Mathematical Tools for Physics

6—Vector Spaces 166

6.33 Show that the sequence of rational numbersan =

∑n
∑n k=1^1 /k is not a Cauchy sequence. What about
k=1^1 /k

(^2)?
6.34 In the vector space of polynomials of the formαx+βx^3 , use the scalar product

〈

f,g

〉

=

∫ 1

0 dxf(x)

∗g(x)

and construct an orthogonal basis for this space.

6.35 You can construct the Chebyshev polynomials by starting from the successive powers,xn,n= 0, 1 , 2 ,...
and applying the Gram-Schmidt process. The scalar product in this case is

〈

f,g

〉

=

∫ 1

− 1

dx

f(x)∗g(x)

√

1 −x^2

The conventional normalization for these polynomials isTn(1) = 1, so you don’t have to make the norm of the
resulting vectors one. Construct the first four of these polynomials, and show that the result isTn(cosθ) =
cos(nθ). These polynomials are used in numerical analysis because they have the property that they oscillate
uniformly between− 1 and+1on the domain− 1 < x < 1. Verify that your results satisfy the recurrence relation:
Tn+1(x) = 2xTn(x)−Tn− 1 (x)

6.36 In spherical coordinates(θ,φ), the angleθis measured from thez-axis, and the functionf 1 (θ,φ) = cosθ
can be written in terms of rectangular coordinates as

f 1 (θ,φ) = cosθ=

z

r

=

z

√

x^2 +y^2 +z^2

Pick up the functionf 1 and rotate it by 90 ◦counterclockwise about the positivey-axis. Do this rotation in terms
of rectangular coordinates, but express the result in terms of spherical. Call itf 2
Now pick up the samef 1 and rotate it by 90 ◦clockwise about the positivex-axis, again finally expressing the
result in terms of spherical coordinates. Call itf 3.
If I ask you to take the originalf 1 and rotate it about some random axis by some random angle, show that the
resulting functionf 4 is a linear combination of the three functionsf 1 ,f 2 , andf 3. I.e., all these possible rotated
functions form only a three dimensional vector space. Again, calculations such as these are easier to demonstrate
in rectangular coordinates.