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Exercises: Complex Integration

Module by: Steven Cox

Summary: (Blank Abstract)

  1. Let us confirm the representation of this Cauchy's Theorem equation in the matrix case. More precisely, if Φz≡zI-B-1 Φ z z I B is the transfer function associated with BB then this Cauchy's Theorem equation states that Φz=∑j=1h∑k=1 d j Φ j , k z- λ j k Φ z j 1 h k 1 d j Φ j , k z λ j k where
    Φ j , k =12πⅈ∫ C j Φzz- λ j k-1dz Φ j , k 1 2 z C j Φ z z λ j k 1 (1)
    Compute the Φ j , k Φ j , k per Equation 1 for the BB in this equation from the discussion of Complex Differentiation. Confirm that they agree with those appearing in this equation from the Complex Differentiation discussion.
  2. Use this inverse Laplace Transform equation to compute the inverse Laplace transform of 1s2+2s+2 1 s 2 2 s 2 .
  3. Use the result of the previous exercise to solve, via the Laplace transform, the differential equation ddtxt+xt=ⅇ-tsint ,    x0=0 t x t x t t t ,    x 0 0 Hint: Take the Laplace transform of each side.
  4. Explain how one gets from r 1 r 1 and p 1 p 1 to x 1 t x 1 t .
  5. Compute, as in fib4.m, the residues of ℒ x 2 s ℒ x 2 s and ℒ x 3 s ℒ x 3 s and confirm that they give rise to the x 2 t x 2 t and x 3 t x 3 t you derived in the discussion of Chapter 1.

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