2. Series Solution and Special Functions ...................................................................................... 2
Series solution of second order ordinary differential equations with variable coefficient ..................................2
Bessel equations and their series solutions ......................................................................................5
Properties of Bessel function ................................................................................................. 11
Properties of Legendre polynomials ............................................................................................... 15
Generating function............................................................................................................................................15
Rodrigues’ Formula.............................................................................................................................................16
Series Solution and Special Functions
Series solution of second order ordinary differential equations with variable
coefficient
We have fully investigated solving second order linear differential equations with constant
coefficients. Now we will explore how to find solutions to second order linear differential
equations whose coefficients are not necessarily constant. Let
P(x)y'' + Q(x)y' + R(x)y = g(x)
Be a second order differential equation with P, Q, R, and g all continuous. Then x0 is a singular
point if P(x0) = 0, but Q and R do not both vanish at x0. Otherwise we say that x0 is an
ordinary point. For now, we will investigate only ordinary points.
Example
Find a solution to
y'' + xy' + y = 0 y(0) = 0 y'(0) = 1
Solution
3. Since the differential equation has non-constant coefficients, we cannot assume that a solution is
in the form y = ert
. Instead, we use the fact that the second order linear differential equation
must have a unique solution. We can express this unique solution as a power series
If we can determine the an for all n, then we know the solution. Fortunately, we can easily take
derivatives
Now we plug these into the original differential equation
We can multiply the x into the second term to get
We would like to combine like terms, but there are two problems. The first is the powers of x do
not match and the second is that the summations begin in differently. We will first deal with the
powers of x. We shift the index of the first summation by letting
u = n - 2 n = u + 2
We arrive at
Since u is a dummy variable, we can call it n instead to get
4. Next we deal with the second issue. The second summation begins at 1 while the first and third
begin at 0. We deal with this by pulling out the 0th
term. We plug in 0 into the first and third
series to get
(0 + 2)(0 + 1)a0+2x0 = 2a2
and
a0x0
= a0
We can write the series as
The initial conditions give us that
a0 = 0 and a1 = 1
Now we equate coefficients. The terms in the series begin with the first power of x, hence the
constant term gives us
2a2 + a0 = 0
Since a0 = 0, so is a2. Now the coefficient in front of xn
is zero for all n. We have
(n + 2)(n + 1)an+2 + (n + 1)an = 0
Solving for an+2 gives
-an
an+2 =
n+2
We immediately see that
an = 0
for n even. Now compute the odd an
5. -1 1 -1
a1 = 1 a3 = a5 = a7 =
3 3.
5 3.
5.
7
In general
(-1)n
2n
(n!)(-1)n
a2n+1 = =
3.
5.
7.
....
(2n+1) (2n + 1)!
The final solution is
This cannot be written in terms of elementary functions, however a computer can graph or
calculate a value with as many decimal places as needed.
Bessel equations and their series solutions
The linear second order ordinary differential equation of type
is called the Bessel equation. The number v is called the order of the Bessel equation.
The given differential equation is named after the German mathematician and astronomer
Friedrich Wilhelm Bessel who studied this equation in detail and showed (in 1824) that its
solutions are expressed through a special class of functions called cylinder functions or Bessel
functions.
Concrete representation of the general solution depends on the number v. Further we consider
separately two cases:
• The order v is non-integer;
• The order v is an integer.
Case 1. The Order v is Non-Integer
Assuming that the number v is non-integer and positive, the general solution of the Bessel
equation can be written as
6. where C1, C2 are arbitrary constants and Jv(x), J−v(x) are Bessel functions of the first kind.
The Bessel function can be represented by a series, the terms of which are expressed through the
so-called Gamma function:
The Gamma function is the generalization of the factorial function from integers to all real
numbers. It has, in particular, the following properties:
The Bessel functions of the negative order (−v) (it's assumed that v > 0) are written in similar
way:
The Bessel functions can be calculated in most mathematical software packages. For example,
the Bessel functions of the 1st kind of orders v = 0 to v = 4 are shown in Figure 1. The
corresponding functions are also available in MS Excel.
Fig.1 Fig.2
Case 2. The Order v is an Integer
If the order v of the Bessel differential equation is an integer, the Bessel functions Jv(x) and
J−v(x) can become dependent from each other. In this case the general solution is described by
another formula:
7. where Yv(x) is the Bessel function of the second kind. Sometimes this family of functions is also
called Neumann functions or Weber functions.
The Bessel function of the second kind Yv(x) can be expressed through the Bessel functions of
the first kind Jv(x) and J−v(x):
The graphs of the functions Yv(x) for several first orders v are shown above in Figure 2.
Note: Actually the general solution of the differential equation expressed through Bessel
functions of the first and second kind is valid for non-integer orders as well.
Some Differential Equations Reducible to Bessel's Equation
1. One of the well-known equations tied with the Bessel's differential equation is the modified
Bessel's equation that is obtained by replacing x to −ix. This equation has the form:
The solution of this equation can be expressed through the so-called modified Bessel functions of
the first and second kind:
where Iv(x) and Kv(x) are modified Bessel functions of the 1st and 2nd kind, respectively.
2. The Airy differential equation known in astronomy and physics has the form:
It can be also reduced to the Bessel equation. Its solution is given by the Bessel functions of the
fractional order :
3. The differential equation of type
differs from the Bessel equation only by a factor a2
before x2
and has the general solution in the
form:
8. 4. The similar differential equation
is reduced to the Bessel equation
by using the substitution
Here the parameter n2
denotes:
As a result, the general solution of the differential equation is given by
The special Bessel functions are widely used in solving problems of theoretical physics, for
example in investigating
• wave propagation;
• heat conduction;
• vibrations of membranes
in the systems with cylindrical or spherical symmetry.
Legendre equations
The Legendre differential equation is the second-order ordinary differential equation
(1)
which can be rewritten
(2)
9. The above form is a special case of the so-called "associated Legendre differential equation"
corresponding to the case . The Legendre differential equation has regular singular points at
, 1, and .
If the variable is replaced by , then the Legendre differential equation becomes
(3)
derived below for the associated ( ) case.
Since the Legendre differential equation is a second-order ordinary differential equation, it has
two linearly independent solutions. A solution which is regular at finite points is called a
Legendre function of the first kind, while a solution which is singular at is called a
Legendre function of the second kind. If is an integer, the function of the first kind reduces to a
polynomial known as the Legendre polynomial.
The Legendre differential equation can be solved using the Frobenius method by making a series
expansion with ,
(4)
(5)
(6)
Plugging in,
(7)
(8)
(9)
(10)
(11)
10. (12)
(13)
(14)
so each term must vanish and
(15)
(16)
(17)
Therefore,
(18)
(19)
(20)
(21)
(22)
so the even solution is
(23)
Similarly, the odd solution is
(24)
If is an even integer, the series reduces to a polynomial of degree with only even powers
of and the series diverges If is an odd integer, the series reduces to a polynomial of
degree with only odd powers of and the series diverges The general solution for an
integer is then given by the Legendre polynomials
11. (25)
(26)
where is chosen so as to yield the normalization and is a hypergeometric
function.
A generalization of the Legendre differential equation is known as the associated Legendre
differential equation.
Moon and Spencer (1961, p. 155) call the differential equation
(27)
Properties of Bessel function
For integer order α = n, Jn is often defined via a Laurent series for a generating function:
an approach used by P. A. Hansen in 1843. (This can be generalized to non-integer order by
contour integration or other methods.) Another important relation for integer orders is the
Jacobi–Anger expansion:
and
which is used to expand a plane wave as a sum of cylindrical waves, or to find the Fourier series
of a tone-modulated FM signal.
12. More generally, a series
is called Neumann expansion of ƒ. The coefficients for ν = 0 have the explicit form
where Ok is Neumann's polynomial.[33]
Selected functions admit the special representation
with
due to the orthogonality relation
More generally, if ƒ has a branch-point near the origin of such a nature that
then
or
13. where is f's Laplace transform.[34]
Another way to define the Bessel functions is the Poisson representation formula and the Mehler-
Sonine formula:
where ν > −1/2 and z ∈ C.[35]
This formula is useful especially when working with Fourier
transforms.
The functions Jα, Yα, Hα
(1)
, and Hα
(2)
all satisfy the recurrence relations:
where Z denotes J, Y, H(1)
, or H(2)
. (These two identities are often combined, e.g. added or
subtracted, to yield various other relations.) In this way, for example, one can compute Bessel
functions of higher orders (or higher derivatives) given the values at lower orders (or lower
derivatives). In particular, it follows that:
Modified Bessel functions follow similar relations :
14. and
The recurrence relation reads
where Cα denotes Iα or eαπi
Kα. These recurrence relations are useful for discrete diffusion
problems.
Because Bessel's equation becomes Hermitian (self-adjoint) if it is divided by x, the solutions
must satisfy an orthogonality relationship for appropriate boundary conditions. In particular, it
follows that:
where α > −1, δm,n is the Kronecker delta, and uα, m is the m-th zero of Jα(x). This orthogonality
relation can then be used to extract the coefficients in the Fourier–Bessel series, where a function
is expanded in the basis of the functions Jα(x uα, m) for fixed α and varying m.
An analogous relationship for the spherical Bessel functions follows immediately:
Another orthogonality relation is the closure equation:[36]
for α > −1/2 and where δ is the Dirac delta function. This property is used to construct an
arbitrary function from a series of Bessel functions by means of the Hankel transform. For the
spherical Bessel functions the orthogonality relation is:
15. for α > −1.
Another important property of Bessel's equations, which follows from Abel's identity, involves
the Wronskian of the solutions:
where Aα and Bα are any two solutions of Bessel's equation, and Cα is a constant independent of
x (which depends on α and on the particular Bessel functions considered). For example, if Aα =
Jα and Bα = Yα, then Cα is 2/π. This also holds for the modified Bessel functions; for example, if
Aα = Iα and Bα = Kα, then Cα is −1.
Properties of Legendre polynomials
Generating function
Let F(x t) be a function of the two variables x and t that can be expressed as a Taylor’s series in
t, ∑ncn(x)tn. The function F is then called a generating function of the functions cn.
Example 11.1:
Show that F(x t)=11−xt is a generating function of the polynomials xn.
Solution:
Look at
11−xt=∑∞n=0xntn( xt 1) (11.16)
Example 11.2:
Show that F(x t)=exp 2ttx−t is the generating function for the Bessel functions,
F(x t)=exp(2ttx−t)=∑∞n=0Jn(x)tn (11.17)
Example 11.3:
16. (The case of most interest here)
F(x t)=1 1−2xt+t2=∑∞n=0Pn(x)tn (11.18)
Rodrigues’ Formula
Pn(x)=12nn!dndxn(x2−1)n (11.19)
A table of properties
1. Pn(x) is even or odd if n is even or odd.
2. Pn(1)=1.
3. Pn(−1)=(−1)n.
4. (2n+1)Pn(x)=P n+1(x)−P n−1(x).
5. (2n+1)xPn(x)=(n+1)Pn+1(x)+nPn−1(x).
6. ∫x−1Pn(x )dx =12n+1 Pn+1(x)−Pn−1(x) .
Let us prove some of these relations, first Rodrigues’ formula. We start from the simple formula
(x2−1)ddx(x2−1)n−2nx(x2−1)n=0 (11.20)
which is easily proven by explicit differentiation. This is then differentiated n+1 times,
dn+1dxn+1 (x2−1)ddx(x2−1)n−2nx(x2−1)n
===n(n+1)dndxn(x2−1)n+2(n+1)xdn+1dxn+1(x2−1)n+(x2−1)dn+2dxn+2(x2−1)n−2n(n+1)dndx
n(x2−1)n−2nxdn+1dxn+1(x2−1)n−n(n+1)dndxn(x2−1)n+2xdn+1dxn+1(x2−1)n+(x2−1)dn+2dxn
+2(x2−1)n− ddx(1−x2)ddx dndxn(x2−1)n +n(n+1) dndxn(x2−1)n =0
We have thus proven that dndxn(x2−1)n satisfies Legendre’s equation. The normalisation
follows from the evaluation of the highest coefficient,
dndxnx2n=n!2n!xn (11.22)
and thus we need to multiply the derivative with 12nn! to get the properly normalised Pn.
Let’s use the generating function to prove some of the other properties: 2.:
17. F(1 t)=11−t=∑ntn (11.23)
has all coefficients one, so Pn(1)=1. Similarly for 3.:
F(−1 t)=11+t=∑n(−1)ntn (11.24)
Property 5. can be found by differentiating the generating function with respect to t:
ddt1
1−2tx+t2x−t(1−2tx+t2)3∕2x−t1−2xt+t2∑∞n=0tnPn(x)∑∞n=0tnxPn(x)−∑∞n=0tn+1Pn(x)∑∞n=0t
n(2n+1)xPn(x)=====ddt∑∞n=0tnPn(x)∑n=0ntn−1Pn(x)∑n=0ntn−1Pn(x)∑∞n=0ntn−1Pn(x)−2
∑∞n=0ntnxPn(x)+∑∞n=0ntn+1Pn(x)∑∞n=0(n+1)tnPn+1(x)+∑∞n=0ntnPn−1(x)
Equating terms with identical powers of t we find
(2n+1)xPn(x)=(n+1)Pn+1(x)+nPn−1(x) (11.26)
