Bessels Differential equation

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Bessels Differential equation
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In this lecture we discuss Bessels Differential
Equation. We also study properties of Bessels
functions, which are solutions of Bessels
equation. We first review the definition and
properties of the celebrated Gamma function
(which is also called the extended factorial
function).
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You may view my lecture slides in the following
site.
http//discovery.bits-pilani.ac.in/discipline/math
/msr/index.html
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Gamma Function ?(x) This is easily one of the
most important functions in Mathematics. Definitio
n For each real no x gt 0, the improper integral
converges and its value is denoted by ?(x).
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Properties of the Gamma function ?(x)

• ?(x 1) x ?(x) --- (Proved by Integrating by
  parts)
• ?(1) 1 -- (immediate from the definition)
• ?(n 1) n ! For all positive integers n
• (follows from the first two properties thus
  the Gamma function is also referred to as the
  extended factorial function)

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• ?( ½ ) ??

We now extend the definition of ?(x) for negative
numbers x as follows
If -1 lt x lt 0, we define
Thus
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Thus we have extended the definition of ?(x) for
-1 lt x lt 0. Note that ?(0), ?(-1) are undefined.
Now for -2 lt x lt -1, we define
Thus
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Proceeding like this, we define ?(x) for all
negative numbers x which are not negative
integers. We also note that for all x (not a
negative integer),
And so on.
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We note that if x is a positive integer,
?(x 1) x ?(x) x (x-1)?(x 1) ..
x!
We now define for all numbers x (not a negative
integer),
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We also note for future reference that
where x is NOT a negative integer. The next slide
shows the graph of the Gamma function.
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The second order homogeneous l.d.e.
where p is a nonnegative constant, is called as
the Bessels differential equation of order p.
Here
and
are analytic at all
points except x 0.
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Hence x 0 is the only singular point. As
and
are both analytic at x 0, x 0 is a regular
singular point.
The indicial equation is
i.e.
or
p, -p.
Hence the exponents are m
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Hence corresponding to the bigger exponent mp,
there always exists a Frobenius Series solution
Hence
Substituting for y, y? , y? we get
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Divide throughout by xp, we get
In the second ?, replace n by n-2. We then get
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Note that the constant term is absent. Thus the
above equation becomes
Thus
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or
Recurrence relation for ans
n3 gives
Putting n 5, 7, we get
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n 2 gives
n 4 gives
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Proceeding like this we get
n 1, 2,
Hence the solution corresponding to the exponent
m p is
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we get the solution
Now choosing
and is
This solution is denoted by
referred to as Bessels function of the first
kind.
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Thus

• If p is a positive integer or zero, we see that
• Jp(x) is a power series
• If p is odd, Jp(x) contains only odd powers of x
• If p is even, Jp(x) contains only even powers of
  x
• Jp(x) converges absolutely for all x gt 0.

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If p is NOT an integer or zero, we easily show
that a second LI solution is
Hence when p is NOT an integer or zero, the
general solution of Bessels equation is
c1, c2 arbitrary constants
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If p is a positive integer m, we formally define
Noting that
we get
Changing n to nm, we get
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Hence Jm(x) and J-m(x) are not LI.
( m, a positive integer)
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Bessels Function of the second kind
We define the standard Bessel function of the
second kind by
p not an integer.
It is obvious that we can write the general
solution of Bessels equation of order p also as
(c1, c2 arbitrary constants)
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It has been shown that
exists, is a solution of Bessels equation of
order m and that Jm(x) and Ym(x) are LI.
Hence for all p (integer or not), the general
solution of Bessels equation of order p is
(c1, c2 arbitrary constants)
The following slides shows the graphs of J0(x) ,
J1(x), .. and Y0(x), Y1(x),
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