THE SECOND SOLUTION OF BESSEL’S EQUATION
13. THE SECOND SOLUTION OF BESSEL’S EQUATION
We have found just one of the two solutions of Bessel’s equation, that is, the one when s = p; we must next find the solution when s = −p. It is unnecessary to go through the details again; we can just replace p by −p in (12.9). In fact, the
solution when s = −p is usually written J −p . From (12.9) we have
x 2n−p (13.1)
(−1) n
J −p (x) =
n=0 Γ(n + 1)Γ(n − p + 1) 2
If p is not an integer, J p (x) is a series starting with x p and J −p is a series starting with x −p . Then J p (x) and J −p (x) are two independent solutions and a linear combination of them is a general solution. But if p is an integer, then the first few terms in J −p are zero because Γ(n − p + 1) in the denominator is Γ of a negative integer, which is infinite. You can show (Problem 2) that J −p (x) starts with the term x p (for integral p) just as J p (x) does, and that
for integral p; thus J −p (x) is not an independent solution when p is an integer. The second solution
−p p (x) = (−1) J p (x)
in this case is not a Frobenius series (11.1) but contains a logarithm. J p (x) is finite at the origin, but the second solution is infinite and so is useful only in applications involving regions not containing the origin.
Although J −p (x) is a satisfactory second solution when p is not an integer, it is customary to use a linear combination of J p (x) and J −p (x) as the second solution.
Section 14 Graphs and Zeros of Bessel Functions 591
This is much as if sin x and (2 sin x − 3 cos x) were used as the two solutions of y ′′ + y = 0 instead of sin x and cos x. Remember that the general solution of
this differential equation is a linear combination of sin x and cos x with arbitrary coefficients. But A sin x + B(2 sin x − 3 cos x) is just as good a linear combination as
c 1 sin x + c 2 cos x. Similarly, any combination of J p (x) and J −p (x) is a satisfactory second solution of Bessel’s equation. The combination which is used is called either the Neumann or the Weber function and is denoted by either N p or Y p :
cos(πp)J p (x) − J (x)
it has a limit (as p tends to an integral value) which gives a second solution. This is why the special form (13.3) is used; it is valid for any p. N p or Y p are called Bessel functions of the second kind. The general solution of Bessel’s equation (12.1) or (12.2) may then be written as
y = AJ p (x) + BN p (x),
where A and B are arbitrary constants.
PROBLEMS, SECTION 13
1. Using equations (12.9) and (13.1), write out the first few terms of J 0 (x), J 1 (x), J −1 (x), J 2 (x), J −2 (x). Show that J −1 (x) = −J 1 (x) and J −2 (x) = J 2 (x).
2. Show that, in general for integral n, J −n (x) = (−1) n J n (x), and J n (−x) = (−1) n J n (x). Use equations (12.9) and (13.1) to show that: 3. pπx/2 J −1/2 (x) = cos x 4. J 3/2 (x) = x −1 J 1/2 (x) − J −1/2 (x). 5. Using equation (13.3), show that N 1/2 (x) = −J −1/2 (x); that N 3/2 (x) = J −3/2 (x). 6. Show from (13.3) that N
(2n+1)/2 (x) = (−1)
n+1
J −(2n+1)/2 (x).
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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