Survival Guide to Bessel Functions

Transcription

1 Survival Guide to Bessel Functions December, 13 1 The Problem (Original by Mike Herman; edits and additions by Paul A.) For cylindrical boundary conditions, Laplace s equation is: [ 1 s Φ ] + 1 Φ s s s s φ + Φ z = (1) The way s mies throughout the equation means that the z and φ behavior will have a combined nontrivial influence on the very equation for the function of s in the mode epansion found by separating variables. The resulting functions of s are called Bessel functions, and the usual notation to denote them shows the z and φ behavior in different places: the φ behavior gives the order of the Bessel function and appears in the subscript, while the z behavior appears in the argument of the Bessel function, multiplied by s. Choices: It will turn out that the type of solution encountered depends crucially upon the domain being considered, including the type of boundary conditions assumed. A flow chart method is sketched in the following paragraphs in order to help decide which functions are appropriate. Behavior with z: The first choice concerns the z behavior. In the separation of variables technique, we need the eigenvalues of the sub-operators corresponding to the different coordinates to all sum to zero, in each of the basic solutions used to construct the general solution. This leaves a choice of sign for the eigenvalue of ( / z), and the sign chosen (to satisfy the necessary boundary conditions) determines whether the ordinary or modifield Bessels appear for the radial part. For a positive eigenvalue (eponential behavior in z), the ordinary Bessels (J ν and Y ν ) appear. For a negative eigenvalue (oscillating behavior in z), we find the modified Bessels (I ν and K ν ). Behavior with φ: The net domain to consider is that of φ. The sign ambiguity usually doesn t arise in the eigenvalue of ( / φ) : we usually choose its eigenvalue to be negative (and equal to ν ) because φ is periodic. (It is conceivable that the positive eigenvalues will appear for special situations, such as zero potential on the surface φ = and a fied nonzero potential at another surface φ = φ, giving sinh(νφ) behavior and thus a positive eigenvalue. These cases lead to Bessels of purely imaginary order, and are relatively rare.) When the entire range of φ is in the domain where Laplace s equation holds, this gives ν = m = an integer. In other cases, ν will be found by fitting the boundary conditions. For instance, if the domain is limited to a slice of pie of angular width β, ν will not be restricted to integers, since we don t require periodicity of length in that case. Instead, ν = m β, with m = 1,,3,..., for Dirichlet boundary conditions on the boundaries. For Neumann boundary conditions, the same values of ν apply, but m = is now allowed (since we have cosines rather than sines). For mied boundary condtions (Dirichlet on one side, and Neumann on the other), we must fit an odd number of quarter-wavelengths into φ, so ν = (m+1) β in that case, with m =,1,,... Resulting equations for s: With the above behavior in φ assumed, we can replace φ derivatives with im and z derivatives with +k (case A) or k (case B) in the cylindrical Laplacian (eq. 1 above) to arrive at an equation for S(s). These choices correspond to passing to a Fourier series in φ and either doing a Laplace transform in z (case A) or Fourier transform in z (case B). Case A gives the Bessel equation, which is eq. (6) below, and that for case B gives the modified Bessel equation, which replaces k by k in the 1

2 equation. Recall that we may write nonlinear functions only of dimensionless variables, so both of these can be put into a standard form by switching to the dimensionless variable = ks: [ 1 d ds ] ( d d 1 S = () where the sign gives the ordinary Bessel equation, and the + sign gives the modified Bessel equation. Some of the properties given below are in terms of s and some in terms of the dimensionless instead, for convenience. (Be careful especially when doing derivatives to watch factors of k when translating from s to or back). Solutions Assuming a solution of (sums of terms of) the form we find for the two general cases: Φ(s, φ, z) = S(s)Ψ(φ)Z(z), (3).1 Case A: Non-Cyclic/Eponential in Z (k R, k > ) giving d S ds + 1 ds s ds + d Z dz k Z = () d Ψ dφ +ν Ψ = (5) (k ν s ) S = (6) Note that k is determined by either z or s boundary conditions; e. g., if cylinder is grounded at top (z = L) and bottom (z = ), then the z dependence determines k: Z(z) sin(k n z), where k n = n L. If the cylinder is grounded (Dirichlet boundary conditions) at its radial edge (s = a) instead, then k will be restricted to k n = νn /a, n = 1,,3,..., where νn is the n th zero of J ν : J ν ( νn ) =. (These are instead called z νn in Cahill.) For Neumann boundary conditions, the zeroes of the derivatives of J ν appear (called νn and/or y νn in Jackson): [(d/d)j ν ()] = νn =. Bessel functions of the first kind of order ±ν are the solution for Case A, where: e kz,e kz Z(z) sinh(kz), cosh(kz) This solution is used when you have non-cyclic behavior in z, as in the case where Φ(z = ) = and Φ(z = L) = V(s,φ). In this case, the solutions to the radial equation are: S(s) = J ±ν (ks) However, the two solutions (for the two signs of ν) are linearly dependent when ν = m Z, so we need another solution: S(s) = Y ν (ks) = J ν(ks)cos(ν) J ν (ks) sin(ν) (with a limit ν integer m understood for integer ν.) The above solutions J ν and Y ν are always linearly independent (even when ν / Z). (Note that Y ν is often instead denoted N ν in some tets, such as Jackson.) The domain in question determines which of these is

3 used: if s = is included in the domain, only the J ν appear. (This follows from the s behavior listed below: the Y ν (s) blow up as s.) If s = is not included, we must keep a linear combination of both the J ν and the Y ν in a general solution. Limiting forms of these functions are given for = ks as: Y ν () J ν () 1 Γ(ν +1) ( Γ(ν) [ ( ln ) ] +.577, ν = (, ν where Γ(ν + 1) is the Gamma function: when ν = m = an integer, Γ(m + 1) = m!, the factorial function. Plugging in, we see that J () = 1 and J ν () =, while Y ν ( ). For the behavior at large argument, we have J m ( ) ( cos ν ) Y m ( ) ( sin ν ) Because of this asymptotic behavior, it is sometimes convenient to introduce instead as the two independent solutions the Hankel functions H 1 and H, defined by which will thus have e ±i / behavior as. H (1) ν (ks) = J ν (ks)+iy ν (ks) H () ν (ks) = J ν (ks) iy ν (ks).1.1 Orthogonality and Completeness Relations On the infinite domain s <, the orthogonality relation is and the completeness relation is sj ν (ks)j ν (k s)ds = δ(k k ) k kj ν (ks)j ν (ks )dk = δ(s s ). s Note the appearance of s ds as the integration measure; this comes naturally from the volume element in cylindrical coordinates. (You ve likely noticed that these are basically the same integral with variables switched; recall that orthogonality and completeness also gave the same integral as one another with variables switched for the modes in Fourier transforms, where the domain was also infinite.) When the domain is instead the interior of a finite cylinder of radius a with Dirichlet conditions at s = a, orthogonality becomes instead a sj ν (z νn s/a)j ν (z νn s/a)ds = a J ν+1(z νn )δ nn. where z νn is the n th zero of J ν, as mentioned above. Note that this making the domain finite has given us discrete modes ( quantized numbers) instead, just as it did when changing from Fourier transforms for infinite domain to Fourier series on a finite domain..1. Generating Function [ ] ep (t 1/t) = m= t m J m () 3

4 .1.3 Recursion Relations All solutions of the ordinary Bessel equation (S can be J, Y, H (1), or H () ) obey. Case B: Solutions Cyclic in Z [S ν 1()+S ν+1 ()] = νs ν () S ν 1 () S ν+1 () = d d S ν() Modified Bessel functions are the solutions for Case B, where: e ikz,e ikz Z(z) sin(kz), cos(kz) This solution is used when you have cyclic behavior in z, as in the case where Φ(z = ) = and Φ(z = L) =. These are equivalent to making k imaginary above, but it is more convenient to introduce eplicit real-valued solutions. Real-valued linearly independent solutions are: I ν (ks) = i ν J ν (iks) (7)..1 Properties As, we have K ν (ks) = iν+1 H (1) ν (iks) (8) I ν ( ) 1 ( Γ(ν +1) [ ln ( ] K ν ( ) ) +.577, ν =, ν Γ(ν) ( and so the I ν behave the same as J ν for small : I () = 1 and I ν () =, while the K ν blow up as goes to zero. The similarity with ordinary Bessels is not shared at large, where: I ν ( ) 1 e K ν ( ) e.. Generating Function [ ] ep (t+1/t) = m= t m I m ()..3 Recursion Relations Solutions of the modified Bessel equation obey slightly different relations: [I ν 1() I ν+1 ()] = νi ν () I ν 1 ()+I ν+1 () = d d I ν()

5 for the I ν, but an etra minus sign for the K ν : 3 A few other useful integrals [K ν 1() K ν+1 ()] = νk ν () K ν 1 ()+K ν+1 () = d d K ν() (See also orthogonality and completeness integrals given above.) J ν(ks)sds = 1 J ν (ks)ds = 1/k (s ν k ) J ν(ks)+ 1 s J ν(ks) When J vanishes on the boundaries, this reduces to the J term only, and we can use the recurrence formulas to replace this by Jν+1, as is done in the orthogonality integrals. For integer m, we have the integral representation which generalizes to arbitrary order ν by J ν () = 1 J m () = 1 cos[νφ sinφ] dφ sin(ν) Another integral which appears surprisingly often in physics is K ν () = cos[mφ sinφ] dφ (9) ep[ νt sinht]dt. (1) e cosht cosh(νt)dt (11) 5