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2.(8%) Evaluate the following integral:
where C denotes a counterclockwise simple closed contour |z| =3.

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1. (7%) Find the Laurent series representation of a function with center at z = j in the domain 1 ＜|z -j|＜ 2, j=.

3. (15%) Compute the Fourier transformdt of a signum function f(t) detined as Each calculation step is required for obtaining the credit.

(a) (4%) If, where N denotes the set of all positive integers, then find positive integers β and such that {a1, a2, a3} is a linearly dependent set.

(b) (4%) If a e N, find real β and γ such that{a1, a2, a3} is a linearly dependent set.

(C) (6%) 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 satisfy ||x||1 + 2||x||∞ + k||x||2 = 0.

(d) (6%) Nowlet a = 2,β = -1, r = -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.

(a) (3%+4%) Compute Ilxll and the angle θ, taken value in [0, π/2, between I andx.

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