12 Chapter in Review Answers to selected odd-numbered problems begin on page ANS-33.

In Problems 1–10, fill in the blank or answer true/false without referring back to the text.

  1. The functions f(x) = x2 – 1 and g(x) = x5 are orthogonal on the interval [–π, π].
  2. The product of an odd function f with an odd function g is an function.
  3. To expand f(x) = |x| + 1, –π < x < π, in an appropriate trigonometric series we would use a series.
  4. y = 0 is never an eigenfunction of a Sturm–Liouville problem.
  5. λ = 0 is never an eigenvalue of a Sturm–Liouville problem.
  6. If the function

    is expanded in a Fourier series, the series will converge to at x = –1, to at x = 0, and to at x = 1.

  7. Suppose the function f(x) = x2 + 1, 0 < x < 3, is expanded in a Fourier series, a cosine series, and a sine series. At x = 0, the Fourier series will converge to , the cosine series will converge to , and the sine series will converge to .
  8. The corresponding eigenfunction for the boundary-value problem

    y″ + λy = 0, y′(0) = 0, y(π/2) = 0

    for λ = 25 is .

  9. The set {P2n (x)}, n = 0, 1, 2, … of Legendre polynomials of even degree is orthogonal with respect to the weight function p(x) = 1 on the interval [0, 1].
  10. The set {Pn (x)}, n = 0, 1, 2, … of Legendre polynomials is orthogonal with respect to the weight function p(x) = 1 on the interval [–1, 1]. Hence, for n > 0, (x) dx = .
  11. Without doing any work, explain why the cosine series of f(x) = cos2 x, 0 < x < π, is the finite series

    f(x) = + cos 2x.

    1. Show that the set

      is orthogonal on the interval [0, L].

    2. Find the norm of each function in part (a). Construct an orthonormal set.
  13. Expand f(x) = |x| – x, –1 < x < 1, in a Fourier series.
  14. Expand f(x) = 2x2 – 1, –1 < x < 1, in a Fourier series.
  15. Expand f(x) = e–x, 0 < x < 1, in a cosine series. In a sine series.
  16. In Problems 13, 14, and 15, sketch the periodic extension of f to which each series converges.
  17. Find the eigenvalues and eigenfunctions of the boundary-value problem

  18. Give an orthogonality relation for the eigenfunctions in Problem 17.
  19. Chebyshev’s differential equation

    (1 – x2)y″ – xy′ + n2y = 0

    has a polynomial solution y = Tn(x) for n = 0, 1, 2, …. Specify the weight function p(x) and the interval over which the set of Chebyshev polynomials {Tn(x)} is orthogonal. Give an orthogonality relation.

  20. Expand the periodic function shown in FIGURE 12.R.1 in an appropriate Fourier series.
    Six parts of a function are graphed on an x y plane. The first part is a solid horizontal line that starts from the left of the second quadrant and ends at the point (negative 4, 2). The second part is a dashed vertical line that starts from the point (negative 4, 2) and ends at the point (negative 4, 0). The third part is a solid line that starts from the point (negative 4, 0), goes to the point (negative 2, 2), then becomes horizontal, and ends at the point (0, 2). The fourth part is a solid line that starts from the point (0, 0), goes to the point (2, 2), then becomes horizontal, and ends at the point (4, 2). The fifth part is a dashed vertical line that starts from the point (4, 2) and ends at the point (4, 0). The sixth part is a solid line that starts from the point (4, 0), goes to the point (6, 2), then becomes horizontal, and exits the right of the first quadrant.

    FIGURE 12.R.1 Graph for Problem 20

  21. Expand

    f(x) =

    in a Fourier–Bessel series, using Bessel functions of order zero that satisfy the boundary condition J0(4α) = 0.

  22. Expand f(x) = x4, –1 < x < 1, in a Fourier–Legendre series.
  23. Suppose the function y = f(x) is defined on the interval (–∞, ∞).
    1. Verify the identity f(x) = fe(x) + fo(x), where

    2. Show that fe is an even function and fo an odd function.
  24. The function f(x) = ex is neither even nor odd. Use Problem 23 to write f as the sum of an even function and an odd function. Identify fe and fo.
  25. Suppose f is an integrable 2p-periodic function. Prove that for any real number a,