Bessel’s equation

Bessel’s equation

The

linear differential equation

x

2

⁢

d

2

⁢

y

d

⁢

x

2

+

x

⁢

d

⁢

y

d

⁢

x

+

(

x

2

-

p

2

)

⁢

y

=

0

,

(1)

in which

p

is a constant (non-negative if it is real), is called the

Bessel’s equation

.  We derive its

general solution

by trying the series form

y

=

x

r

⁢

∑

k

=

0

∞

a

k

⁢

x

k

=

∑

k

=

0

∞

a

k

⁢

x

r

+

k

,

(2)

due to Frobenius.  Since the

parameter

r

is

indefinite

, we may regard

a

0

as distinct from 0.

We substitute (2) and the derivatives of the series in (1):

x

2

⁢

∑

k

=

0

∞

(

r

+

k

)

⁢

(

r

+

k

-

1

)

⁢

a

k

⁢

x

r

+

k

-

2

+

x

⁢

∑

k

=

0

∞

(

r

+

k

)

⁢

a

k

⁢

x

r

+

k

-

1

+

(

x

2

-

p

2

)

⁢

∑

k

=

0

∞

a

k

⁢

x

r

+

k

=

0

.

Thus the coefficients of the powers

x

r

,

x

r

+

1

,

x

r

+

2

and so on must vanish, and we get the system of equations

{

[

r

2

-

p

2

]

⁢

a

0

=

0

,

[

(

r

+

1

)

2

-

p

2

]

⁢

a

1

=

0

,

[

(

r

+

2

)

2

-

p

2

]

⁢

a

2

+

a

0

=

0

,

…

[

(

r

+

k

)

2

-

p

2

]

⁢

a

k

+

a

k

-

2

=

0

.

(3)

The last of those can be written

(

r

+

k

-

p

)

⁢

(

r

+

k

+

p

)

⁢

a

k

+

a

k

-

2

=

0

.

Because

a

0

≠

0

,  the first of those (the

indicial equation

) gives

r

2

-

p

2

=

0

,  i.e. we have the roots

r

1

=

p

,

r

2

=

-

p

.

Let’s first look the the solution of (1) with

r

=

p

;  then

k

⁢

(

2

⁢

p

+

k

)

⁢

a

k

+

a

k

-

2

=

0

,  and thus

a

k

=

-

a

k

-

2

k

(

2

p

+

k

)

.

From the system (3) we can solve one by one each of the coefficients

a

1

,

a

2

,

…

and express them with

a

0

which remains arbitrary.  Setting for

k

the integer values we get

{

a

1

=

0

,

a

3

=

0

,

…

,

a

2

⁢

m

-

1

=

0

;

a

2

=

-

a

0

2

⁢

(

2

⁢

p

+

2

)

,

a

4

=

a

0

2

⋅

4

⁢

(

2

⁢

p

+

2

)

⁢

(

2

⁢

p

+

4

)

,

…

,

a

2

⁢

m

=

(

-

1

)

m

⁢

a

0

2

⋅

4

⋅

6

⁢

⋯

⁢

(

2

⁢

m

)

⁢

(

2

⁢

p

+

2

)

⁢

(

2

⁢

p

+

4

)

⁢

…

⁢

(

2

⁢

p

+

2

⁢

m

)

(4)

(where

m

=

1

,

2

,

…

).
Putting the obtained coefficients to (2) we get the particular solution

y

1

:=

a

0

x

p

[

x

2

2

⁢

(

2

⁢

p

+

2

)

+

x

4

2

⋅

4

⁢

(

2

⁢

p

+

2

)

⁢

(

2

⁢

p

+

4

)

-

x

6

2

⋅

4

⋅

6

⁢

(

2

⁢

p

+

2

)

⁢

(

2

⁢

p

+

4

)

⁢

(

2

⁢

p

+

6

)

+

-

…

]

(5)

In order to get the coefficients

a

k

for the second root

r

2

=

-

p

we have to look after that

(

r

2

+

k

)

2

-

p

2

≠

0

,

or

r

2

+

k

≠

p

=

r

1

.  Therefore

r

1

-

r

2

=

2

⁢

p

≠

k

where

k

is a positive integer.  Thus, when

p

is not an integer and not an integer added by

1

2

, we get the second particular solution, gotten of (5) by replacing

p

by

-

p

:

y

2

:=

a

0

x

-

p

[

1

-

x

2

2

⁢

(

-

2

⁢

p

+

2

)

+

x

4

2

⋅

4

⁢

(

-

2

⁢

p

+

2

)

⁢

(

-

2

⁢

p

+

4

)

-

x

6

2

⋅

4

⋅

6

⁢

(

-

2

⁢

p

+

2

)

⁢

(

-

2

⁢

p

+

4

)

⁢

(

-

2

⁢

p

+

6

)

+

-

…

]

(6)

The

power series

of (5) and (6) converge for all values of

x

and are

linearly independent

(the ratio

y

1

/

y

2

tends to 0 as

x

→

∞

).  With the appointed value

a

0

=

1

2

p

⁢

Γ

⁢

(

p

+

1

)

,

the solution

y

1

is called the

Bessel function

of the

first kind

and of order

p

and denoted by

J

p

.  The

similar

definition is set for the first kind Bessel function of an arbitrary order

p

∈

ℝ

(and

ℂ

).
For

p

∉

ℤ

the general solution of the Bessel’s

differential equation

is thus

y

:=

C

1

⁢

J

p

⁢

(

x

)

+

C

2

⁢

J

-

p

⁢

(

x

)

,

where

J

-

p

⁢

(

x

)

=

y

2

with

a

0

=

1

2

-

p

⁢

Γ

⁢

(

-

p

+

1

)

.

The explicit expressions for

J

±

p

are

J

±

p

⁢

(

x

)

=

∑

m

=

0

∞

(

-

1

)

m

m

!

⁢

Γ

⁢

(

m

±

p

+

1

)

⁢

(

x

2

)

2

⁢

m

±

p

,

(7)

which are obtained from (5) and (6) by using the last for

gamma function

.

E.g. when

p

=

1

2

the series in (5) gets the form

y

1

=

x

1

2

2

⁢

Γ

⁢

(

3

2

)

[

1

-

x

2

2

⋅

3

+

x

4

2

⋅

4

⋅

3

⋅

5

-

x

6

2

⋅

4

⋅

6

⋅

3

⋅

5

⋅

7

+

-

…

]

=

2

π

⁢

x

(

x

-

x

3

3

!

+

x

5

5

!

-

+

…

)

.

Thus we get

J

1

2

⁢

(

x

)

=

2

π

⁢

x

⁢

sin

⁡

x

;

analogically (6) yields

J

-

1

2

⁢

(

x

)

=

2

π

⁢

x

⁢

cos

⁡

x

,

and the general solution of the equation (1) for

p

=

1

2

is

y

:=

C

1

⁢

J

1

2

⁢

(

x

)

+

C

2

⁢

J

-

1

2

⁢

(

x

)

.

In the case that

p

is a non-negative integer

n

, the “+” case of (7) gives the solution

J

n

⁢

(

x

)

=

∑

m

=

0

∞

(

-

1

)

m

m

!

⁢

(

m

+

n

)

!

⁢

(

x

2

)

2

⁢

m

+

n

,

but for

p

=

-

n

the expression of

J

-

n

⁢

(

x

)

is

(

-

1

)

n

⁢

J

n

⁢

(

x

)

, i.e. linearly dependent on

J

n

⁢

(

x

)

.  It can be shown that the other solution of (1) ought to be searched in the form

y

=

K

n

⁢

(

x

)

=

J

n

⁢

(

x

)

⁢

ln

⁡

x

+

x

-

n

⁢

∑

k

=

0

∞

b

k

⁢

x

k

.  Then the general solution is

y

:=

C

1

⁢

J

n

⁢

(

x

)

+

C

2

⁢

K

n

⁢

(

x

)

.

Other formulae

The first kind Bessel functions of integer order have the generating function

F

:

F

⁢

(

z

,

t

)

=

e

z

2

⁢

(

t

-

1

t

)

=

∑

n

=

-

∞

∞

J

n

⁢

(

z

)

⁢

t

n

(8)

This

function

has an

essential singularity

at

t

=

0

but is

analytic

elsewhere in

ℂ

; thus

F

has the

Laurent expansion

in that point.  Let us prove (8) by using the general expression

c

n

=

1

2

⁢

π

⁢

i

⁢

∮

γ

f

⁢

(

t

)

(

t

-

a

)

n

+

1

⁢

𝑑

t

of the

coefficients of Laurent series

.  Setting to this

a

:=

0

,

f

⁢

(

t

)

:=

e

z

2

⁢

(

t

-

1

t

)

,

ζ

:=

z

⁢

t

2

gives

c

n

=

1

2

⁢

π

⁢

i

⁢

∮

γ

e

z

⁢

t

2

⁢

e

-

z

2

⁢

t

t

n

+

1

⁢

𝑑

t

=

1

2

⁢

π

⁢

i

⁢

(

z

2

)

n

⁢

∮

δ

e

ζ

⁢

e

-

z

2

4

⁢

ζ

ζ

n

+

1

⁢

𝑑

ζ

=

∑

m

=

0

∞

(

-

1

)

m

m

!

⁢

(

z

2

)

2

⁢

m

+

n

⁢

1

2

⁢

π

⁢

i

⁢

∮

δ

ζ

-

m

-

n

-

1

⁢

e

ζ

⁢

𝑑

ζ

.

The paths

γ

and

δ

go once round the origin anticlockwise in the

t

-plane and

ζ

-plane, respectively.  Since the

residue

of

ζ

-

m

-

n

-

1

⁢

e

ζ

in the origin is

1

(

m

+

n

)

!

=

1

Γ

⁢

(

m

+

n

+

1

)

,  the

residue theorem

(

http://planetmath.org/CauchyResidueTheorem

) gives

c

n

=

∑

m

=

0

∞

(

-

1

)

m

m

!

⁢

Γ

⁢

(

m

+

n

+

1

)

⁢

(

z

2

)

2

⁢

m

+

n

=

J

n

⁢

(

z

)

.

This that

F

has the Laurent expansion (8).

By using the generating function, one can easily derive other formulae, e.g.
the of the Bessel functions of integer order:

J

n

⁢

(

z

)

=

1

π

⁢

∫

0

π

cos

⁡

(

n

⁢

φ

-

z

⁢

sin

⁡

φ

)

⁢

𝑑

φ

Also one can obtain the

addition formula

J

n

⁢

(

x

+

y

)

=

∑

ν

=

-

∞

∞

J

ν

⁢

(

x

)

⁢

J

n

-

ν

⁢

(

y

)

and the series of cosine and sine:

cos

z

=

J

0

(

z

)

-

2

J

2

(

z

)

+

2

J

4

(

z

)

-

+

…

sin

z

=

2

J

1

(

z

)

-

2

J

3

(

z

)

+

2

J

5

(

z

)

-

+

…

References

1

N. Piskunov:

Diferentsiaal- ja integraalarvutus kõrgematele tehnilistele õppeasutustele

.  Kirjastus Valgus, Tallinn (1966).

2

K. Kurki-Suonio:

Matemaattiset apuneuvot

.  Limes r.y., Helsinki (1966).

Title

Bessel’s equation

Canonical name

BesselsEquation

Date of creation

2013-03-22 16:34:57

Last modified on

2013-03-22 16:34:57

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

26

Author

pahio (2872)

Entry type

Definition

Classification

msc 34A05

Classification

msc 33C10

Synonym

Bessel’s differential equation

Synonym

Bessel equation

Related topic

LaplaceEquationInCylindricalCoordinates

Related topic

CauchyResidueTheorem

Related topic

PropertiesOfEntireFunctions

Related topic

FrobeniusMethod

Related topic

TableOfLaplaceTransforms

Related topic

BesselFunctionsAndHelicalStructureDiffractionPatterns

Defines

Bessel’s function

Defines

Bessel function

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