Survival Guide to Bessel Functions

December 4, 2013

(Original by Mike Herman; edits and additions by Paul A.)

1 The Problem

For cylindrical boundary conditions, Laplace’s equation is:

1

s

∂

∂s

[

s∂Φ

∂s

]

+1

s2∂2Φ

∂φ2+

∂2Φ

∂z2= 0 (1)

The way s mixes throughout the equation means that the z and φ behavior will have a combined nontrivialinfluence on the very equation for the function of s in the mode expansion found by separating variables.The resulting functions of s are called Bessel functions, and the usual notation to denote them shows the zand φ behavior in different places: the φ behavior gives the order of the Bessel function and appears in thesubscript, 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 beingconsidered, including the type of boundary conditions assumed. A “flow chart” method is sketched in thefollowing 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, weneed the eigenvalues of the sub-operators corresponding to the different coordinates to all sum to zero, in eachof the basic solutions used to construct the general solution. This leaves a choice of sign for the eigenvalue of(∂/∂z)2, and the sign chosen (to satisfy the necessary boundary conditions) determines whether the ordinaryor modifield Bessels appear for the radial part. For a positive eigenvalue (exponential behavior in z), theordinary Bessels (Jν and Yν) appear. For a negative eigenvalue (oscillating behavior in z), we find themodified Bessels (Iν and Kν).

Behavior with φ: The next domain to consider is that of φ. The sign ambiguity usually doesn’t arisein the eigenvalue of (∂/∂φ)2: we usually choose its eigenvalue to be negative (and equal to −ν2) becauseφ is periodic. (It is conceivable that the positive eigenvalues will appear for special situations, such as zeropotential on the surface φ = 0 and a fixed nonzero potential at another surface φ = φ0, giving sinh(νφ)behavior and thus a positive eigenvalue. These cases lead to Bessels of purely imaginary order, and arerelatively rare.) When the entire range of φ is in the domain where Laplace’s equation holds, this givesν = m = an integer. In other cases, ν will be found by fitting the boundary conditions. For instance, if thedomain is limited to a “slice of pie” of angular width β, ν will not be restricted to integers, since we don’trequire periodicity of length 2π in that case. Instead, ν = mπ

β, with m = 1, 2, 3, . . ., for Dirichlet boundary

conditions on the boundaries. For Neumann boundary conditions, the same values of ν apply, but m = 0 isnow allowed (since we have cosines rather than sines). For mixed boundary condtions (Dirichlet on one side,

and Neumann on the other), we must fit an odd number of quarter-wavelengths into φ, so ν = (2m+1)π2β in

that case, with m = 0, 1, 2, . . ..Resulting equations for s: With the above behavior in φ assumed, we can replace φ derivatives with

im and z derivatives with +k2 (case A) or −k2 (case B) in the cylindrical Laplacian (eq. 1 above) to arrive atan equation for S(s). These choices correspond to passing to a Fourier series in φ and either doing a Laplacetransform in z (case A) or Fourier transform in z (case B). Case A gives the Bessel equation, which iseq. (6) below, and that for case B gives the modified Bessel equation, which replaces k2 by −k2 in the

1

equation. Recall that we may write nonlinear functions only of dimensionless variables, so both of these canbe put into a standard form by switching to the dimensionless variable x = ks:

1

x

d

dx

[

xdS

dx

]

−(

ν2

x2∓ 1

)

S = 0 (2)

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 x instead, forconvenience. (Be careful especially when doing derivatives to watch factors of k when translating from s tox or back).

2 Solutions

Assuming a solution of (sums of terms of) the form

Φ(s, φ, z) = S(s)Ψ(φ)Z(z), (3)

we find for the two general cases:

2.1 Case A: Non-Cyclic/Exponential in Z (k ∈ R, k > 0)

d2Z

dz2− k2Z = 0 (4)

d2Ψ

dφ2+ ν2Ψ = 0 (5)

giving

d2S

ds2+

1

s

dS

ds+

(

k2 − ν2

s2

)

S = 0 (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 = 0), then the z dependence determines k: Z(z) ∝ sin(knz), where kn = nπ

L. If

the cylinder is grounded (Dirichlet boundary conditions) at its radial edge (s = a) instead, then k will berestricted to kn = xνn/a, n = 1, 2, 3, . . ., where xνn is the nth zero of Jν : Jν(xνn) = 0. (These are insteadcalled zνn in Cahill.) For Neumann boundary conditions, the zeroes of the derivatives of Jν appear (calledx′νn and/or yνn in Jackson): [(d/dx)Jν(x)]x=x′

νn

= 0.Bessel functions of the first kind of order ±ν are the solution for “Case A”, where:

Z(z) ∝{

ekz, e−kz

sinh(kz), cosh(kz)

This solution is used when you have non-cyclic behavior in z, as in the case where Φ(z = 0) = 0 andΦ(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 needanother solution:

S(s) = Yν(ks) =Jν(ks) cos(νπ)− J−ν(ks)

sin(νπ)

(with a limit ν → integer m understood for integer ν.)The above solutions Jνand Yν are always linearly independent (even when ν /∈ Z). (Note that Yν is often

instead denoted Nν in some texts, such as Jackson.) The domain in question determines which of these is

2

used: if s = 0 is included in the domain, only the Jν appear. (This follows from the s → 0 behavior listedbelow: the Yν(s) blow up as s → 0.) If s = 0 is not included, we must keep a linear combination of both theJν and the Yν in a general solution.

Limiting forms of these functions are given for x = ks → 0 as:

Jν(x) →1

Γ(ν + 1)

(x

2

)ν

Yν(x) →{

2π

[

ln(

x2

)

+ 0.5772]

, ν = 0

−Γ(ν)π

(

2x

)ν, ν 6= 0

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 J0(0) = 1 and Jν 6=0(0) = 0, while Yν(x → 0) → ∞.

For the behavior at large argument, we have

Jm(x → ∞) −→√

2

πxcos

(

x− νπ

2− π

4

)

Ym(x → ∞) −→√

2

πxsin

(

x− νπ

2− π

4

)

Because of this asymptotic behavior, it is sometimes convenient to introduce instead as the two indepen-dent solutions the Hankel functions H1 and H2, defined by

H(1)ν (ks) = Jν(ks) + iYν(ks)

H(2)ν (ks) = Jν(ks)− iYν(ks)

which will thus have e±ix/√x behavior as x → ∞.

2.1.1 Orthogonality and Completeness Relations

On the infinite domain 0 ≤ s < ∞, the orthogonality relation is∫ ∞

0

s Jν(ks) Jν(k′s) ds =

δ(k − k′)

k

and the completeness relation is∫ ∞

0

k Jν(ks) Jν(ks′) dk =

δ(s− s′)

s.

Note the appearance of s ds as the integration measure; this comes naturally from the volume elementin cylindrical coordinates. (You’ve likely noticed that these are basically the same integral with variablesswitched; recall that orthogonality and completeness also gave the same integral as one another with variablesswitched 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

0

s Jν(zνn′s/a) Jν(zνns/a) ds =a2

2J2ν+1(zνn) δnn′ .

where zνn is the nth zero of Jν , as mentioned above. Note that this making the domain finite has givenus discrete modes (“quantized” numbers) instead, just as it did when changing from Fourier transforms forinfinite domain to Fourier series on a finite domain.

2.1.2 Generating Function

exp[x

2(t− 1/t)

]

=

∞∑

m=−∞

tmJm(x)

3

2.1.3 Recursion Relations

All solutions of the ordinary Bessel equation (S can be J , Y , H(1), or H(2)) obey

x

2[Sν−1(x) + Sν+1(x)] = νSν(x)

Sν−1(x)− Sν+1(x) = 2d

dxSν(x)

2.2 Case B: Solutions Cyclic in Z

Modified Bessel functions are the solutions for “Case B,” where:

Z(z) ∝{

eikz, e−ikz

sin(kz), cos(kz)

This solution is used when you have cyclic behavior in z, as in the case where Φ(z = 0) = 0 andΦ(z = L) = 0. These are equivalent to making k imaginary above, but it is more convenient to introduceexplicit real-valued solutions. Real-valued linearly independent solutions are:

Iν(ks) = i−νJν(iks) (7)

Kν(ks) =π

2iν+1H(1)

ν (iks) (8)

2.2.1 Properties

As x → 0, we have

Iν(x → 0) → 1

Γ(ν + 1)

(x

2

)ν

Kν(x → 0) →{

−[

ln(

x2

)

+ 0.5772]

, ν = 0Γ(ν)2

(

2x

)ν, ν 6= 0

and so the Iν behave the same as Jν for small x: I0(0) = 1 and Iν 6=0(0) = 0, while the Kν blow up as x goesto zero. The similarity with ordinary Bessels is not shared at large x, where:

Iν(x → ∞) −→ 1√2πx

ex

Kν(x → ∞) −→√

π

2xe−x

2.2.2 Generating Function

exp[x

2(t+ 1/t)

]

=∞∑

m=−∞

tmIm(x)

2.2.3 Recursion Relations

Solutions of the modified Bessel equation obey slightly different relations:

x

2[Iν−1(x)− Iν+1(x)] = νIν(x)

Iν−1(x) + Iν+1(x) = 2d

dxIν(x)

4

for the Iν , but an extra minus sign for the Kν :

x

2[Kν−1(x)−Kν+1(x)] = −νKν(x)

Kν−1(x) +Kν+1(x) = −2d

dxKν(x)

3 A few other useful integrals

(See also orthogonality and completeness integrals given above.)

∫ ∞

0

Jν(ks) ds = 1/k

∫

J2ν (ks) s ds =

1

2

(

s2 − ν2

k2

)

J2ν (ks) +

1

2s2J ′2

ν(ks)

When J vanishes on the boundaries, this reduces to the J ′2 term only, and we can use the recurrence formulasto replace this by J2

ν+1, as is done in the orthogonality integrals.For integer m, we have the integral representation

Jm(x) =1

π

∫ π

0

cos [mφ− x sinφ] dφ (9)

which generalizes to arbitrary order ν by

Jν(x) =1

π

∫ π

0

cos [νφ− x sinφ] dφ − sin(νπ)

π

∫ ∞

0

exp[−νt− x sinh t] dt . (10)

Another integral which appears surprisingly often in physics is

Kν(x) =

∫ ∞

0

e−x cosh t cosh(νt) dt (11)

5