Jim LambersMAT 415/515

Fall Semester 2013-14Lecture 15 Notes

These notes correspond to Section 14.2 in the text.

Orthogonality of Bessel Functions

Since Bessel functions often appear in solutions of PDE, it is necessary to be able to computecoefficients of series whose terms include Bessel functions. Therefore, we need to understand theirorthogonality properties.

Consider the Bessel equation

ρ2d2Jν(kρ)

dρ2+ ρ

dJν(kρ)

dρ+ (k2ρ2 − ν2)Jν(kρ) = 0,

where ν ≥ −1. Rearranging yields

−(d2

dρ2+

1

ρ

d

dρ− ν2

ρ2

)Jν(kρ) = k2Jν(kρ).

Thus Jν(kρ) is an eigenfunction of the linear differential operator

L = −(d2

dρ2+

1

ρ

d

dρ− ν2

ρ2

)with eigenvalue k2.

The operator L is not self-adjoint with respect to the standard scalar product, as the coefficientsp0(ρ) = −1 and p1(ρ) = −1/ρ do not satisfy the condition p′1(ρ) = p0(ρ), so we use the weightfunction

w(ρ) = p0(ρ)e∫ p1(ρ)p0(ρ)

dρ= e

∫1/ρ dρ = eln ρ = ρ.

It follows from the relation

(λu − λv)∫ b

av∗(x)u(x)w(x) dx =

(w(x)p0(x)(v∗(x)u′(x)− (v∗)′(x)u(x)

)∣∣ba

that ∫ a

0ρJν(kρ)Jν(k′ρ) dρ =

a[k′Jν(ka)J ′ν(k′a)− kJν(k′a)J ′ν(ka)]

k2 − k′2.

Therefore, in order to ensure orthogonality, we must have ka and k′a be zeros of Jν . Thus we havethe orthogonality relation ∫ a

0ρJν

(ανi

ρ

a

)Jν

(ανj

ρ

a

)dρ = 0, i 6= j,

where ανj is the jth zero of Jν .It is worth noting that because of the weight function ρ being the Jacobian of the change of

variable to polar coordinates, Bessel functions that are scaled as in the above orthogonality relationare also orthogonal with respect to the unweighted scalar product over a circle of radius a.

1

Normalization

Now that we have orthogonal Bessel functions, we seek orthonormal Bessel functions. From∫ a

0ρ[Jν(kρ)]2 dρ = lim

k′→k

a[k′Jν(ka)J ′ν(k′a)− kJν(k′a)J ′ν(ka)]

k2 − k′2.

Substituting ka = ανi and applying l’Hospital’s Rule yields∫ a

0ρ[Jν(kρ)]2 dρ = lim

k′→k−kaJν(k′a)J ′ν(ανi)

k2 − k′2=a2

2[J ′ν(ανi)]

2.

Using the recurrence relation

Jν(x) = ±J ′ν±1(x) +ν ± 1

xJν±1(x),

we then obtain ∫ a

0ρ[Jν(kρ)]2 dρ =

a2

2[Jν+1(ανi)]

2.

Bessel Series

Now we can easily describe functions as series of Bessel functions. If f(ρ) has the expansion

f(ρ) =

∞∑j=1

cνjJν

(ανj

ρ

a

), 0 ≤ ρ ≤ a, ν > −1,

Then, the coefficients cνj are given by

cνj =

⟨Jν(ανj

ρa

)|f(ρ)

⟩⟨Jν(ανj

ρa

)|Jν(ανj

ρa

)⟩ =2

a2[Jν+1(ανj)]2

∫ a

0ρJν

(ανj

ρ

a

)f(ρ) dρ.

It is worth noting that orthonormal sets of Bessel functions can also be obtained by imposingNeumann boundary conditions J ′ν(kρ) = 0 at ρ = a, in which case ka = βνj , where βνj is the jthzero of J ′ν .

We now consider an example in which a Bessel series is used to describe a solution of a PDE.

Example Consider Laplace’s equation in a hollow cylinder of radius a with endcaps at z = 0 andz = h,

∇2V = 0,

with boundary conditions

V (a, ϕ, z) = 0, V (ρ, ϕ, 0) = 0, V (ρ, ϕ, h) = f(ρ, ϕ)

for a given potential function f(ρ, ϕ). In cylindrical coordinates, Laplace’s equation becomes

1

ρ

∂

∂ρ

(ρ∂V

∂ρ

)+

1

ρ2∂2V

∂ϕ2+∂2V

∂z2= 0.

Using separation of variables, we assume a solution of the form

V (ρ, ϕ, z) = P (ρ)Φ(ϕ)Z(z).

2

Substituting this form into the PDE and dividing by V yields

1

ρP

d

dρ

(ρdP

dρ

)+

1

ρ2Φ

d2Φ

dϕ2+

1

Z

d2Z

dz2= 0.

Due to periodicity, we require that Φ satisfy

d2Φ

dϕ2= −m2Φ,

where m is an integer, and therefore Φ(ϕ) = eimϕ. We also have

d2Z

dz2= `2Z,

which has solutions e`z. Because of the boundary condition at z = 0, we take linear combinationsof solutions so that

Z(z) = sinh lz.

Then, we have

ρ2d2P

dρ2+ ρ

dP

dρ+ (`2ρ2 −m2)P = 0.

This is the Bessel equation of order m, which has solutions Jm(`ρ). To satisfy the boundarycondition at ρ = a, we set ` = αmj/a, where αmj is the jth zero of Jm. We then have

V (ρ, ϕ, z) = Jm

(αmj

ρ

a

)eimϕ sinh

(αmj

z

a

).

To satisfy the boundary condition at z = h, we take a linear combination of solutions of thisand seek the coefficients of the expansion

f(ρ, ϕ) =∞∑

m=−∞

∞∑j=1

cmjJm

(αmj

ρ

a

)eimϕ sinh

(αmj

h

a

).

From the orthogonality relation for Bessel functions, as well as the orthogonality relation∫ 2π

0e−imϕeim

′ϕ dϕ = 2πδmm′ ,

we obtain

cmj =1

πa2 sinh(αmj

ha

)J2m+1(αmj)

∫ 2π

0

∫ a

0ρe−imϕJm

(αmj

ρ

a

)f(ρ, ϕ) dρ dϕ.

Thus the PDE and all boundary conditions are satisfied. 2

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