An Introduction to Bessel Functions

Transcription

1 An Introduction to R. C. Trinity University Partial Differential Equations March 25, 2014

2 Bessel s equation Given p 0, the ordinary differential equation x 2 y +xy +(x 2 p 2 )y = 0, x > 0 (1) is known as Bessel s equation of order p. Solutions to (1) are known as Bessel functions. Since (1) is a second order homogeneous linear equation, the general solution is a linear combination of any two linearly independent (i.e. fundamental) solutions. We will describe and give the basic properties of the most commonly used pair of fundamental solutions.

3 The method of Frobenius We begin by assuming the solution has the form y = a m x r+m (a 0 0) m=0 and try to determine r and a m. Substituting into Bessel s equation and collecting terms with common powers of x gives a 0 (r 2 p 2 )x r ( +a 1 (r +1) 2 p 2) x r+1 + ( ( am (r +m) 2 p 2) ) +a m 2 x r+m = 0. m=2

4 Setting the coefficients equal to zero gives the equations a 0 (r 2 p 2 ) = 0 r = ±p, a 1 ( (r +1) 2 p 2) = 0 a 1 = 0, ( a m (r +m) 2 p 2) +a m 2 = 0 a m 2 a m = (r +m) 2 p 2 = a m 2 m(m+2p) (m 2). This means that a 1 = a 3 = a 5 = = a 2k+1 = 0 and a 0 a 2 = 2(2+2p) = a (1+p), a 2 a 4 = 4(4+2p) = a (2+p) = a (1+p)(2+p), a 4 a 6 = 6(6+2p) = a (3+p) = a !(1+p)(2+p)(3+p), a 6 a 8 = 8(8+2p) = a (4+p) = a !(1+p)(2+p)(3+p)(4+p).

5 In general, we see that a 2k = ( 1) k a 0 2 2k k!(1+p)(2+p) (k +p). Setting r = p and m = 2k in the original series gives y = = k=0 k=0 ( 1) k a 0 2 2k k!(1+p)(2+p) (k +p) x2k+p ( 1) k 2 p a ( 0 x ) 2k+p. k!(1+p)(2+p) (k +p) 2 The standard way to choose a 0 involves the so-called Gamma function.

6 Interlude The Gamma function The Gamma function is defined to be Γ(x) = 0 e t t x 1 dt (x > 0). One can use integration by parts to show that Γ(x +1) = x Γ(x). Applying this repeatedly, we find that for k N Γ(x +k) = (x +k 1)Γ(x +k 1) = (x +k 1)(x +k 2)Γ(x +k 2) = (x +k 1)(x +k 2)(x +k 3)Γ(x +k 3). = (x +k 1)(x +k 2)(x +k 3) x Γ(x).

7 This has two nice consequences. According to the definition, one has Γ(1) = 0 e t dt = 1. Setting x = 1 above: Γ(k +1) = k(k 1)(k 2) 1 Γ(1) = k! This is why Γ(x) is called the generalized factorial. Setting x = p +1 above: Γ(p +1+k) = (p +k)(p +k 1) (p +1)Γ(p +1) or 1 Γ(p +1) = (1+p)(2+p) (k +p) Γ(k +p +1).

8 Bessel functions of the first and second kind Returning to Bessel s equation, choosing a 0 = that is one solution. x 2 y +xy +(x 2 p 2 )y = 0, x > p Γ(p +1) y = J p (x) = k=0 in the Frobenius solution, we now see ( 1) k ( x ) 2k+p, k!γ(k +p +1) 2 J p (x) is called the Bessel function of the first kind of order p.

9 Remarks A second linearly independent solution can be found via reduction of order. When (appropriately normalized), it is denoted by Y p (x), and is called the Bessel function of the second kind of order p. The general solution to Bessel s equation is y = c 1 J p (x)+c 2 Y p (x). In Maple, the functions J p (x) and Y p (x) are called by the commands BesselJ(p,x) and BesselY(p,x).

10 Graphs of Bessel functions

11 Properties of Bessel functions J 0 (0) = 1, J p (0) = 0 for p > 0 and lim x 0 +Y p(x) =. The values of J p always lie between 1 and 1. J p has infinitely many positive zeros, which we denote by 0 < α p1 < α p2 < α p3 < J p is oscillatory and tends to zero as x. More precisely, J p (x) 2 (x πx cos pπ 2 π ). 4 lim α pn α p,n+1 = π. n

12 For 0 < p < 1, the graph of J p has a vertical tangent line at x = 0. For 1 < p, the graph of J p has a horizontal tangent line at x = 0, and the graph is initially flat. For some values of p, the Bessel functions of the first kind can be expressed in terms of familiar functions, e.g. 2 J 1/2 (x) = πx sinx, (( 2 3 J 5/2 (x) = )sinx πx x 2 1 3x ) cosx.

13 Differentiation identities Using the series definition of J p (x), one can show that: d dx (xp J p (x)) = x p J p 1 (x), d ( x p J p (x) ) = x p J p+1 (x). dx The product rule and cancellation lead to (2) xj p (x)+pj p(x) = xj p 1 (x), xj p(x) pj p (x) = xj p+1 (x). Addition and subtraction of these identities then yield J p 1 (x) J p+1 (x) = 2J p(x), J p 1 (x)+j p+1 (x) = 2p x J p(x).

14 Integration identities Integration of the differentiation identities (2) gives x p+1 J p (x)dx = x p+1 J p+1 (x)+c x p+1 J p (x)dx = x p+1 J p 1 (x)+c. Exercises and give similar identities. Identities such as these can be used to evaluate certain integrals of the form a 0 f(r)j m (λ mn r)r dr, which will occur frequently in later work.

15 Example Evaluate x p+5 J p (x)dx. We integrate by parts, first taking u = x 4 du = 4x 3 dx dv = x p+1 J p (x)dx v = x p+1 J p+1 (x), which gives x p+5 J p (x)dx = x p+5 J p+1 (x) 4 x p+4 J p+1 (x)dx.

16 Now integrate by parts again with u = x 2 du = 2x dx dv = x p+2 J p+1 (x)dx v = x p+2 J p+2 (x), to get x p+5 J p (x)dx = x p+5 J p+1 (x) 4 x p+4 J p+1 (x)dx ( ) = x p+5 J p+1 (x) 4 x p+4 J p+2 (x) 2 x p+3 J p+2 (x)dx = x p+5 J p+1 (x) 4x p+4 J p+2 (x)+8x p+3 J p+3 (x)+c.

17 The parametric form of Bessel s equation For p 0, consider the parametric Bessel equation x 2 y +xy +(λ 2 x 2 p 2 )y = 0 (λ > 0). (3) If we let ξ = λx, then the chain rule implies Hence (3) becomes y = dy dx = dy dξ dξ dx = λẏ, y = dy dx = λdẏ dx = λdẏ dξ dξ dx = λ2 ÿ. ξ 2 ÿ +ξẏ +(ξ 2 p 2 )y = 0, which is Bessel s equation in the variable ξ.

18 It follows that y = c 1 J p (ξ)+c 2 Y p (ξ) = c 1 J p (λx)+c 2 Y p (λx) gives the general solution to the parametric Bessel equation. Because lim x 0 +Y p(x) =, we find that y(0) is finite c 2 = 0, so that the only solutions that are defined at x = 0 are y = c 1 J p (λx). This will be important in later work.