12.10. EXERCISES 299

10. Let f (x) be the odd 2π periodic extension of f (x) = X[0,π] (x) . Explain why itsFourier series is of the form ∑

∞k=1 an sin(nx). Doing minimal computations, why can

you say that nan cannot converge to 0 as n→∞. Hint: See Problem 55 on Page 226.

11. Find the Fourier series expansion for the above function and use it to find interesting

summation formulas, for example ∑∞n=1

(−1)n−1

2n−1 = π

4 .

12. This and the remaining problems will require a beginning course in linear alge-bra. All that is needed is in any of my linear algebra books. Fill in the details.A fundamental matrix for the n× n matrix A is an n× n matrix Φ(t) having func-tions as entries such that Φ′ (t) = AΦ(t) ,Φ(0) = I. That is, you have a differen-tial equation involving a dependent variable which is a matrix along with an initialcondition Φ(0) = I, the identity matrix. Using the properties of the Laplace trans-form, we can take the Laplace transform of both sides and get sF (s)− I = AF (s)so (sI−A)F (s) = I and so the Laplace transform of Φ(t) denoted here as F (s) isF (s) = (sI−A)−1. Now from linear algebra, there is a formula for this inverse validfor all s large enough which comes as 1

det(sI−A) (cofactor matrix )T . If |s| is largeenough, the inverse does indeed exist because there are only finitely many eigenval-ues. Now each term in (sI−A)−1 is a rational function for which the degree of thenumerator is at least one more than the degree of the denominator. Thus it satisfiesthe necessary conditions for the Bromwich integral and thus there exists a uniquesuch Φ(t).

13. Next show that AΦ(t) = Φ(t)A. Fill in the details. To do this, let Ψ(t) ≡ AΦ(t)−Φ(t)A. Thus Ψ(0) = 0. Also

Ψ′ (t) = AΦ

′ (t)−Φ′ (t)A = A2

Φ(t)−AΦ(t)A = A(AΦ(t)−Φ(t)A) = AΨ(t)

In short, Ψ′ (t) = AΨ(t) ,Ψ(0) = 0. Use the Laplace transform method to show thatΨ(t) = 0. Also show that Φ(−t)Φ(t) = I. To do this last one, define the functionΨ(t)≡Φ(−t)Φ(t) ,

Ψ′ (t) = −Φ

′ (−t)Φ(t)+Φ(−t)Φ′ (t) =−AΦ(−t)Φ(t)+Φ(−t)AΦ(t)

= −Φ(−t)AΦ(t)+Φ(−t)AΦ(t) = 0

thus Ψ(0) = I and Ψ′ (0) = 0 so each entry of Ψ(t) is a constant from the mean valuetheorem. Thus Ψ(t) = I for all t. Also show that Φ(t)Φ(s) = Φ(t + s) using similarconsiderations using Laplace transforms.

14. All linear equations in an undergraduate differential equations course, which includesthe vast majority of what is discussed in these courses, can be written as x′ (t) =Ax(t)+ f(t) ,x(0) = x0. Fill in details. x′−Ax = f(t) . Multiply by Φ(−t) and youget d

dt (Φ(−t)x(t)) = Φ(−t) f(t). Note that the left side equals the following by theproduct and chain rule.

−Φ′ (−t)x+Φ(−t)x′ = Φ(−t)x′−AΦ(−t)x = Φ(−t)

(x′−Ax

)Now it follows from considering the individual entries of the matrices and vectors