102年 - 102 國立中山大學_碩士班招生考試_電機系(甲、丙、丁、戊、己組)：工程數學甲#110510

1. (7%) Find the Laurent series representation of a function
  
with center at a = 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 transform of a signum function f(t) defined as  Each calculation step is required for obtaining the credit. 

(a) (4%) Suppose a is not a positive integer. Find real β and γsuch that{a₁, a₂, a₃} is a linearly dependent set.

(b) (5%) 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 satisty ||x|| ₁ + 2||x||∞ + k||x||₂ = 0.

(c) (6%) Now let a = 2, β = -1, γ= -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) (4%) Compute the angle θ, taken value in [0, π /2), between 1 and x.

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

(b) (10%) Let the initial conditions be equal to zero. For the values , = -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).

(a) (3% + 2%) Find the flux of = (1 + y²)j out of R through the two sides C₁ (the horizontal segment) and C₂ (the vertical segment).

(b) (10%) Find the flux of = (1 + y²)j out of the third side C₃.

(a) (5%) Suppose all eigenvalues of A have negative real parts. Show that X =dτ a solution to the Lyapunov equation.