Math 311-Special Functions — Exercises

Section	Exercises
1.1	9, 12, 14, 25, 26
1.2	25, 30, 32
1.3	3, 5, 9, 13, 14, 18, 19
1.4	8, 15, 18, 21, 23
1.5	13, 19, 27
1.6	7, 9, 16
2.1A	5, 9, 12
2.2C	9, 13, 19, 24, 28
2.2D	2, 4, 10, 11 (see errors)
2.3	6, 10, 17, 33, 39
2.4	8, 9, 19, 21, 23
2.5	5, 6, 8, 9, 11 (use #12), 17, 18
3.1	1, 4, 5, 8, 9, 10, 13, 19, 23
3.2	7-11, 13, 16, 17, 20, 26, 27
3.3	6-8, 11, 15-17, 20, 24, 27
3.4	5, 6, 9, 13, 17, 19, 20, 21
3.5B	3, 8, 10, 13, 17, 27, 35, 40, 41
3.5C	1, 3, 4, 11, extra problems 1-3
3.6A	3, 5, 6, 8, 15, 17
3.6C	2, 3, 8, 11, 13, 16, 19, 21
3.7A	2-5, 7, 8, 11, 13
3.7B	3-5, 7, 8, 11, 12, 17, extra problem 4
3.7C	1-3, 5 (see errors)
14.6	3, 7
14.7	2, 5, 7, 8, extra problem 5
14.8	3, 5, 9, 22, 23, 31, 32
14.9	2, 5, 7, 13, 15 16
14.10B	3, 5, 9, 15, 16, 23, 30, extra problem 6

Extra problems

1. Let L : P₂ → P₂ be the linear transformation defined by

   L[p] = (x²+2)p″ + (x-1)p′ - 4p .

   Find the matrix for L. Find bases of polynomials for the image and null space of L.

2. Let A be the matrix given below. Find the dimension of the image of A. Use it and problem 11, § 3.5C to find the dimension of the null space of A.

   1	-2	3	3
   2	-5	7	3
   -1	3	-4	3

3. Suppose that B is a 7×10 matrix, and that the dimension of the null space of B is 5. What is the dimension of the image (column space) of B?

4. Expand p(x) =3x³ - 2x + 1 in Legendre polynomials up to degree 3.

5. Consider the differential equation u″ + x^-1u′ + λ² u = 0, where λ > 0.
   1. Show that u=J₀(λx) satisfies the equation.
   2. Use your favorite software (MATLAB, MAPLE, etc.) to plot J₀(x), starting at x = 0 and running through x = 100. (In MATLAB, besselj(0,x) is J₀(x)). Observe that J₀(x) has oscillations similar to trig functions. In particular, it has lots of zeros.
   3. Suppose that we list the zeros of J₀ in increasing order, λ₁ < λ₂ ... . Show that if j ≠ k, then J₀(λ_jx) and J₀(λ_kx) are orthogonal in the inner product
      < f,g > = ∫₀¹ x f(x)g(x)dx;
      that is, that they satisfy
      ∫₀¹ x J₀(λ_jx) J₀(λ_kx)dx = 0.

   4. If f(x) = a₁J₀(λ₁x) + a₂J₀(λ₂x) + a₃J₀(λ₃x) + ..., use the orthogonality relation derived above to show that the coefficients are given by
      a_k = < f(x), J₀(λ_kx)> (< J₀(λ_kx), J₀(λ_kx)>)^-1.
      You may assume that interchanging sum and integral is permissible.

6. In polar coordinates r, θ, the diffusion equation has the form,

   ∂u/∂ t = ∂²u/∂r² + r^-1 ∂u/∂r + r^-1∂²u/∂θ².

   Given that u=u(r,t) (no angular dependence), u(a,t) = 0, and assuming that u is continuous at r = 0, separate variables.

Updated 8/24/05 (fjn).