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# ch6 - Chapter 6 Bessel functions 6.1 Motivation The...

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Chapter 6: Bessel functions 6.1. Motivation The 2-dimensional wave equation in polar coordinates is v tt = c 2 v rr + v r r + v θθ r 2 . If we apply separation of variables, v = R ( r )Θ( θ ) T ( t ), the wave equation becomes T c 2 T = R R + R rR + Θ r 2 Θ . Both sides must be equal to a constant which we call - μ 2 . Setting the expres- sion on the right-hand side equal to - μ 2 and multiplying by r 2 , we obtain r 2 R R + rR R + μ 2 r 2 = - Θ Θ . Here both sides must be equal to another constant, which we call ν 2 , so we arrive at the ODE’s T + c 2 μ 2 T = 0 , and Θ + ν 2 Θ = 0 , (1) which are familiar, and r 2 R ( r ) + rR ( r ) + ( μ 2 r 2 - ν 2 ) R ( r ) = 0 , (2) which is new. The last equation can be simplified a bit by the change of variable x := μr . That is, we substitute R ( r ) = f ( μr ). Then (2) becomes x 2 f ( x ) + xf ( x ) + ( x 2 - ν 2 ) f ( x ) = 0 . (3) This is Bessel’s equation of order ν . It and its variants arise in many problems in physics and engineering, in particular where some sort of circu- lar symmetry is involved. For this reason, its solutions are sometimes called cylinder functions , but we shall use the more common term Bessel func- tions . We will state some of their properties in section 6.2, without proving them, and then apply these properties to some boundary value problems in section 6.3. For a more detailed study of Bessel’s equation we refer to the book of G. Folland, ”Fourier analysis and its applications”, and G.N. Watson, ”A treatise on the theory of Bessel functions”. 44

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6.2. Solutions of Bessel’s equation If ν is not a negative integer then Bessel’s equation (3) has solutions, ( c IR), f ( x ) = c k =0 ( - 1) k x 2 k + ν 2 2 k k !(1 + ν )(2 + ν ) · · · ( k + ν ) (4) We now make use of the definition and some properties of the Gamma func- tion Γ: It is defined as Γ( z ) := 0 t z - 1 e - t dt, ( z IC , Re z > 0) . (5) and satisfies the functional equation Γ( z + 1) = z Γ( z ) , (6) and in particular Γ( n ) = 1 · 2 · · · n Γ(1) = n ! , ( n IN) . (7) Choosing c = (2 - ν / Γ( ν + 1)), and using the fact that Γ( k + ν + 1) = ( k + ν ) · · · (1 + ν )Γ( ν + 1) , (4) becomes f ( x ) = k =0 ( - 1) k k !Γ( k + ν + 1) x 2 2 k + ν =: J ν ( x ) . (8) It can be shown easily that this series converges absolutely for all x = 0 (and also for x = 0 when Re ( ν ) > 0 or ν = 0). The function J ν thus defined is called the Bessel function (of the first kind) of order ν . In this chapter we will be only interested in the case that x > 0 and ν IR. Then J ν ( x ) is a real function. Moreover, we then have that lim x 0 J ν ( x ) = 0 whenever ν > 0, and blows up as x 0 whenever ν < 0 and ν is not an integer. If ν is not an integer , then J ν and J - ν are linear independent solutions of (3), so that the general solution of (3) becomes f ( x ) = c 1 J ν ( x ) + c 2 J - ν ( x ) , ( c 1 , c 2 IR) . 45
On the other hand, when ν IN, we can use the fact that that j ! = Γ( j + 1) to write J n ( x ) = k =0 ( - 1) k k !( n + k )! x 2 2 k + n , ( n = 0 , 1 , 2 , . . . ) . It also can be proved that J - n ( x ) = ( - 1) n J n ( x ) , ( n = 0 , 1 , . . . ) , (9) so that J n and J - n are not linear independent functions. However, it can be shown that if ν is not an integer, then another solution of (3), which is linearly independent from J ν , is given by the Bessel function of the second kind or Neumann function , Y

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ch6 - Chapter 6 Bessel functions 6.1 Motivation The...

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