非選
VIP與詳解卡管理
= (1 + y2)j out of the third side C3.
dτ
a solution to the Lyapunov equation.
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-阿摩線上測驗
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
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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.
4.(15%)下面的問題共有三個子題,只要寫出每個子題的答案即可(不需寫出計算過程),例如:
(a) β=1,γ=2。 Let a1 = [1 0 1 α]T,a2=[I β 2 2]T, and a3 = [-2 3 γ -4]T be three vectors in R4,
where a, β, and γ are three real parameters, and denote A := [a1 a2 a3] 
.
(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(AT), the range space of AT. Compute the value of d.
5.(10%)下面的問題共有二個子題,只要寫出每個子題的答案即可(不需寫出計算過程),例如:
(a)θ=30或θ=π/6。
Consider the inner product space C[0, 1] with (f, g) :=
f(x)g(x)dc and the nomm 
.
Denote S := span (1, x} as a subspace of C[0, 1].
(a) (4%) Compute the angle θ, taken value in [0, π /2), between 1 and x.
6. (20%) Consider the following system of differential equations:
where
are constant coefficients.
(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).
7. (15%) Consider the region R enclosed by the x-axis, x = I and y = x3, as illustrated below
(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).