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Bessel Differential Equation and Bessel Functions

DEFINITION. The differential equation x2y!! + xy! + (x2 - "2)y = 0 on an interval 0 < x < b is called

Bessel's differential equation of order ".

The Bessel differential equation has a regular singular point at x = 0. For every " the equation

has at least one solution y of the form

y = xr#anxn.

THEOREM. The series part of the following (with "$ 0)

%"(x) = x" 1 +

(-1)n

n! " + 1(n)

x

2

2n

#n = 1

&

has infinite radius of convergence and is a solution of the Bessel differential equation

x2y!! + xy! + (x

2- "

2)y = 0

for x > 0. If 2" is not an integer, then {%", %'"} is a fundamental set of solutions. Here

(" + 1)(n) = (" + 1)(("+ 2)(((" + n).

PROOF. We have that p(x) = 1 and q(x) = x2 - "2. So we have that

p0 = 1, p1 = p2 = = 0, q0 = -"2, q2 = 1, q1 = q3 = q4 = = 0.

The indicial equation is

F(r) = r(r - 1) + p0r + q0 = r(r - 1) + r - "2 = r2 - "2 = 0

and the roots of the indicial equation are

r = " and r = '".

We have thatF(r + 1)a1 + (p1r + q1)a0 = 0.

The recurrence relation is

an = -

(r + k)p n-k + q n-k ak#k = 0

n - 1

F(r + n)= -

an - 2

r + n + " r + n - ".

Now assume that "$ 0 for the sake of being definite. For the root r = " we have that we have

an = -an - 2

n + 2" n

and for r = '", we have

an = -an - 2

n 2" n.

Both of these are valid when the denominator does not vanish. We see that

(n + 2")n ) 0

for every n = 1, 2, while

Bessel Differential Equation and Bessel Functions page 1


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n(n - 2") = 0

if

" - (-") = n

for an integer n. Then setting

a0 = 1,

we have

a1 = 0

a2 = -a0

2 2 + 2"= -

1

22

1 + ",

a3 = 0,

a4 = -a2

4 4 + 2"=

12

4 4 + 2" 22

1 + "=

1

24

2 1 + " 2 + ",

a5 = 0,

a6 = -a4

6 6 + 2"= -

(-1)3

26

3 + " 2 + " 1 + " 3!,

etc. So we have that the solution is

%"(x) = x"

1 +(-1)

n

n! " + 1 (n)

x

2

2n

#n = 1

&

.

We compute the radius of convergence for the integer case later. Q.E.D.

Solutions for " equal to an Integer

If" is an integer " = m, then a0is usually chosen so that

Jm(x) =x

2

m (-1)n

n!*(m + n + 1)

x

2

2n

#n = 0

&

where * is the gamma function. The function Jm(x) is called a Bessel function of the first kind. We

also write

J"(x) =x

2

" (-1)n

n!*(" + n + 1)

x

2

2n

#n = 0

&

for non integer values with "$ 0.

We note that the series %"(x) and the series J"(x) differ by a constant multiple.

PROPOSITION.The series

x

2

"

1 +(-1)

n

n!*(" + n + 1)

x

2

2n

#n = 1

&

and the series



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x"

1 +(-1)

n

n! " + 1 (n)

x

2

2n

#n = 1

&

differ by a constant multiple.

PROOF. We have that

1

(v + 1)(" + 2) (" + n) =

1

*(" + n + 1 )

*(" + n + 1 )

(v + 1)(" + 2) (" + n)

=1

*(" + n + 1)

(v + 1)(" + 2) (" + n) *(")

(v + 1)(" + 2) (" + n)

=1

(v + 1)(" + 2) (" + n)=

1

*(" + n + 1)*(") .

The ratio of the terms of the respective series is

(-1)

n

n! " + 1 (n)

x

2

2n

x" x

2

" (-1)n

n!*(" + n + 1)

x

2

2n1

=*(" + n + 1)

" + 1 (n)2"

= 2"*(")

which is independent of n. Thus, both series converge for the same x due to the limit comparison test.

Q.E.D.

The series for Jm(x) has infinite radius of convergence since the ratio

1

(k + 1)!*(n + k + 2)

x2k+2

22k+2

((k + 1)!*(n + k + 2)

1

22k

x2k

=1

k + 1

x2

22

1

n + k + 1

goes to 0 for every choice of x.

COROLLARY. The Bessel function satisfies the relation

J1 = - J0!, xJ2 = J1 - xJ2! = 2J1 - xJ0, xJ3 = J2 - xJ2!,

and in general

xJn+1 = nJn - xJn! = 2nJn - xJn-1for positive integers n.

PROOF. By direct computation using the power series.

Q.E.D.

REMARK. The preceding relationships can also be summarized as

Jn(x) = x2nJn + 1(x) + Jn 1(x)

and



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Jn!(x) =

1

2Jn 1(x) Jn + 1(x) .

These are recurssion relations from which one can calculate Jn from Jn -1 and Jn - 2.

PROPOSITION. The Bessel function Jn(x) satisfies the differential equation

y! + nx

y = Jn 1(x)

for n = 0, 1, and the differential equationy! n

xy = Jn + 1(x)

for n = 1, 2, .

PROOF. These equations are obtained by the preceding results. Q.E.D.

We also obtain the following by direct computation.

PROPOSITION. The Bessel function Jn satisfy the following differential equations:

1)d

dxx

nJn(x) = x

nJn 1(x), and

2)d

dxx

nJn(x) = x

nJn + 1(x).

PROOF. These follow by direct computation.

Q.E.D.

The preceding proposition is often useful for computing integrals with Bessel functions. In fact,

we have the following proposition:

PROPOSITION. For all integers m and n with m + n $ 0 and m + n odd, the integral

xm

Jn(x) dx

0

1

can be written in closed form.

PROOF. One can complete several steps of integration by parts using the preceding differential

equations. Q.E.D.

EXAMPLE. Show that u = 2ex/2 (i.e., ex =u

2

4) transforms the DE

y!! + (ex - m2)y = 0

into the Bessel DE

u2d2y

du2+ u

dy

du+ (u2 - 4m2)y = 0.

SOLUTION. We have that

dy

dx=

dy

du

du

dx=

dy

du

d

dx2ex/2 = ex/2

dy

du=

u

2

dy

du, and



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d

2y

dx2

=d

dx

u

2

dy

du=

d

dx

u

2

dy

du+

u

2

d

dx

dy

du=

dx

du

1

2

dy

du+

u

2

d2

y

du2

=u

4

dy

du+

u2

4

d2

y

du2

so that the original DE becomes

u

2 d2y

du2

+ udy

du+ u

2- 4m

2y = 0

which is a Bessel equation since after we multiply both sides by 4 to getu

2 d2

y

du2

+ udy

du+ u

2- 4m

2y = 0 .

EXAMPLE. Find the general solution

y!! + (ex - 9)y = 0

n terms of Bessel functions.

SOLUTION. We identify this the preceding example by letting m = 3. The general solution the DE

u2 d

2y

du

2+ u

dy

du+ u

2- 36 y = 0

is

y(u) = c1J6(u) + c2Y6(u),

where J6 is the solution of the Bessel function using the standard series method for regular singular

points (a Bessel function of the first kind) and Y6 is the solution of the Bessel equation (" = 6)

completes the fundamental set when taken in conjunction with J6 found by the exceptional methods

(with a certain normalization for the a0 term called the Neumann function). Then the general

solution of the original equation is obtained by putting back the original variable x as

y(x) = c1J6(2ex/2) + c2Y6(2e

x/2).

EXAMPLE. The substitution u = x1/2y transforms

u!! + 1 +1 - 4k

2

4x2

u = 0

(k = constant)

into the DE

x2y!! + xy! + (x2 - k2)y = 0.

SOLUTION. We have that

y = x-1/2u

y! = - 12

x-3/2u + x-1/2u

y!! =3

4x-5/2u - x-3/2u!+ x

-1/2u!

Substituting the preceding in the original differential equation, we get



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x3

4x-5/2u - x-3/2u!+ x

-1/2u!! + x -1

2x-3/2u + x-1/2u! + (x2 - k2)x-1/2u = 0.

By clearing terms, we get that

u!! + 1 +1 - 4k2

4x2u = 0 (k = constant).

EXAMPLE. The function u = x1/2J1(x) is a solution of

u!! + 1 -3

4x2

u = 0.

SOLUTION. Here we can see that l - 4k2 = -3 or k = 1. The general solution of the differential

equation

x2y!! + xy! + (x2 - 1)y = 0

is

y = c1J1(x) + c2Y1(x).

So the solution of the differential equation

u!! + 1 -3

4x2

u = 0

is

u = c1x1/2J1(x) + c2x

1/2Y1(x).

EXAMPLE. The general solution of the differential equation

u!! + bxmu = 0

is

u = c1x1/2J 1

m+2

2 b

m + 2x

m + 2

+ c2x1/2J

1

m+2

2 b

m + 2x

m + 2

for m > 0.

SOLUTION. The substitution u = x1/2y followed by

w =2 b

m + 2xm + 2

to transform the given differential equation into a Bessel differential equation.

EXAMPLE. The general solution of the differential equation

y!!+ 1x

y! + +"2

x2y = 0

for + > 0 is

y = AJ" x + + BY" x +

where A and B are constants.



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SOLUTION. Let ,(x) be a solution of the Bessel equation

y!! +

1

xy! + 1

"2

x2

y = 0.

Then the function

(x)= ,(x +)

is a solution of the equation y!! +

1

xy! + +

"2

x2

y = 0 .

In fact, we have that

!(x)= +,!(x +)

and

!!(x)= +,!!(x +).

Now substituting for-!!,-! and- in the the equation

,!! x + +1

x + ,! x + + 1

"2

x +2 , x + = 0 .

or the equivalent equation

+ ,!! x + +1

x+,! x + + +

"2

x2

, x + = 0 ,

obtained by multiplying the previous equation by +, we get that

!! x + +1

x-! x + + +

"2

x2

- x + = 0

Since ,(x) has the form,(x) = A J"(x) + B Y"(x)

due to the fact it is the solution of the Bessel equation of order ", we get that all functions of the form

-(x) = AJ" x + + B Y" x +

are solutions of the equation

(1)

y!! +

1

xy! + +

"2

x2

y = 0 .

We can reverse the process by showing that every solution-(x) of (1) produces a solution

,(x) = (x/ +) of the Bessel equation of order ". We do not carry out this step. This means that allsolutions-(x) of (1) are of the form -(x) = AJ" x + + B Y" x + .



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Orthogonality

We develop two orthogonality conditions that relate to boundary condtions of the Bessel

equation.

PROPOSITION. Let + and be positive numbers with +) . Then

J"(+x)J"(x) x dx

0

1=+J"()J"!(+) J"(+)J"!()

2

+2.

PROOF. We have that

,(x) = J"(+x)

is a solution of the differential equation

(2)

y!! +

1

xy! + +2

"2

x2

y = 0

and

-(x) = J"(x)is a solution of the differential equation

(3)

y!! +

1

xy! + 2

"2

x2

y = 0

by the previous example. Now we get that

(x) ,!!(x) +1

x,!(x) + 2

"2

x2,(x) ,(x) -!!(x) +

1

x-!(x) + 2

"2

x2-(x) = 0 .

Combining this equation, we get that

(x),!!(x) ,(x)-!!(x)+

1

x -(x),!(x) ,(x)-!(x) + +2 2 ,(x)-(x) = 0 .

Multiplying both sides of this last equation by x, we get

x -(x),!!(x) ,(x)-!!(x) + -(x),!(x) ,(x)-!(x) + +2 2 x,(x)-(x) = 0

Then we may rewrite the preceding equation as

d

dxx -(x),!(x) ,(x)-!(x + +2

2x,(x)-(x) = 0 .

Integrating both sides of the preceding, we get that

xW(-, ,)(x)

0

1

= 2 +2 x,(x)-(x)

0

1

= 0

where W denotes the Wronskian

W(-, ,)(x) = -(x),!(x) - -!(x),(x) = +J"(x)J"!(+x) - J"!(x)J"(+x).

So we get that



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xW(-, ,)(x)

0

1

= -(1),!(1) -!(1),(1) = +J"()J"!(+) J"!()J"(+) .

So we get that

xJ"(x+)J"(x) dx =

0

1 +J"()J"!(+) J"!()J"(+)

2

+2. Q.E.D.

Now we get the desired orthogonaliy relation

COROLLARY. Let + and be two distinct zeroes of J"(x); then J"(x+) and J"(x) are orthogonal on

[0, 1] with respect to the weight x dx.

PROOF. We have that

xJ"(x+)J"(x) dx =

0

1 +J"()J"!(+) J"!()J"(+)

2

+2= 0 . Q.E.D.

We use the functions J"(x+n) where +n are the positive zeroes of J"(x). These functions form a

complete orthogonal set of functions on [0, 1] with respect to the integration x dx. We can calculate

the normalization from the preceding proposition.

PROPOSITION. x J"2

(x+) dx =

0

11

2J"!(+)

2+ 1

"2

+2J"

2(+) .

PROOF. We have that

x J"2(x+) dx

0

1

= lim. +

x J"(x+) J"(x) dx

0

1

= lim.+

+J"()J"!(+) J"!()J"(+)

2 +2

.

The last limit is a form 0/0 and so we can apply lHpitals Rule by taking the derivatives of both

numerator and denominator to give the limits

lim. +

+J"()J"!(+) J"!()J"(+)

2

+2= lim

. +

+J"!()J"!(+) J"!()J"(+) J"!!()J"(+)

2

=+J"!(+)

2 J "!(+)J"(+) +J"!!(+)J"(+)

2+.

But we have that

+2J"!!(+) + +J"(+) + (+2 - "2)J"(+) = 0.

This means that



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+J"!(+)2 - J"(+)J"!(+) - +J"(+)J"!!(+) = +J"!(+)

2 - J"(+)J"!(+) +

+J"(+)+2 "2

+2J"(+) +

1

+J"!(+)

= +J"!(+)2 +

+2 "2

+J"

2(+)

So we get that

xJ"

2(x+) dx =

0

11

2J"!(+)

2+ 1

"2

+2J"

2(+) Q.E.D.

Recalling that Jn(x) satisfies the differential equations y!nx

y = Jn 1(x) and , we have that

x Jn

2(x+) dx =

0

11

2J n!(+)

2=

1

2J

n 1

2(+) =

1

2J

n + 1

2(+) .

whenever + is a zero of of Jn(x) and n is an integer.

A second orthogonality condition is the following.

PROPOSITION. Let + and be two different roots of the equation

RJ"(x) + SxJ"!(x) = 0

with R and S constants not both equal to 0. Then

xJn(x+)Jn(x) d x =

0

1

0 .

PROOF. Substituting x = + and x = , we get that

RJ"(+) + S+J"!(+) = 0RJ"() + SJ"!() = 0.

Then the system of linear equations

J"(+)/1 + +J"!(+)/2 = 0

J"()/1 + J"!()/2 = 0

has a non trivial solution /1 = R and /2 = S. So the determinant of the system is 0, i.e.,

J"(+)J"!() - +J"()J"!(+) = 0.

By the previous proposition we get that

xJ n(x+)Jn(x) dx =

0

1

0.


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