18.4 CHEBYSHEV FUNCTIONS
T 0
T 1
T 2
T 3
− 1
− 1
− 0. 5
− 0. 5
0. 5
0. 5
1
1
Figure 18.3 The first four Chebyshev polynomials of the first kind.From (18.56) and (18.57), we see immediately thatUn(x)isapolynomialof order
n, but thatWn(x)isnota polynomial. In practice, it is usual to work entirely in
terms ofTn(x)andUn(x), which are known, respectively, asChebyshev polynomials
of the first and second kind. In particular, we note that the general solution to
Chebyshev’s equation can be written in terms of these polynomials as
y(x)=
c 1 Tn(x)+c 2√
1 −x^2 Un− 1 (x)forn=1, 2 , 3 ,...,c 1 +c 2 sin−^1 x forn=0.Then= 0 solution could also be written asd 1 +c 2 cos−^1 xwithd 1 =c 1 +^12 πc 2.
The first few Chebyshev polynomials of the first kind are easily constructedand are given by
T 0 (x)=1, T 1 (x)=x,T 2 (x)=2x^2 −1, T 3 (x)=4x^3 − 3 x,T 4 (x)=8x^4 − 8 x^2 +1, T 5 (x)=16x^5 − 20 x^3 +5x.The functionsT 0 (x),T 1 (x),T 2 (x)andT 3 (x) are plotted in figure 18.3. In general,
the Chebyshev polynomialsTn(x) satisfyTn(−x)=(−1)nTn(x), which is easily
deduced from (18.56). Similarly, it is straightforward to deduce the following