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6.(15%) Using the theory of Residues, compute the inverse f(t), -∞＜t ＜∞, of the Fourier transform

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(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%)

(4.4) Now suppose a = and β =0. Find all possible values of γ such that the set ofeigenvalues of B is.(6%)

(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).

(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).

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(AT), the range space of AT. Compute the value of d.

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