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101年 - 101 國立中山大學_碩士班招生考試_電機系(乙組)：工程數學乙#110508
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題組內容
5.
(b)(8%) Does the derivative of f(z) at z =0 ,i.e., f'(0), exist? Give your reason (no credit will be given if there is no explanation).
其他申論題
(4.2) Let's define an inner product for P2 by ＜p(x),q(x),for arbitrary and γ≠I an undecided parameter. Find the orthonormal basis, denoted by F := [f1, f2], of P2, generated from basis E given above to satisfy the subspace equality constraints Span(f1)=Span(I) and Span(f1,f2)=Span(1,x). (8%)
#473297
(4.3) Let B denote the matrix representation of transformation L with respect to the ordered bases F computed in (4.2) and F'=for P2 and R2 , respectively. Find the matrix B. (6%)
#473298
(4.4) Now suppose a = and β =0. Find all possible values of γ such that the set ofeigenvalues of B is.(6%)
#473299
(a)(7%)Let f(2) be a complex function defined by where denotes the complex conjugate of the complex variable z. Does the function f(z) satisfy the Cauchy-Riemann equations? Give your reason (no credit will be given if there is no explanation).
#473300
6.(15%) Using the theory of Residues, compute the inverse f(t), -∞＜t ＜∞, of the Fourier transform
#473302
1. (7%) Find the Laurent series representation of a function with center at a = j in the domain 1 ＜|z -j |＜ 2, j = .
#473303
2. (8%) Evaluate the following integral:where C denotes a counterclockwise simple closed contour |z| = 3.
#473304
3. (15%) Compute the Fourier transform of a signum function f(t) defined as Each calculation step is required for obtaining the credit.
#473305
(a) (4%) Suppose a is not a positive integer. Find real β and γsuch that{a1, a2, a3} is a linearly dependent set.
#473307
(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.
#473308
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