Mathematical Methods for Physics and Engineering : A Comprehensive Guide

SPECIAL FUNCTIONS

Using (18.65) and settingx=cosθimmediately gives a rearrangement of the required
result (18.69). Similarly, adding the plus and minus cases of result (18.68) gives

sin(n+1)θ+sin(n−1)θ=2sinnθcosθ.

Dividing through on both sides by sinθand using (18.66) yields (18.70).

The recurrence relations (18.69) and (18.70) are extremely useful in the practical

computation of Chebyshev polynomials. For example, given the values ofT 0 (x)

andT 1 (x) at some pointx, the result (18.69) may be used iteratively to obtain

the value of anyTn(x) at that point; similarly, (18.70) may be used to calculate

the value of anyUn(x) at some pointx, given the values ofU 0 (x)andU 1 (x)at

that point.

Further recurrence relations satisfied by the Chebyshev polynomials are

Tn(x)=Un(x)−xUn− 1 (x), (18.71)

(1−x^2 )Un(x)=xTn+1(x)−Tn+2(x), (18.72)

which establish useful relationships between the two sets of polynomialsTn(x)

andUn(x). The relation (18.71) follows immediately from (18.68), whereas (18.72)

follows from (18.67), withnreplaced byn+ 1, on noting that sin^2 θ=1−x^2.

Additional useful results concerning the derivatives of Chebyshev polynomials

may be obtained from (18.65) and (18.66), as illustrated in the following example.

Show that

Tn′(x)=nUn− 1 (x),

(1−x^2 )Un′(x)=xUn(x)−(n+1)Tn+1(x).

These results are most easily derived from the expressions (18.65) and (18.66) by noting
thatd/dx=(− 1 /sinθ)d/dθ. Thus,

Tn′(x)=−

1

sinθ

d(cosnθ)

dθ

=

nsinnθ

sinθ

=nUn− 1 (x).

Similarly, we find

U′n(x)=−

1

sinθ

d

dθ

[

sin(n+1)θ

sinθ

]

=

sin(n+1)θcosθ

sin^3 θ

−

(n+1)cos(n+1)θ

sin^2 θ

=

xUn(x)

1 −x^2

−

(n+1)Tn+1(x)

1 −x^2

,

which rearranges immediately to yield the stated result.

18.5 Bessel functions

Bessel’s equation has the form

x^2 y′′+xy′+(x^2 −ν^2 )y=0, (18.73)

which has a regular singular point atx= 0 and an essential singularity atx=∞.

The parameterνis a given number, which we may take as≥0 with no loss of