THE ASSOCIATED LEGENDRE FUNCTIONS
10. THE ASSOCIATED LEGENDRE FUNCTIONS
A differential equation closely related to the Legendre equation is
(10.1) (1 − x 2 )y ′′
− 2xy + l(l + 1) −
2 1−x y=0
with m 2 ≤l 2 . We could solve this equation by series; however, it is more useful to know how the solutions are related to Legendre polynomials, so we shall simply verify the known solution. First we substitute
y = (1 − x 2 ) m/2 u
into (10.1) and obtain (Problem 1) (10.3)
(1 − x 2 )u ′′ − 2(m + 1)xu ′ + [l(l + 1) − m(m + 1)]u = 0.
For m = 0, this is Legendre’s equation with solutions P l (x). Differentiate (10.3), obtaining (Problem 1)
(10.4) (1 − x 2 )(u ′ ) ′′ − 2[(m + 1) + 1]x(u ′ ) ′ + [l(l + 1) − (m + 1)(m + 2)]u ′ = 0. But this is just (10.3) with u ′ in place of u, and (m + 1) in place of m. In other
words, if P l (x) is a solution of (10.3) with m = 0, P ′ l (x) is a solution of (10.3) with m = 1, P l ′′ (x) is a solution with m = 2, and in general for integral m, 0 ≤ m ≤ l, (d m /dx m )P l (x) is a solution of (10.3). Then
is a solution of (10.1). The functions in (10.5) are called associated Legendre func- tions are are denoted by
(10.6) P m
Associated Legendre functions
dx m
[Some authors include a factor (−1) m in the definition of P m l (x).]
A negative value for m in (10.1) does not change m 2 , so a solution of (10.1) for positive m is also a solution for the corresponding negative m. Thus many references define P m
l (x) for −l ≤ m ≤ l as equal to P l (x). Alternatively, we may use Rodrigues’ formula (4.1) for P l (x) in (10.6) to get
|m|
1 d l+m
(10.7) P m l (x) =
l+m (x − 1) 2 . l! dx It can be shown that (10.7) is a solution of (10.1) for either positive or nega-
2 l (1 − x ) m/2
tive m; however P −m l (x) and P m l (x) are then proportional rather than equal (see Problem 8).
584 Series Solutions of Differential Equations Chapter 12
For each m, the functions P m l (x) are a set of orthogonal functions on (−1, 1) (Problem 3). The normalization constants can be evaluated; for the definition (10.7) we find (Problem 10)
2 2 (10.8) (l + m)! [P
l (x)] dx =
2l + 1 (l − m)!
The associated Legendre functions arise in many of the same problems in which Legendre polynomials appear (see the first paragraph of Section 2); in fact, the Legendre polynomials are just the special case of the functions P m l (x) when m = 0.
PROBLEMS, SECTION 10
1. Verify equations (10.3) and (10.4). 2. The equation for the associated Legendre functions (and for Legendre functions
when m = 0) usually arises in the form (see, for example, Chapter 13, Section 7)
Make the change of variable x = cos θ, and obtain (10.1). 3. Show that the functions P l m (x) for each m are a set of orthogonal functions on
(-1,1), that is, show that
Z 1 P l m (x)P n m (x) dx = 0, −1
Hint: Use the differential equations (10.1) and follow the method of Section 7. Substitute the P l (x) you found in Problems 4.3 or 5.3 into equation (10.6) to find P m l (x);
then let x = cos θ to evaluate: 1 1 4. 2 P
1 (cos θ)
5. P 4 (cos θ)
6. P 3 (cos θ)
7. Show that
− 1) l = (x − 1) (x + 1) l and find the derivatives by Leibniz’ rule.
Hint: Write (x 2 l
8. Write (10.7) with m replaced by −m; then use Problem 7 to show that
P l (x).
(l + m)!
Comment: This shows that (10.7) is a solution of (10.1) when m is negative. 9. Use Problem 7 to show that
m (l + m)! (1 − x 2 ) −m/2 d P l−m
(x) = (−1)
(x 2 − 1) .
dx l−m 10. Derive (10.8) as follows: Multiply together the two formulas for P m l (x) given in
(l − m)!
2 l l!
(10.7) and Problem 9. Then integrate by parts repeatedly lowering the l + m deriva- tive and raising the l − m derivative until both are l derivatives. Then use (8.1).
Section 11 Generalized Power Series or the Method of Frobenius 585
Parts
» CONVERGENCE TESTS FOR SERIES OF POSITIVE TERMS; ABSOLUTE CONVERGENCE
» CONDITIONALLY CONVERGENT SERIES
» POWER SERIES; INTERVAL OF CONVERGENCE
» EXPANDING FUNCTIONS IN POWER SERIES
» TECHNIQUES FOR OBTAINING POWER SERIES EXPANSIONS
» ACCURACY OF SERIES APPROXIMATIONS
» COMPLEX POWER SERIES; DISK OF CONVERGENCE
» ELEMENTARY FUNCTIONS OF COMPLEX NUMBERS
» POWERS AND ROOTS OF COMPLEX NUMBERS
» THE EXPONENTIAL AND TRIGONOMETRIC FUNCTIONS
» LINEAR COMBINATIONS, LINEAR FUNCTIONS, LINEAR OPERATORS
» LINEAR DEPENDENCE AND INDEPENDENCE
» SPECIAL MATRICES AND FORMULAS
» EIGENVALUES AND EIGENVECTORS; DIAGONALIZING MATRICES
» APPLICATIONS OF DIAGONALIZATION
» A BRIEF INTRODUCTION TO GROUPS
» POWER SERIES IN TWO VARIABLES
» APPROXIMATIONS USING DIFFERENTIALS
» CHAIN RULE OR DIFFERENTIATING A FUNCTION OF A FUNCTION
» APPLICATION OF PARTIAL DIFFERENTIATION TO MAXIMUM AND MINIMUM PROBLEMS
» MAXIMUM AND MINIMUM PROBLEMS WITH CONSTRAINTS; LAGRANGE MULTIPLIERS
» ENDPOINT OR BOUNDARY POINT PROBLEMS
» DIFFERENTIATION OF INTEGRALS; LEIBNIZ’ RULE
» APPLICATIONS OF INTEGRATION; SINGLE AND MULTIPLE INTEGRALS
» CHANGE OF VARIABLES IN INTEGRALS; JACOBIANS
» APPLICATIONS OF VECTOR MULTIPLICATION
» DIRECTIONAL DERIVATIVE; GRADIENT
» SOME OTHER EXPRESSIONS INVOLVING ∇
» GREEN’S THEOREM IN THE PLANE
» THE DIVERGENCE AND THE DIVERGENCE THEOREM
» THE CURL AND STOKES’ THEOREM
» SIMPLE HARMONIC MOTION AND WAVE MOTION; PERIODIC FUNCTIONS
» APPLICATIONS OF FOURIER SERIES
» COMPLEX FORM OF FOURIER SERIES
» LINEAR FIRST-ORDER EQUATIONS
» OTHER METHODS FOR FIRST-ORDER EQUATIONS
» SECOND-ORDER LINEAR EQUATIONS WITH CONSTANT COEFFICIENTS AND ZERO RIGHT-HAND SIDE
» SECOND-ORDER LINEAR EQUATIONS WITH CONSTANT COEFFICIENTS AND RIGHT-HAND SIDE NOT ZERO
» OTHER SECOND-ORDER EQUATIONS
» SOLUTION OF DIFFERENTIAL EQUATIONS BY LAPLACE TRANSFORMS
» A BRIEF INTRODUCTION TO GREEN FUNCTIONS
» THE BRACHISTOCHRONE PROBLEM; CYCLOIDS
» SEVERAL DEPENDENT VARIABLES; LAGRANGE’S EQUATIONS
» TENSOR NOTATION AND OPERATIONS
» KRONECKER DELTA AND LEVI-CIVITA SYMBOL
» PSEUDOVECTORS AND PSEUDOTENSORS
» VECTOR OPERATORS IN ORTHOGONAL CURVILINEAR COORDINATES
» ELLIPTIC INTEGRALS AND FUNCTIONS
» GENERATING FUNCTION FOR LEGENDRE POLYNOMIALS
» COMPLETE SETS OF ORTHOGONAL FUNCTIONS
» NORMALIZATION OF THE LEGENDRE POLYNOMIALS
» THE ASSOCIATED LEGENDRE FUNCTIONS
» GENERALIZED POWER SERIES OR THE METHOD OF FROBENIUS
» THE SECOND SOLUTION OF BESSEL’S EQUATION
» OTHER KINDS OF BESSEL FUNCTIONS
» ORTHOGONALITY OF BESSEL FUNCTIONS
» SERIES SOLUTIONS; FUCHS’S THEOREM
» HERMITE FUNCTIONS; LAGUERRE FUNCTIONS; LADDER OPERATORS
» LAPLACE’S EQUATION; STEADY-STATE TEMPERATURE IN A RECTANGULAR PLATE
» THE DIFFUSION OR HEAT FLOW EQUATION; THE SCHR ¨ODINGER EQUATION
» THE WAVE EQUATION; THE VIBRATING STRING
» STEADY-STATE TEMPERATURE IN A CYLINDER
» VIBRATION OF A CIRCULAR MEMBRANE
» STEADY-STATE TEMPERATURE IN A SPHERE
» INTEGRAL TRANSFORM SOLUTIONS OF PARTIAL DIFFERENTIAL EQUATIONS
» EVALUATION OF DEFINITE INTEGRALS BY USE OF THE RESIDUE THEOREM
» THE POINT AT INFINITY; RESIDUES AT INFINITY
» SOME APPLICATIONS OF CONFORMAL MAPPING
» THE NORMAL OR GAUSSIAN DISTRIBUTION
» STATISTICS AND EXPERIMENTAL MEASUREMENTS
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