BESSEL’S EQUATION
12. BESSEL’S EQUATION
Like Legendre’s equation, Bessel’s equation is another of the “named” equations which have been studied extensively. There are whole books on Bessel functions, and you will find numerous formulas, graphs, and numerical values available in your computer program and in reference tables. You can think of Bessel functions as being something like damped sines and cosines. In fact, if you had first learned
about sin nx and cos nx as power series solutions of y ′′ = −n 2 y instead of in ele- mentary trigonometry, you would not feel that Bessel functions were appreciably more difficult or strange than trigonometric functions. Like sines and cosines, Bessel functions are solutions of a differential equation; they can be represented by power series, their graphs can be drawn, and many formulas involving them (compare trigonometric identities) are known. Of special interest to science students is the fact that they occur in many applications. The following list of some of the prob- lems in which they arise will give you an idea of the great range of topics which may involve Bessel functions: problems in electricity, heat, hydrodynamics, elas- ticity, wave motion, quantum mechanics, etc., involving cylindrical symmetry (for this reason Bessel functions are sometimes called cylinder functions); the motion of a pendulum whose length increases steadily; the small oscillations of a flexible chain; railway transition curves; the stability of a vertical wire or beam; Fresnel integrals in optics; the current distribution in a conductor; Fourier series for the arc of a circle. We shall discuss some of these applications later (see Section 18, and Chapter 13, Sections 5 and 6).
588 Series Solutions of Differential Equations Chapter 12
Bessel’s equation in the usual standard form is
x 2 y ′′ + xy ′ + (x 2 −p 2 )y = 0,
where p is a constant (not necessarily an integer) called the order of the Bessel func- tion y which is the solution of (12.1). You can easily verify that x(xy ′ ) ′ =x 2 y ′′ +xy ′ , so we can write (12.1) in the simpler form
x(xy ′ ) ′ + (x 2 −p 2 )y = 0.
We find a generalized power series for (12.2) in the same way that we solved (11.2). [In fact, (11.2) is a form of Bessel’s equation! See Problems 16.1 and 17.1.] Writing only the general terms in the series for y and the derivatives we need in (12.2), we have
a n (n + s)x n+s−1
a n (n + s) 2 x n+s−1
n=0 ∞
x(xy ′ ) ′ =
a n (n + s) 2 x n+s
n=0
We substitute (12.3) into (12.2) and tabulate the coefficients of powers of x:
x s+n x(xy ′ ) ′
The coefficient of x s gives the indicial equation and the values of s:
s 2 −p 2 = 0,
s = ±p.
The coefficient of x s+1 gives a 1 = 0. The coefficient of x s+2 gives a 2 in terms of a 0 , etc., but we may as well write the general formula from the last column at this point. We get
[(n + s) 2 −p 2 ]a n +a n−2 =0
or
a n n−2 =−
(n + s) 2 −p 2
Section 12 Bessel’s Equation 589
First we shall find the coefficients for the case s = p. From (12.4) we have
a a (12.5)
n(n + 2p) Since a 1 = 0, all odd a’s are zero. For even a’s it is convenient to replace n by 2n;
(n + p) 2 −p 2
n + 2np
then from (12.5) we have
2n =−
a 2n−2
2n−2
2n(2n + 2p)
2 2 n(n + p)
The formulas for the coefficients can be simplified by the use of the Γ function notation (Chapter 11, Sections 2 to 5) as you can see by examining (12.7) below. Recall that Γ(p + 1) = pΓ(p) for any p, so,
Γ(p + 2) = (p + 1)Γ(p + 1), Γ(p + 3) = (p + 2)Γ(p + 2) = (p + 2)(p + 1)Γ(p + 1),
and so on. Then from (12.6) we find
(1 + p)(2 + p)
a 6 =−
3!2(3 + p) 3!2 6 (1 + p)(2 + p)(3 + p)
and so on. Then the series solution (for the s = p case) is
2!Γ(3 + p) 2 3!Γ(4 + p) 2
(12.8) p x p
=a 0 2 Γ(1 + p)
2 Γ(1)Γ(1 + p) − Γ(2)Γ(2 + p) 2
Γ(3)Γ(3 + p) 2 − Γ(4)Γ(4 + p) 2 +··· . We have inserted Γ(1) and Γ(2) (which are both equal to 1) in the first two terms
and written x p =2 p (x/2) p to make the series appear more systematic. If we take
then y is called the Bessel function of the first kind of order p, and written J p (x).
590 Series Solutions of Differential Equations Chapter 12
1 x 2+p J p (x) =
Γ(1)Γ(1 + p) 2 Γ(2)Γ(2 + p) 2
1 x 6+p (12.9)
1 x 4+p
+··· Γ(3)Γ(3 + p) 2 Γ(4)Γ(4 + p) 2
(−1) n
x 2n+p
n=0 Γ(n + 1)Γ(n + 1 + p) 2
PROBLEMS, SECTION 12
1. Show by the ratio test that the infinite series (12.9) for J p (x) converges for all x. Use (12.9) to show that:
2. J 2 (x) = (2/x)J 1 (x) − J 0 (x)
3. J 1 (x) + J 3 (x) = (4/x)J 2 (x)
4. (d/dx)J 0 (x) = −J 1 (x)
5. (d/dx)[xJ 1 (x)] = xJ 0 (x)
6. J 0 (x) − J 2 (x) = 2(d/dx)J 1 (x)
7. lim J
1 (x)/x = 1 2
x→0
8. x→0 lim x −3/2 J 3/2 (x) = 3 −1 p2/π Hint: See Chapter 11, equations (3.4) and (5.3). 9. pπx/2 J 1/2 (x) = sin 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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