88 academic year Show all works 1.
(a) [10%] Show that the integral Z ∞
0
sin x
x dx = π 2. (b) [5%] Evaluate the integral
Z ∞
0
sin x cos xy
x dx.
2. [10%] Let (x, y, z) and (ρ, θ, φ) be the rectangular coordinates and the spherical coordinates, respectively, for R3. Compute ∂(x, y, z)
∂(ρ, θ, φ).
3. [15%] Explain the identity 1
1 + x2 = Σ∞n=0(−1)nx2n
why the left side is defined on R1 while the right side is only defined on the interval
−1 < x < 1?
4. [15%] Suppose that the series Σ∞n=1an converges and for each an ≥ 0. Discuss the convergence of the series
Σ∞n=1√
ann−p, p ∈ R,
on which interval the series converges and on which interval the series may or may not diverge. If it is in the latter case, please give examples.
5. Let A =
3 0 0 0
a 3 0 0
b c −2 0
d e f 5
, where a, b, c, d, e, f ∈ C.
(a) [4%] Find all possible characteristic and minimal polynomials for A.
(b) [8%] Find all possible Jordan forms of A.
(c) [3%] Find all possible diagonal matrix that are similar to A.
6. Let A ∈ M (n, C), set of all n × n matrices with complex entries, such that A∗ =
−A, and let B = eA. (Recall that the joint matrix, A∗, of the matrix A is given by (Ax, y) = (x, A∗y); B is unitary if BB∗ = I.) Show that
(a) [5%] det B = etrA; (b) [5%] B∗ = e−A;
(c) [5%] B is unitary.
7. [15%] Evaluate the area enclosed by the curve 13x2+ 10xy + 13y2− 72 = 0. ( Hint:
use Green’s Theorem. )
