Chapter 4
Power Series Solutions
Alberto J. Benavides,
Xiaohong Cui,
Adewale Awoniyi
Department of Chemical Engineering
Texas A&M University, College Station, TX
1
Agenda
4.4
4.5
4.6
Legendre Functions
Single Integrals; Gamma Function
Bessel Functions
2
4.4. Legendre Functions
Alberto J. Benavides,
Xiaohong Cui,
Adewale Awoniyi
Department of Chemical Engineering
Texas A&M University, College Station, TX
3
4.4.1 Legendre Polynomials
A special form a second order differential equation which is encountered often when
solving physical problems is:
This equation is known as the Legendre Equation. The general solution to the Legendre
equation is given as a function of two Legendre functions as follows:
where
Adapted from:
http://www.mhtl.uwaterloo.ca/courses/m
e755/web_chap5.pdf
4/9/2015
4
4.4.1 Legendre Polynomials
If n = 0,1,2,3,…the Pn(x) functions are called Legendre Polynomials of order n and are
given by Rodrigues’s formula:
The first four Legendre Polynomials are:
P0 ( x )  1
P1 ( x )  x
1
(3 x 2  1)
2
1
P3 ( x )  (5 x 3  3 x )
2
1
P4 ( x )  (35x 4  30x 2  3)
8
P2 ( x ) 
Plot of the first four Legendre Polynomials
From: http://www.mae.ufl.edu/~uhk/MATHFUNC.htm
Adapted from:
http://www.mhtl.uwaterloo.ca/courses/m
e755/web_chap5.pdf
4/9/2015
5
4.4.2 Orthogonality of Legendre Polynomials
Proof of the orthogonality relation
If Pm and Pn are solutions of Legendre’s equation, then
(1)
(2)
Integrating the combination Pm (1) - Pn (2)
Using integration by parts
`
As Pm,n and their derivatives are finite at x±1 (i.e. they are regular there). Hence if n ≠ m
From:
http://www.ucl.ac.uk/~ucahdrb/MATHM242/OutlineCD2.pdf
4/9/2015
6
4.4.3 Generating Function and Properties
The above equation is called the generating function for the Pn’s
What is
?
We can evaluate the integral using Rodrigues’s formula. We have
Integrating by parts we get
Note that differentiating (x2-1)n anything less than n times leaves an expression
that has (x2-1) as a factor so that the rest of the terms vanishes. Similarly,
integrating by parts n times gives
From:
http://www.ucl.ac.uk/~ucahdrb/MATHM242/OutlineCD2.pdf
4/9/2015
7
4.4.3 Generating Function and Properties
The (2n)th derivative of the polynomial (x2-1)n, which has degree 2n is (2n)!. Thus
Using the transformation s= (x+1)/2 to transform the integral and then use a
reduction formula to show that
The final result is:
From:
http://www.ucl.ac.uk/~ucahdrb/MATHM242/OutlineCD2.pdf
4/9/2015
8
Example
Consider the Legendre Equation:
(1  x 2 ) y' '2xy' (  1) y  0
1. Find 2 linearly independent solutions in the form of power series about x= 0.
2. Compute the radius of convergence of the series. Explain why it is possible to predict
the radius without actually deriving the series.
3. Show that α = 2n, with n an integer and n≥0, the series for one of the solutions reduces
to an even polynomial of degree 2n
4. Show that α = 2n+1, with n an integer and n≥0, the series for one of the solutions
reduces to an odd polynomial of degree 2n+1
5. Show that the first 4 polynomial solutions Pn(x) (known as Legendre polynomials)
ordered by their degree and normalized so that Pn(1) =1 are
P0  1
P1  x
1
P2  (3x 2  1)
2
1
P3  (5 x 3  3x)
2
From: Introduction to Applied Mathematics
or Advanced Mathematical Methods for
Scientists and Engineers - Mauch
4/9/2015
9
Solution
1. First we write the differential equation in the general form:
(1)
Since the coefficients of y’ and y are analytic in a neighborhood of x = 0, We can find two
Taylor series solutions about that point. We find the Taylor series for y and its derivatives.
From: Introduction to Applied Mathematics
or Advanced Mathematical Methods for
Scientists and Engineers.-Mauch
4/9/2015
10
Solution
Index shifting was used to explicitly write two forms that we will need for y’’. Note that we
can take the lower bound of summation to be n=0 for all the sums. The terms added by
this operation are zero. We substitute the Taylor series in to Equation (1).
We equate coefficients of xn to obtain the recurrence relation
From: Introduction to Applied Mathematics
or Advanced Mathematical Methods for
Scientists and Engineers.-Mauch
4/9/2015
11
Solution
We can solve this difference equation to determine the an’s (a0 and a1 are arbitrary)
We will find the fundamental solutions at x=0, that is the set {y1,y2} that satisfies
For y1 we take a0=1 and a1=0; for y2 we take a0=0 and a1=1. The rest of the coefficients
are determined from the recurrence relation.
From: Introduction to Applied Mathematics
or Advanced Mathematical Methods for
Scientists and Engineers.-Mauch
4/9/2015
12
Solution
2. We determine the radius of convergence of the series solutions with the ratio test.
=
=
Therefore.
Thus we see that the radius of convergence of the series is 1. We knew that the radius of
convergence would be at least one, because the nearest singularities of the coefficients of
(1) occur at x=±1, a distance of 1 from the origin. This implies that the solutions of the
equation are analytic in the unit circle about x=0. The radius of convergence of the Taylor
series expansion of an analytic function is the distance to the nearest singularity.
From: Introduction to Applied Mathematics
or Advanced Mathematical Methods for
Scientists and Engineers-Mauch.
4/9/2015
13
Solution
3. If α=2n then a2n+2=0 in our first solution. From the recurrence relation, we see that all
subsequent coefficients are also zero. The solution becomes an even polynomial.
4. If α=2n+1 then a2n+3=0 in our second solution. From the recurrence solution, we see
that all subsequent coefficients are also zero. The solution becomes an odd polynomial.
From: Introduction to Applied Mathematics
or Advanced Mathematical Methods for
Scientists and Engineers-Mauch.
4/9/2015
14
Solution
5. From our solution above, the first four polynomials are
To obtain the Legendre polynomials we normalize these to have unity at x=1
From: Introduction to Applied Mathematics
or Advanced Mathematical Methods for
Scientists and Engineers-Mauch.
4/9/2015
15
4.5. Single Integrals; Gamma Function
Alberto J. Benavides,
Xiaohong Cui,
Adewale Awoniyi
Department of Chemical Engineering
Texas A&M University, College Station, TX
16
4.5.1 Singular Integrals; Gamma Function
An integral is said to be singular (or improper) if one or both integration limits are infinite
and/or if the integrand is unbounded on the interval; otherwise, it is regular (or proper).
We can group improper integrals into two:
and
Adapted from:
http://www.mcae.umn.edu/acadsupport/documents/ptest_000.pdf
4/9/2015
17
4.5.1 Singular Integrals
The “p-test” is used to determine whether or not the integral converges an
diverges and it can be summarized as follows.
The following series is called the p-series:
The p-series acts similar to the Type I improper integral. Therefore:
Adapted from:
http://www.mcae.umn.edu/acadsupport/docu
ments/p-test_000.pdf
4/9/2015
18
The Integral Test can be summarized as follows:
For proof of this, consider the following graphs,
If integral diverges, so does series
4/9/2015
If integral converges, so does series
Adapted from:
http://faculty.otterbein.edu/TJames/math
1800/presentations/9.3%20The%20Integ
ral%20Test.pdf
19
4.5.1 Singular Integrals
The Comparison Tests: Suppose that ∑ an and ∑ bn are series with positive terms
and suppose that an ≤ bn for all n. Then
Example 1: Determine if the following series converges or diverges.
Solution: We have
and we know that
converges. Hence by the comparison test, it also converges.
From:
http://www.math.northwestern.edu/~mlerma/cour
ses/math214-2-04f/notes/c2-convertest.pdf
4/9/2015
20
4.5.1 Singular Integrals
Example 2: Determine if the following converges or diverges.
Solution: We consider
. Here p=3. Since p >1. This integral converges.
Looking at our integral, we see that for positive x,
From the comparison test, since
converges, so does
.
From:
http://www.mcae.umn.edu/acadsupport/documents/ptest_000.pdf
4/9/2015
21
4.5.1 Singular Integrals
Absolute Convergence
A series is
values
said to be absolutely convergent if the series of absolute
converges.
Example: Consider the series
The series absolutely converges since
converges.
From:
http://de2de.synechism.org/c
5/sec56.pdf
4/9/2015
22
4.5.2 Gamma Function
The gamma function is given by the improper integral
The gamma function satisfied the recursive property:
The recursive relationship can be used to compute the gamma function of all real
numbers (except the non-positive integers) by knowing only the value of the
gamma function between 1 and 2.
Example 1: For α = 5, and using the recursive property:
Values of
can be found using this sort of Table
Adapted from:
jekyll.math.byuh.edu/courses/m321/han
douts/gammaproperties.pdf
4/9/2015
23
4.5.2 Gamma Function
Example 2: Now with α = 23/3, and using the recursive property
Adapted from:
jekyll.math.byuh.edu/courses/m321/han
douts/gammaproperties.pdf
4/9/2015
24
4.5.2 Gamma Function
When α = n and n is a positive integer, then the gamma function is related to the
factorial function by
The gamma function evaluated at α = ½ is
Next, we can extend the usefulness of the gamma function so that it can be
computed for negative non-integer numbers. Substituting α+1 with α in the
recursive relationship, we get:
and solving for
Adapted from:
jekyll.math.byuh.edu/courses/m321/han
douts/gammaproperties.pdf
4/9/2015
25
4.5.2 Gamma Function
Example 3: Suppose that α = -5/6
Using linear interpolation between 1.16 and 1.17 yields
.
Hence,
which is close to the actual value of -6.679579.
Many integrals can be evaluated by making suitable changes of variables so as to
reduce them to gamma function integrals.
Example 4:
Solution:
Adapted from:
jekyll.math.byuh.edu/courses/m321/han
douts/gammaproperties.pdf
4/9/2015
26
4.5.2 Gamma Function
Evaluating the integral, we get
4/9/2015
27
4.5.2 Gamma Function
Table of the Gamma Function
From:
jekyll.math.byuh.edu/courses/m321/han
douts/gammaproperties.pdf
4/9/2015
28
4.6. Bessel Functions
Alberto J. Benavides,
Xiaohong Cui,
Adewale Awoniyi
Department of Chemical Engineering
Texas A&M University, College Station, TX
29
4.6 Bessel Functions
One of the varieties of special functions which are encountered in the solution of physical
problems is the class of functions called Bessel functions. They are solutions to a very
important differential equation, the Bessel equation:
Bessel's equation arises when finding separable solutions to Laplace's equation and the
Helmholtz equation in cylindrical or spherical coordinates. In solving problems in cylindrical
coordinate systems, one obtains Bessel functions of integer order (α = n); in spherical
problems, one obtains half-integer orders (α = n + ½). For example:
•Electromagnetic waves in a cylindrical waveguide
•Heat conduction in a cylindrical object
•Modes of vibration of a thin circular (or annular) artificial membrane
•Diffusion problems on a lattice
•Solutions to the radial Schrödinger equation (in spherical and cylindrical coordinates) for a
free particle
•Solving for patterns of acoustical radiation
Adapted from: http://en.wikipedia.org/wiki/Bessel_function
4/9/2015
30
4.6 Bessel Functions of the first kind
Since this is a second-order differential equation, there must be two linearly
independent solutions. Depending upon the circumstances, however, various formulations
of these solutions are convenient, and the different variations are described below.
Bessel functions of the first kind, denoted as Jv(x), are solutions of Bessel's differential
equation that are finite at the origin (x = 0) for integer v, and diverge as x approaches zero
for negative non-integer v. The solution type (e.g., integer or non-integer) and normalization
of Jv(x) are defined by its properties below. It is possible to define the function by its Taylor
series expansion around x = 0:
Γ(z) is the gamma function, a generalization of the factorial function to non-integer values.
Adapted from: http://en.wikipedia.org/wiki/Bessel_function
4/9/2015
31
4.6.1 v ≠ integer
Shaped like oscillating sine or
cosine
functions
that
decay
proportionally to 1/√x.
Taken from: http://www.dplot.com/fct_besselj.htm
For non-integer v, the functions Jv(x) and J−v(x) are linearly independent, and are
therefore the two solutions of the differential equation. That is y(x) = AJv(x)+BJ-v(x). On
the other hand, for integer order v=n, the following relationship is valid (note that the
Gamma function becomes infinite for negative integer arguments).
This means that the two solutions are no longer linearly independent. In this case, the
second linearly independent solution is then found to be the Bessel function of the second
kind, as discussed below.
Adapted from: Calculus:
A Physical & Intuitive
Approach - Morris Kline
4/9/2015
32
4.6.2 v = integer
Since for the case n = 1,2,… the two solutions are no longer linearly independent, therefore
we must use the Frobenius regular singular point solution theorem in order to find a
solution. For v = n = integer the indicial roots r ±n differ by an integer; y2(x) can be found
from y1(x) which results in what is know as the Weber’s Bessel function of the second kind,
or order n:
That is, for v = n = integer(x), the solution
of the Bessel equation will be given by:
y(x) = AJn(x)+BYn(x).
Taken from: http://www.dplot.com/fct_besseljyhtm
Adapted from: Calculus:
A Physical & Intuitive
Approach - Morris Kline
4/9/2015
33
4.6.2 Bessel Functions of the second kind
If we define:
For non-integer v, then the limit of Yv(x) as v  n gives the same result as Yn(x).
That is Nn(x) = Yn(x). Nv(x) is known as the Neumann function of order n
Adapted from: Calculus:
A Physical & Intuitive
Approach - Morris Kline
4/9/2015
34
4.6.3 Hankel functions
Another important formulation of the two linearly independent solutions to Bessel's
equation are the Hankel functions of the first and second kind Hv(1)(x) and Hv(2)(x),
defined by:
where j is the imaginary unit. These linear combinations are also known as Bessel
functions of the third kind; they are two linearly independent solutions of Bessel's
differential equation.
Adapted from: Calculus:
A Physical & Intuitive
Approach - Morris Kline
4/9/2015
35
4.6.4 General solution of the Bessel Equation
As expected any two of the following functions are linearly independent
solutions of the Bessel equation:
Thus, the general solution can be written as a linear combination of any two of
the above functions.
Usually the general solution is written as one of these forms:
Adapted from: Calculus:
A Physical & Intuitive
Approach - Morris Kline
4/9/2015
36
4.6.5 Modified Bessel Equation
The Bessel functions are valid even for complex arguments x, and an important special
case is that of a purely imaginary argument. In this case, the solutions to the Bessel
equation are called the modified Bessel functions (or occasionally the hyperbolic Bessel
functions) of the first and second kind, and are defined by any of these equivalent
alternatives:
These are chosen to be real-valued for real and positive arguments x. The series expansion
for In(x) is thus similar to that for Jn(x), but without the alternating (−1)k factor. In(x) and Kn(x)
are the two linearly independent solutions to the modified Bessel's equation:
Adapted from: Calculus:
A Physical & Intuitive
Approach - Morris Kline
4/9/2015
37
4.6.5 Modified Bessel Equation
Unlike the ordinary Bessel functions, which are oscillating as functions of a real argument,
In and Kn are exponentially growing and decaying functions, respectively. Like the ordinary
Bessel function Jn, the function In goes to zero at x = 0 for α > 0 and is finite at x = 0 for α
= 0. Analogously, Kn diverges at x = 0.
Modified Bessel Functions of the First Kind
Taken from:
http://mathworld.wolfram.com/ModifiedBesselFunctionoft
heFirstKind.html
4/9/2015
Modified Bessel Functions of the Second Kind
Taken from:
http://mathworld.wolfram.com/ModifiedBesselFunctionoftheS
econdKind.html
38
4.6.6 Equations reducible to Bessel Equations
There are many equation which can be reduced to Bessel or modified Bessel equations
by changes of variables and then solved in closed form in terms of Bessel or modified
Bessel functions. There it is the true power of Bessel equations!
Example 1. The Airy differential
equation known in astronomy and
physics has the form:
y’’ – xy = 0
Example 2. The differential equation
of type
It can be also reduced to the Bessel
equation. Its solution is given by the
Bessel functions of the fractional order
+/- 3
differs from the Bessel equation only
by a factor a2 before x2 and has the
general solution in the form:
y(x) = C1Jv(x)+C1Yv(x).
Adapted from: http://www.math24.net/bessel-equation.html
4/9/2015
39
4.6.6 Equations reducible to Bessel Equations
(1)
Example 3.
(2)
Taken from: Textbook Of Engineering
Mathematics By Debashis Dutta
4/9/2015
40
4.6.6 Equations reducible to Bessel Equations
Examples 3.i, 3.ii
Taken from: Textbook Of Engineering
Mathematics By Debashis Dutta
4/9/2015
41