2. (20 pts) evaluate (a) ∫|z|=1 √9 - z 2dz; (b) ∫|z|=2 (z + 1) -2e zdz; (c) ∫|z-2i|=2 (z - i) -3arcsin zdz; (d) ∫|z|=2 z 3 (z - 1) -4dz; (e) let γ = [-1,1 + i] + β + [-1 + i, 1] where β(t) = i + e it for 0 ≤ t ≤ 3π. compute ∫γ (z 2 + 1) -1dz
Ex. 1 - Consider the two-state DTMC with the transition matrix P = [ 1 - a a b 1 - b ] a, b ∈ [0,1]. Discuss the reducibility, periodicity and limiting/stationary/occupancy distributions to classify the DTMC for different a, b.
Ex. 2 - Prove the transition matrix used by Metropolis [pij] satisfies the detailed balance condition with πi.
Ex. 3 - Define the matrix P˜ whose (i,j) entry is πi 1/2 pijπj -1/2 , i.e., P˜: = D 1/2PD -1/2 , where D ≜ diag{π1, π2, … , πN} is the diagonal matrix with diagonal entry {πi }. Verify that the detailed balance condition is equivalent to the symmetry of P˜ : P˜ ⊤ = P˜.