[PDF] Power Series Solutions to the Bessel Equation

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Power Series Solutions to the Bessel Equation

Power Series Solutions to the Bessel Equation Note:The ratio test shows that the power series formula converges for all x 2R For x

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Power Series Solutions to the Bessel Equation

Power Series Solutions to the Bessel Equation

Department of Mathematics

IIT Guwahati

RA/RKSMA-102 (2016)

Power Series Solutions to the Bessel Equation

The Bessel equation

The equation

x

2y00+xy0+ (x22)y= 0;(1)

whereis a nonnegative constant, is called theBessel equation The pointx0= 0 is a regular singular point. We shall use the method of Frobenius to solve this equation.

Thus, we seek solutions of the form

y(x) =1X n=0a nxn+r;x>0;(2) witha06= 0.RA/RKSMA-102 (2016)

Power Series Solutions to the Bessel Equation

Dierentiation of (2) term by term yields

y 0=1X n=0(n+r)anxn+r1:

Similarly, we obtain

y

00=xr21X

n=0(n+r)(n+r1)anxn:

Substituting these into (1), we obtain

1 X n=0(n+r)(n+r1)anxn+r+1X n=0(n+r)anxn+r 1X n=0a nxn+r+21X n=0

2anxn+r= 0:RA/RKSMA-102 (2016)

Power Series Solutions to the Bessel Equation

This implies

x r1X n=0[(n+r)22]anxn+xr1X n=0a nxn+2= 0: Now, cancelxr, and try to determinean's so that the coecient of each power ofxwill vanish. For the constant term, we require (r22)a0= 0. Since a

06= 0, it follows that

r

22= 0;

which is the indicial equation. The only p ossiblevalues of r areand.RA/RKSMA-102 (2016)

Power Series Solutions to the Bessel Equation

Case I.

F orr=, the equations for determining the

coecients are: [(1 +)22]a1= 0 and; [(n+)22]an+an2= 0;n2:

Since0, we havea1= 0. The second equation yields

a n=an2(n+)22=an2n(n+ 2):(3)

Sincea1= 0, we immediately obtain

a

3=a5=a7== 0:RA/RKSMA-102 (2016)

Power Series Solutions to the Bessel Equation

For the coecients with even subscripts, we have

a

2=a02(2 + 2)=a02

2(1 +);

a

4=a24(4 + 2)=(1)2a02

42!(1 +)(2 +);

a

6=a46(6 + 2)=(1)3a02

63!(1 +)(2 +)(3 +);

and, in general a

2n=(1)na02

2nn!(1 +)(2 +)(n+):

Therefore, the choicer=yields the solution

y(x) =a0x 1 +1X n=1(1)nx2n2

2nn!(1 +)(2 +)(n+)!

RA/RKSMA-102 (2016)

Power Series Solutions to the Bessel Equation

Note: The ratio test sho wsthat the p owerseries fo rmula converges for allx2R. Forx<0, we proceed as above withxrreplaced by (x)r.

Again, in this case, we nd thatrsatises

r

22= 0:

Takingr=, we obtain the same solution, withxis

replaced by (x). Therefore, the functiony(x) is given by y (x) =a0jxj 1 +1X n=1(1)nx2n2

2nn!(1 +)(2 +)(n+)!

(4) is a solution of the Bessel equation valid for all realx6= 0.RA/RKSMA-102 (2016)

Power Series Solutions to the Bessel Equation

Case II.

Forr=, determine the coecients from

[(1)22]a1= 0 and [(n)22]an+an2= 0:

These equations become

(12)a1= 0 andn(n2)an+an2= 0:

If 2is not an integer, these equations give us

a

1= 0 andan=an2n(n2);n2:

Note that this formula is same as (3), withreplaced by.

Thus, the solution is given by

y (x) =a0jxj 1 +1X n=1(1)nx2n2

2nn!(1)(2)(n)!

(5) which is valid for all realx6= 0.RA/RKSMA-102 (2016)

Power Series Solutions to the Bessel Equation

Euler's gamma function and its properties

Fors2Rwiths>0, we dene (s) by

(s) =Z 1 0+ ts1etdt:

The integral converges ifs>0 and diverges ifs0.

Integration by parts yields the functional equation (s+ 1) =s(s):

In general,

(s+n) = (s+n1)(s+ 1)s(s);for everyn2Z+: Since (1) = 1, we nd that (n+ 1) =n!:Thus, the gamma function is an extension of the factorial function from integers to positive real numbers. Therefore, we write (s) =(s+ 1)s;s2R:RA/RKSMA-102 (2016)

Power Series Solutions to the Bessel Equation

Using this gamma function, we shall simplify the form of the solutions of the Bessel equation. Withs= 1+, we note that (1 +)(2 +)(n+) =(n+ 1 +)(1 +):

Choosea0=2(1+)in (4), the solution forx>0 can be

written J (x) =x2 1X n=0(1)nn!(n+ 1 +) x2 2n:

The functionJdened above forx>0 and0 is called

the

Bessel function of t herst kind

of o rder.RA/RKSMA-102 (2016)

Power Series Solutions to the Bessel Equation

Whenis a nonnegative integer, say=p, the Bessel

functionJp(x) is given by J p(x) =1X n=0(1)nn!(n+p)! x2

2n+p;(p= 0;1;2;:::):

This is a solution of the Bessel equation forx<0.Figure :The Bessel functions J0andJ1.RA/RKSMA-102 (2016)

Power Series Solutions to the Bessel Equation

If =2Z+, dene a new functionJ(x) (replacingby)

J (x) =x2 1X n=0(1)nn!(n+ 1) x2 2n:

Withs= 1, we note that

(n+ 1) = (1)(2)(n)(1): Thus, the series forJ(x) is the same as that fory(x) in (5) witha0=2(1);x>0. Ifis not positive integer,Jis a solution of the Bessel equation forx>0.

If =2Z+,J(x) andJ(x) are linearly independent on

x>0. The general solution of the Bessel equation forx>0 is y(x) =c1J(x) +c2J(x):RA/RKSMA-102 (2016)

Power Series Solutions to the Bessel Equation

Useful recurrence relations forJ

ddx (xJ(x)) =xJ1(x): ddx (xJ(x)) =ddx x 1X n=0(1)nn!(1 ++n) x2 2n+) ddx 1X n=0(1)nx2n+2n!(1 ++n)22n+) 1X n=0(1)n(2n+ 2)x2n+21n!(1 ++n)22n+:

Since (1 ++n) = (+n)(+n), we have

ddx (xJ(x)) =1X n=0(1)n2x2n+21n!(+n)22n+ =x1X n=0(1)nn!(1 + (1) +n) x2 2n+1 =xJ1(x):RA/RKSMA-102 (2016)

Power Series Solutions to the Bessel Equation

The other relations involvingJare:

ddx (xJ(x)) =xJ+1(x): x

J(x) +J0(x) =J1(x):

x

J(x)J0(x) =J+1(x):

J1(x) +J+1(x) =2x

J(x):

J1(x)J+1(x) = 2J0(x):

Note:

W orkoutthese relations.

*** End ***

RA/RKSMA-102 (2016)

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