	303
Creating matrices with calculator functions	303
Functions GET and PUT	306
Functions GETI and PUTI	306
Function SIZE	307
Function TRN	308
Function CON	308
Function IDN	309
Function RDM	310
Function RANM	311
Function SUB	311
Function REPL	312
Function →DIAG	313
Function DIAG→	313
Function VANDERMONDE	314
Function HILBERT	314
A program to build a matrix out of a number of lists	315
Lists represent columns of the matrix	315
Lists represent rows of the matrix	317
Manipulating matrices by columns	318
Function →COL	318
Function COL→	319
Function COL+	320
Function COL-	320
Function CSWP	321
Manipulating matrices by rows	321
Function →ROW	322
Function ROW→	323
Function ROW+	324
Function ROW-	324
Function RSWP	325
Function RCI	325
Function RCIJ	326
Chapter 11 Matrix Operations and Linear Algebra	327
Operations with matrices	327
Addition and subtraction	328
Multiplication	328
Characterizing a matrix (The matrix NORM menu)	332
Function ABS	333
Function SNRM	333
Functions RNRM and CNRM	334
Function SRAD	335
Function COND	335
Function RANK	336
Function DET	337
Function TRACE	339
Function TRAN	340
Additional matrix operations (The matrix OPER menu)	340
Function AXL	341
Function AXM	341
Function LCXM	341
Solution of linear systems	342
Using the numerical solver for linear systems	343
Least-square solution (function LSQ)	349
Solution with the inverse matrix	352
Solution by “division” of matrices	352
Solving multiple set of equations with the same coefficient matrix	353
Gaussian and Gauss-Jordan elimination	354
Step-by-step calculator procedure for solving linear systems	363
Solution to linear systems using calculator functions	366
Residual errors in linear system solutions (Function RSD)	369
Eigenvalues and eigenvectors	370
Function PCAR	370
Function EGVL	371
Function EGV	372
Function JORDAN	373
Function MAD	373
Matrix factorization	374
Function LU	375
Orthogonal matrices and singular value decomposition	375
Function SCHUR	376
Function LQ	377
Function QR	377
Matrix Quadratic Forms	377
The QUADF menu	378
Linear Applications	380
Function IMAGE	380
Function ISOM	380
Function KER	381
Function MKISOM	381
Chapter 12 Graphics	382
Graphs options in the calculator	382
Plotting an expression of the form y = f(x)	386
Saving a graph for future use	388
Graphics of transcendental functions	389
Graph of ln(X)	389
Graph of the exponential function	391
The PPAR variable	392
Inverse functions and their graphs	393
Summary of FUNCTION plot operation	394
Plots of trigonometric and hyperbolic functions	397
Generating a table of values for a function	398
The TPAR variable	399
Plots in polar coordinates	400
Plotting conic curves	402
Parametric plots	404
Generating a table for parametric equations	407
Plotting the solution to simple differential equations	407
Truth plots	410
Plotting histograms, bar plots, and scatter plots	411
Bar plots	411
Scatter plots	413
Slope fields	415
Fast 3D plots	416
Wireframe plots	418
Ps-Contour plots	420
Y-Slice plots	422
Gridmap plots	423
Pr-Surface plots	424
The VPAR variable	425
Interactive drawing	425
DOT+ and DOT-	426
MARK	427
LINE	427
TLINE	427
BOX	428
CIRCL	428
LABEL	428
DEL	428
ERASE	429
MENU	429
SUB	429
REPL	429
PICT →	429
X,Y →	429
Zooming in and out in the graphics display	430
ZFACT, ZIN, ZOUT, and ZLAST	430
BOXZ	431
ZDFLT, ZAUTO	431
HZIN, HZOUT, VZIN and VZOUT	431
CNTR	431
ZDECI	431
ZINTG	432
ZSQR	432
ZTRIG	432
The SYMBOLIC menu and graphs	432
The SYMB/GRAPH menu	433
Function DRAW3DMATRIX	435
Chapter 13 Calculus Applications	436
The CALC (Calculus) menu	436
Limits and derivatives	436
Function lim	437
Derivatives	438
Functions DERIV and DERVX	438
The DERIV&INTEG menu	438
Calculating derivatives with	439
The chain rule	441
Derivatives of equations	441
Implicit derivatives	442
Application of derivatives	442
Analyzing graphics of functions	442
Function DOMAIN	444
Function TABVAL	444
Function SIGNTAB	445
Function TABVAR	445
Using derivatives to calculate extreme points	447
Higher order derivatives	448
Anti-derivatives and integrals	449
Functions INT, INTVX, RISCH, SIGMA and SIGMAVX	449
Definite integrals	450
Step-by-step evaluation of derivatives and integrals	451
Integrating an equation	453
Techniques of integration	453
Substitution or change of variables	453
Integration by parts and differentials	454
Integration by partial fractions	455
Improper integrals	456
Integration with units	456
Infinite series	458
Taylor and Maclaurin’s series	458
Taylor polynomial and reminder	458
Functions TAYLR, TAYLR0, and SERIES	459
Chapter 14 Multi-variate Calculus Applications	462
Multi-variate functions	462
Partial derivatives	464
The chain rule for partial derivatives	465
Total differential of a function z = z(x,y)	466
Determining extrema in functions of two variables	466
Using function HESS to analyze extrema	467
Multiple integrals	469
Jacobian of coordinate transformation	470
Double integral in polar coordinates	470
Chapter 15 Vector Analysis Applications	472
Definitions	472
Gradient and directional derivative	472
A program to calculate the gradient	473
Using function HESS to obtain the gradient	473
Potential of a gradient	474
Divergence	475
Laplacian	475
Curl	476
Irrotational fields and potential function	476
Vector potential	477
Chapter 16 Differential Equations	479
Basic operations with differential equations	479
Entering differential equations	479
Checking solutions in the calculator	480
Slope field visualization of solutions	481
The CALC/DIFF menu	482
Solution to linear and non-linear equations	482
Function LDEC	483
Function DESOLVE	485
The variable ODETYPE	486
Laplace Transforms	488
Definitions	488
Laplace transform and inverses in the calculator	489
Laplace transform theorems	490
Dirac’s delta function and Heaviside’s step function	493
Applications of Laplace transform in the solution of linear ODEs	495
Fourier series	505
Function FOURIER	506
Fourier series for a quadratic function	507
Fourier series for a triangular wave	513
Fourier series for a square wave	517
Fourier series applications in differential equations	520
Fourier Transforms	521
Definition of Fourier transforms	524
Properties of the Fourier transform	526
Fast Fourier Transform (FFT)	527
Examples of FFT applications	528
Solution to specific second-order differential equations	531
The Cauchy or Euler equation	531
Legendre’s equation	532
Bessel’s equation	533
Chebyshev or Tchebycheff polynomials	535
Laguerre’s equation	536
Weber’s equation and Hermite polynomials	537
Numerical and graphical solutions to ODEs	538
Numerical solution of first-order ODE	538
Graphical solution of first-order ODE	540

Match case Limit results 1 per page

Page 16-54

Legendre’s equation

An equation of the form (1-x

)

⋅

y/dx

)-2

⋅

(dy/dx)+n

⋅

(n+1)

⋅

y = 0, where n

is a real number, is known as the Legendre’s differential equation.

Any

solution for this equation is known as a Legendre’s function.

When n is a

nonnegative integer, the solutions are called Legendre’s polynomials.

Legendre’s polynomial of order n is given by

(

)

(

)

(

∑

⋅

−

⋅

−

⋅

−

⋅

−

...

(

)

(

−

⋅

−

⋅

−

⋅

where M = n/2 or (n-1)/2, whichever is an integer.

Legendre’s polynomials are pre-programmed in the calculator and can be

recalled by using the function LEGENDRE given the order of the polynomial, n.

The function LEGENDRE can be obtained from the command catalog

(

‚N

) or through the menu ARITHMETIC/POLYNOMIAL menu (see

Chapter 5).

In RPN mode, the first six Legendre polynomials are obtained as

follows:

LEGENDRE, result: 1,

i.e.,

(x) = 1.0.

LEGENDRE, result: ‘X’,

i.e.,

(x) = x.

LEGENDRE, result: ‘(3*X^2-1)/2’,

i.e.,

(x) = (3x

-1)/2.

LEGENDRE, result: ‘(5*X^3-3*X)/2’,

i.e.,

(x) =(5x

-3x)/2.

LEGENDRE, result: ‘(35*X^4-30*X^2+3)/8’,

i.e.,

(x) =(35x

-30x

+3)/8.

LEGENDRE, result: ‘(63*X^5-70*X^3+15*X)/8’, i.e.,

(x) =(63x

-70x

+15x)/8.

The ODE

(1-x

)

⋅

y/dx

)-2

⋅

(dy/dx)+[n

⋅

(n+1)-m

/(1-x

)]

⋅

y = 0, has for

solution the function y(x) = P

(x)= (1-x

)

m/2

⋅

Pn/dx

This function is

referred to as an associated Legendre function