An important example of such a class of polynomial functions is the Cheby- v polynomials denoted by the symbol T(x). They are defined by the re- rence formula (;) Tk+2 – xTk+1 + T = 0, (7.59) ere |æ| < 1, and To = 2, T1 = x. In this section, we will investigate certain perties of these functions. Using the above equation and the given values for To and T1, the first eral Chebyshev polynomials can be easily calculated; they are 1 T2(x) = x² – 2' 3x T2(x) = x³ 4' (7.60) 1 T4(x) = xª – x² + 8' oceeding in this way, we can obtain T(x) for any finite integer k. How- r, this procedure is very laborious and it would be much better to have Difference Equations ompact expression giving Tx(x) explicitly in terms of x and k. This can easily done because equation (7.59) is a second-order, linear difference ation, and its general solution can be determined by the techniques of apter 4. The characteristic equation corresponding to equation (7.59) is 1 p2 - xr + = 0. (7.61) 4 _s equation has the two solutions T'1,2 = (x± / Va2 - - 1]. (7.62) erefore, the kth Chebyshev polynomial takes the form [A(r1)* + B(r2)*], (7.63) ere A and B are constants that can be determined by requiring To = 2 and = x; doing this gives A+B = 2, (7.64) rịA+ r2B = 2x, A = B = 1. (7.65) ostitution of these values for A and B into equation (7.63) gives an explicit -ression for T(æ): (ne) - g2 – 1)*). TR(x) = ) [(x + /x² – 1)* + (x – (7.66) check of equation (7.66) for k = 2,3, and 4 shows that it reproduces the ults previously given by equation (7.60).
An important example of such a class of polynomial functions is the Cheby- v polynomials denoted by the symbol T(x). They are defined by the re- rence formula (;) Tk+2 – xTk+1 + T = 0, (7.59) ere |æ| < 1, and To = 2, T1 = x. In this section, we will investigate certain perties of these functions. Using the above equation and the given values for To and T1, the first eral Chebyshev polynomials can be easily calculated; they are 1 T2(x) = x² – 2' 3x T2(x) = x³ 4' (7.60) 1 T4(x) = xª – x² + 8' oceeding in this way, we can obtain T(x) for any finite integer k. How- r, this procedure is very laborious and it would be much better to have Difference Equations ompact expression giving Tx(x) explicitly in terms of x and k. This can easily done because equation (7.59) is a second-order, linear difference ation, and its general solution can be determined by the techniques of apter 4. The characteristic equation corresponding to equation (7.59) is 1 p2 - xr + = 0. (7.61) 4 _s equation has the two solutions T'1,2 = (x± / Va2 - - 1]. (7.62) erefore, the kth Chebyshev polynomial takes the form [A(r1)* + B(r2)*], (7.63) ere A and B are constants that can be determined by requiring To = 2 and = x; doing this gives A+B = 2, (7.64) rịA+ r2B = 2x, A = B = 1. (7.65) ostitution of these values for A and B into equation (7.63) gives an explicit -ression for T(æ): (ne) - g2 – 1)*). TR(x) = ) [(x + /x² – 1)* + (x – (7.66) check of equation (7.66) for k = 2,3, and 4 shows that it reproduces the ults previously given by equation (7.60).
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Explain the determine blue
![7.2.3
Chebyshev Polynomials
An Nth-order polynomial in the variable is defined as
PN (x) = .
anaN
+ aN-12N-1
+...+ a1x + ao, aN + 0,
(7.56)
where the a; (for i = 0,1, 2,..., N) are constants, and N is a nonnegative
integer. For example, the following polynomials are, respectively, of zero, first,
and third order:
Po(x) = 3,
P1 (x) = x – 1,
P3(x) = 5x3 + x² – 2x + 4.
(7.57)
A number of special classes of polynomials are of great value for the inves-
tigation of certain problems that arise in both pure and applied mathematics.
In particular, it can be shown that any "reasonable" function of x, defined
over the interval –1 <x < +1, can be written as a sum of the members of any
one of these special classes of polynomials. An example is the class of power
functions whose Nth member is PN = xN. Hence, any (reasonable) function
f (x) has the representation
f(x) = boPo(x) + 61P1 (x) + · · ·+ bN PN (x) + · ..
(7.58)
E bmam.
m=0
The series defined by the second line on the right side of equation (7.58) is
known as a Taylor series. In general, the series expansion of an arbitrary
function contains an unlimited (infinite) number of terms.
An important example of such a class of polynomial functions is the Cheby-
shev polynomials denoted by the symbol T(x). They are defined by the re-
currence formula
() .
Tk-+2 – xT+1 +
Tk = 0,
(7.59)
where |x| < 1, and To = 2, T1 = x. In this section, we will investigate certain
properties of these functions.
Using the above equation and the given values for To and T1, the first
several Chebyshev polynomials can be easily calculated; they are
1
T2(x)
2'
3x
T2(x)
= x3
(7.60)
4'
1
T4(x) = x4 – x² +
8'
Proceeding in this way, we can obtain T(x) for any finite integer k. How-
ever, this procedure is very laborious and it would be much better to have
222
Difference Equations
a compact expression giving Tk(x) explicitly in terms of x and k. This can
be easily done because equation (7.59) is a second-order, linear difference
equation, and its general solution can be determined by the techniques of
Chapter 4.
The characteristic equation corresponding to equation (7.59) is
1
= 0.
4
p2
xr +
(7.61)
This equation has the two solutions
(;)
T'1,2 =
(x +
x2
1].
(7.62)
Therefore, the kth Chebyshev polynomial takes the form
G) [A(r1)* + B(r2)*],
(7.63)
2k
where A and B are constants that can be determined by requiring To = 2 and
T1 = x; doing this gives
А+В- 2.
(7.64)
rịA + r2B = 2x,
and
A = B = 1.
(7.65)
Substitution of these values for A and B into equation (7.63) gives an explicit
expression for T; (x):
(Te(2) = () (z + V2 - 1)* + (# – VP - 1)*].
T (x) =
(7.66)
A check of equation (7.66) for k = 2,3, and 4 shows that it reproduces the
results previously given by equation (7.60).
Let us now examine equation (7.66) in more detail. The results of this](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F57a26049-260e-45aa-910f-dd852150673b%2Fb0ed9b04-d2d3-4775-8ebb-1593ccd4759f%2Fii8bk89_processed.png&w=3840&q=75)
Transcribed Image Text:7.2.3
Chebyshev Polynomials
An Nth-order polynomial in the variable is defined as
PN (x) = .
anaN
+ aN-12N-1
+...+ a1x + ao, aN + 0,
(7.56)
where the a; (for i = 0,1, 2,..., N) are constants, and N is a nonnegative
integer. For example, the following polynomials are, respectively, of zero, first,
and third order:
Po(x) = 3,
P1 (x) = x – 1,
P3(x) = 5x3 + x² – 2x + 4.
(7.57)
A number of special classes of polynomials are of great value for the inves-
tigation of certain problems that arise in both pure and applied mathematics.
In particular, it can be shown that any "reasonable" function of x, defined
over the interval –1 <x < +1, can be written as a sum of the members of any
one of these special classes of polynomials. An example is the class of power
functions whose Nth member is PN = xN. Hence, any (reasonable) function
f (x) has the representation
f(x) = boPo(x) + 61P1 (x) + · · ·+ bN PN (x) + · ..
(7.58)
E bmam.
m=0
The series defined by the second line on the right side of equation (7.58) is
known as a Taylor series. In general, the series expansion of an arbitrary
function contains an unlimited (infinite) number of terms.
An important example of such a class of polynomial functions is the Cheby-
shev polynomials denoted by the symbol T(x). They are defined by the re-
currence formula
() .
Tk-+2 – xT+1 +
Tk = 0,
(7.59)
where |x| < 1, and To = 2, T1 = x. In this section, we will investigate certain
properties of these functions.
Using the above equation and the given values for To and T1, the first
several Chebyshev polynomials can be easily calculated; they are
1
T2(x)
2'
3x
T2(x)
= x3
(7.60)
4'
1
T4(x) = x4 – x² +
8'
Proceeding in this way, we can obtain T(x) for any finite integer k. How-
ever, this procedure is very laborious and it would be much better to have
222
Difference Equations
a compact expression giving Tk(x) explicitly in terms of x and k. This can
be easily done because equation (7.59) is a second-order, linear difference
equation, and its general solution can be determined by the techniques of
Chapter 4.
The characteristic equation corresponding to equation (7.59) is
1
= 0.
4
p2
xr +
(7.61)
This equation has the two solutions
(;)
T'1,2 =
(x +
x2
1].
(7.62)
Therefore, the kth Chebyshev polynomial takes the form
G) [A(r1)* + B(r2)*],
(7.63)
2k
where A and B are constants that can be determined by requiring To = 2 and
T1 = x; doing this gives
А+В- 2.
(7.64)
rịA + r2B = 2x,
and
A = B = 1.
(7.65)
Substitution of these values for A and B into equation (7.63) gives an explicit
expression for T; (x):
(Te(2) = () (z + V2 - 1)* + (# – VP - 1)*].
T (x) =
(7.66)
A check of equation (7.66) for k = 2,3, and 4 shows that it reproduces the
results previously given by equation (7.60).
Let us now examine equation (7.66) in more detail. The results of this
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