102年 - 102 國立中山大學_碩士班招生考試_電機系(乙組)：工程數學乙#110521

1. (7%) Find the Laurent series representation of a function
  with center at z = j in the domain 1 ＜|z -j|＜ 2, j=.

2.(8%) Evaluate the following integral:
where C denotes a counterclockwise simple closed contour |z| =3. 

3. (15%) Compute the Fourier transformdt of a signum function f(t) detined as Each calculation step is required for obtaining the credit.

(b) (4%) If a e N, find real β and γ such that{a₁, a₂, a₃} is a linearly dependent set.

(C) (6%) Now let a = 2,β = -1, γ = -5, and let x be a nonzero vector in the null space N(A) of A. Find the value of k to satisfy ||x||₁ + 2||x||∞ + k||x||2 = 0.

(d) (6%) Nowlet a = 2,β = -1, r = -5, and let d denote the distance between vector [1 4 0]^T and R(A^T), the range space of A^T. Compute the value of d. 

(a) (3%+4%) Compute Ilxll and the angle θ, taken value in [0, π/2, between I andx.

(c) (6%) Find the vector p(x) in S that is closest to on [0, 1].

(d) (6%) Let q(x) be the vector in S^⊥ that is closest to   . Compute||q(x)||².

(a) (5%) Suppose u  O and the equations are driven by non-zero initial conditions. Determine the conditions on the coefficients such that =0.

(b) (10%) Let the initial conditions be equal to zero. For the values1 = 0, = -1, and calculate the response y(t) = 2x₁(t) - x₂(t). Determine at what time the peak value of y occurs.

(c) (5%) For the values = -2, and u(t) = sin(t), calculate the steady-state response of y(t) =x1(t) - x2(t).

(b) (2%) Suppose the Laplace transform of a function y(t) is equal to Y(s). Then the Laplace transform of y(t - a) is equal to .