103年 - 103 國立中山大學_碩士班招生考試_電機系(甲、丁、戊、己組)：工程數學甲#110234

1. (15%) Evaluate the following integral

(2) (7%6) Compute the quantity A given below where the Fourier transform of f₁(t) is given in the following figure (a).

(b) (8%) Compute the quantity B given belowwhere the Fourier transform of f₂(t) is given in the following figure (b).

(a) (6%) Please continue the argument to derive the result rank(A) = rank(R).
(接下來前段是背景知識介紹,之後才是提問)Insthq given, instead of using the elementary row operations (i.e. the Gauss eliminations) to manipulate the equation, we may also apply the QR factorization to the equation to get QRx = b, which implies further Q^T QRx = Q^Tb. Since Q^TQ = _In, it gives Rx = Q^Tb. Thus, according to the result of (a), when all columns of A are linearly independent, the square matrix R is nonsingular and so the solution x =b is obtained. It seems that we may summarize the above argument as the following statement: 

Given, where all columns of A are assumed linearly independent, then solution to the equation Ax = 6 can always be computed from x =, where Q and R are matrices obtained from the QR factorization of A.

 However, the simple example shows that the summary is incorrect because, according to the summary, the solution is x ==0 and obviously it does not satisfy the original equation.

(a) (10%) Let . Suppose 4 0 and u ≡ 0. Find the solution of x₁ and x₂ for the initial conditions x₁(0) = .

(b) (4%)Suppose  Find the range of k: such that the solution of x₁ and x₂ converges to zero for any initial condition.

(c) (5%) SupposeFind the range of k such that the solution of at and ta exhibits oscillatory behavior for any nonzero initial condition.

(d) (6%) For the values , calculate the steady-state response of.

(c) (2%) Find, where C is the straight line going from the points (2,2,1) to the point (1,-1,2).

7. (5%) Evaluate the following integral