Frobenius method

Frobenius method

Let us consider the linear

homogeneous differential equation

∑

ν

=

0

n

k

ν

⁢

(

x

)

⁢

y

(

n

-

ν

)

⁢

(

x

)

=

0

of order (

http://planetmath.org/DifferentialEquation

)

n

.  If the

coefficient

functions

k

ν

⁢

(

x

)

are

continuous

and the coefficient

k

0

⁢

(

x

)

of the highest order

derivative

(

http://planetmath.org/HigherOrderDerivatives

) does not vanish on a certain

interval

(resp. a domain (

http://planetmath.org/Domain2

) in

ℂ

), then all solutions

y

⁢

(

x

)

are continuous on this interval (resp. ).  If all coefficients have the continuous derivatives up to a certain , the same concerns the solutions.

If, instead,

k

0

⁢

(

x

)

vanishes

in a point

x

0

, this point is in general a

singular point

.  After dividing the

differential equation

by

k

0

⁢

(

x

)

and then getting the form

y

(

n

)

⁢

(

x

)

+

∑

ν

=

1

n

c

ν

⁢

(

x

)

⁢

y

(

n

-

ν

)

⁢

(

x

)

=

0

,

some new coefficients

c

ν

⁢

(

x

)

are

discontinuous

in the singular point.  However, if the

discontinuity

is so, that the products

(

x

-

x

0

)

⁢

c

1

⁢

(

x

)

,

(

x

-

x

0

)

2

⁢

c

2

⁢

(

x

)

,

…

,

(

x

-

x

0

)

n

⁢

c

n

⁢

(

x

)

are continuous, and

analytic

in

x

0

, the point

x

0

is a

regular singular point

of the differential equation.

We introduce the so-called

Frobenius method

for finding solution functions in a neighbourhood of the regular singular point

x

0

, confining us to the case of a second order (

http://planetmath.org/DifferentialEquation

) differential equation.  When we use the

quotient

(

http://planetmath.org/Division

) forms

(

x

-

x

0

)

c

1

(

x

)

:=

p

⁢

(

x

)

r

⁢

(

x

)

,

(

x

-

x

0

)

2

c

2

(

x

)

:

=

q

⁢

(

x

)

r

⁢

(

x

)

,

where

r

⁢

(

x

)

,

p

⁢

(

x

)

and

q

⁢

(

x

)

are analytic in a neighbourhood of

x

0

and

r

⁢

(

x

)

≠

0

,  our differential equation reads

(

x

-

x

0

)

2

⁢

r

⁢

(

x

)

⁢

y

′′

⁢

(

x

)

+

(

x

-

x

0

)

⁢

p

⁢

(

x

)

⁢

y

′

⁢

(

x

)

+

q

⁢

(

x

)

⁢

y

⁢

(

x

)

=

0

.

(1)

Since a change

x

-

x

0

↦

x

of variable brings to the case that the singular point is the origin, we may suppose such a starting situation.  Thus we can study the equation

x

2

⁢

r

⁢

(

x

)

⁢

y

′′

⁢

(

x

)

+

x

⁢

p

⁢

(

x

)

⁢

y

′

⁢

(

x

)

+

q

⁢

(

x

)

⁢

y

⁢

(

x

)

=

0

,

(2)

where the coefficients have the converging

power series

expansions

r

⁢

(

x

)

=

∑

n

=

0

∞

r

n

⁢

x

n

,

p

⁢

(

x

)

=

∑

n

=

0

∞

p

n

⁢

x

n

,

q

⁢

(

x

)

=

∑

n

=

0

∞

q

n

⁢

x

n

(3)

and

r

0

≠

0

.

In the Frobenius method one examines whether the equation (2) allows a series solution of the form

y

⁢

(

x

)

=

x

s

⁢

∑

n

=

0

∞

a

n

⁢

x

n

=

a

0

⁢

x

s

+

a

1

⁢

x

s

+

1

+

a

2

⁢

x

s

+

2

+

…

,

(4)

where

s

is a constant and

a

0

≠

0

.

Substituting (3) and (4) to the differential equation (2) converts the left hand to

[

r

0

⁢

s

⁢

(

s

-

1

)

+

p

0

⁢

s

+

q

0

]

⁢

a

0

⁢

x

s

+

[

[

r

0

⁢

(

s

+

1

)

⁢

s

+

p

0

⁢

(

s

+

1

)

+

q

0

]

⁢

a

1

+

[

r

1

⁢

s

⁢

(

s

-

1

)

+

p

1

⁢

s

+

q

1

]

⁢

a

0

]

⁢

x

s

+

1

+

[

[

r

0

⁢

(

s

+

2

)

⁢

(

s

+

1

)

+

p

0

⁢

(

s

+

2

)

+

q

0

]

⁢

a

2

+

[

r

1

⁢

(

s

+

1

)

⁢

s

+

p

1

⁢

(

s

+

1

)

+

q

1

]

⁢

a

1

+

[

r

2

⁢

s

⁢

(

s

-

1

)

+

p

2

⁢

s

+

q

2

]

⁢

a

0

]

⁢

x

s

+

2

+

…

Our equation seems clearer when using the notations

f

ν

⁢

(

s

)

:=

r

ν

⁢

s

⁢

(

s

-

1

)

+

p

ν

⁢

s

+

q

n

⁢

u

:

f

0

⁢

(

s

)

⁢

a

0

⁢

x

s

+

[

f

0

⁢

(

s

+

1

)

⁢

a

1

+

f

1

⁢

(

s

)

⁢

a

0

]

⁢

x

s

+

1

+

[

f

0

⁢

(

s

+

2

)

⁢

a

2

+

f

1

⁢

(

s

+

1

)

⁢

a

1

+

f

2

⁢

(

s

)

⁢

a

0

]

⁢

x

s

+

2

+

…

=

0

(5)

Thus the condition of satisfying the differential equation by (4) is the

infinite

system of equations

{

f

0

⁢

(

s

)

⁢

a

0

=

0

f

0

⁢

(

s

+

1

)

⁢

a

1

+

f

1

⁢

(

s

)

⁢

a

0

=

0

f

0

⁢

(

s

+

2

)

⁢

a

2

+

f

1

⁢

(

s

+

1

)

⁢

a

1

+

f

2

⁢

(

s

)

⁢

a

0

=

0

⋯

⋯

⋯

(6)

In the first , since

a

0

≠

0

,  the

indicial equation

f

0

⁢

(

s

)

≡

r

0

⁢

s

2

+

(

p

0

-

r

0

)

⁢

s

+

q

0

=

0

(7)

must be satisfied.  Because

r

0

≠

0

,  this

quadratic equation

determines for

s

two values, which in special case may coincide.

The first of the equations (6) leaves

a

0

(

≠

0

)

arbitrary.  The next

linear equations

in

a

n

allow to solve successively the constants

a

1

,

a

2

,

…

provided that the first coefficients

f

0

⁢

(

s

+

1

)

,

f

0

⁢

(

s

+

2

)

,

…

do not vanish; this is evidently the case when the roots (

http://planetmath.org/Equation

) of the indicial equation don’t differ by an integer (e.g. when the are

complex conjugates

or when

s

is the having

greater

real part

).  In any case, one obtains at least for one of the of the indicial equation the definite values of the coefficients

a

n

in the series (4).  It is not hard to show that then this series converges in a neighbourhood of the origin.

For obtaining the solution of the differential equation (2) it suffices to have only one solution

y

1

⁢

(

x

)

of the form (4), because another solution

y

2

⁢

(

x

)

,

linearly independent

on

y

1

⁢

(

x

)

, is gotten via mere integrations; then it is possible in the cases

s

1

-

s

2

∈

ℤ

that

y

2

⁢

(

x

)

has no expansion of the form (4).

References

1

Pentti Laasonen:

Matemaattisia erikoisfunktioita

.  Handout No. 261. Teknillisen Korkeakoulun Ylioppilaskunta; Otaniemi, Finland (1969).

Title

Frobenius method

Canonical name

FrobeniusMethod

Date of creation

2013-03-22 17:43:49

Last modified on

2013-03-22 17:43:49

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

18

Author

pahio (2872)

Entry type

Topic

Classification

msc 15A06

Classification

msc 34A05

Synonym

method of Frobenius

Related topic

FuchsianSingularity

Related topic

BesselsEquation

Related topic

SpecialCasesOfHypergeometricFunction

Defines

indicial equation

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