NORMALIZATION OF THE LEGENDRE POLYNOMIALS
8. NORMALIZATION OF THE LEGENDRE POLYNOMIALS
If we take the scalar product of a vector with itself, A · A = A 2 , we get the square of the length (or norm) of the vector. If we divide A by its length, we get a unit vector. In Chapter 3, Section 14 we showed that we can think of functions as the vectors of a vector space and we defined the norm N of a function A(x) on (a, b) by [see Chapter 3, equation (14.2)]
A ∗ (x)A(x) dx =
|A(x)| 2 dx = N 2 .
We also say that the function N −1 A(x) is normalized ; like a unit vector, a normal- ized function has norm = 1. The factor N −1 is called the normalization factor. For
Section 8 Normalization of the Legendre Polynomials 579 example, π
0 sin 2 nx dx = π/2. Then the norm of sin nx on (0, π) is
functions normalized orthogonal functions is called orthonormal. For example,
is an orthonormal set on (0, π). Such a set of orthonormal functions may remind us of i, j, k; like these unit vectors, the functions are orthogonal and have norm = 1. If the elements of a vector space are functions, we can then use a (complete) orthonormal subset of the functions as the basis vectors of the space. We think of expanding other functions in terms of them (by analogy with writing a three-dimensional vector in terms of
i, j, k). For example, suppose we have expanded a given function f (x) on (0, π) in
a Fourier sine series:
f (x) =
sin nx.
We call f (x) a vector with components B n in terms of the basis vectors Thus, in quantum mechanics, we often refer to a function which describes the state of a physical system as either a state function or a state vector. Just as we can write a three-dimensional vector in terms of i, j, k, or in terms of another basis, say
e r ,e θ ,e φ , so we can expand a given f (x) in terms of another orthonormal set of functions and find its components relative to this new basis. In Section 9, we shall see how to expand functions in Legendre series.
Just as we needed the norm of sin nx in Fourier series, so we shall need the norm of P l (x) in expanding functions in Legendre series. We shall prove that
[P l (x)] 2 dx = 2 .
Then the functions l (x) are an orthonormal set of functions on (−1, 1). To prove (8.1), we use the recursion relation (5.8b), namely,
lP l (x) = xP ′ l (x) − P ′ l−1 (x).
Multiply (8.2) by P l (x) and integrate to get
(8.3) l [P
2 l (x)] dx =
xP l (x)P ′ l (x) dx −
P l (x)P ′ l−1 (x) dx.
The last integral is zero by Problem 7.4. To evaluate the middle integral in (8.3), we integrate by parts:
xP l (x)P
l (x) dx = [P l (x)]
[P l (x)] dx
[P l (x)] 2 dx
(see Problem 2.2). Then (8.3) gives
2 1 l 1 [P l (x)] dx = 1 − [P l (x)] 2 dx
580 Series Solutions of Differential Equations Chapter 12
which simplifies to (8.1). We can combine (7.1) and (8.1) to write
P l (x)P m (x) dx =
PROBLEMS, SECTION 8
Find the norm of each of the following functions on the given interval and state the normalized function.
1. cos nx on (0, π) 2. P 2 (x) on (−1, 1) 3. xe −x/2 on (0, ∞)
4. e −x 2 /2 on (−∞, ∞) 5. xe −x 2 /2 on (0, ∞) Hint: See Chapter 4, Section 12. 6. Give another proof of (8.1) as follows. Multiply (5.8e) by P l (x) and integrate from
−1 to 1. To evaluate the middle term, integrate by parts. Then use Problem 7.4. 7. Using (8.1), write the first four normalized Legendre polynomials and compare with
the answers we found by a different method in Chapter 3, Section 14, Example 6.
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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