7.4: Exercises- Complex Integration

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Sep 17, 2022
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- 7.3: The Inverse Laplace Transform- Complex Integration
- 8: The Eigenvalue Problem

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Exercise $\PageIndex{1}$

Let us confirm the representation of this Cauchy's Theorem equation in the matrix case. More precisely, if $\Phi(z) \equiv (zI-B)^{-1}$ is the transfer function associated with $B$ then this Cauchy's Theorem equation states that

$\Phi(z)= \sum_{j = 1}^{h} \sum_{k = 1}^{d_{j}} \frac{\Phi_{j,k}}{(z-\lambda_{j})^{k}} \nonumber$

where

$\Phi_{j,k} = \frac{1}{2 \pi i} \int \frac{\Phi(z)}{(z-\lambda_{j})^{k-1}} dz \nonumber$

Compute the $\Phi_{j,k}$ per Equation for the $B$ in this equation from the discussion of Complex Differentiation. Confirm that they agree with those appearing in this equation from the Complex Differentiation discussion.

Exercise $\PageIndex{2}$

Use the inverse Laplace Transform equation to compute the inverse Laplace transform of $\frac{1}{s^2+2s+2}$ .

Exercise $\PageIndex{3}$

Use the result of the previous exercise to solve, via the Laplace transform, the differential equation

$\begin{array}{cc} {\frac{d}{dt} (x)(t)+x(t) = e^{-t \sin(t)},}&{x(0) = 0} \end{array} \nonumber$

Hint: Take the Laplace transform of each side.

Exercise $\PageIndex{4}$

Explain how one gets from $r_{1}$ and $p_{1}$ to $x_{1}(t)$ .

Exercise $\PageIndex{5}$

Compute, as in fib4.m, the residues of $\mathscr{L}(x_{2}(s))$ and $\mathscr{L}(x_{3}(s))$ and confirm that they give rise to the $x_{2}(t)$ and $x_{3}(t)$ you derived in the discussion of Chapter 1.1.

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Exercise 7.4.1\PageIndex{1}

Exercise 7.4.2\PageIndex{2}

Exercise 7.4.3\PageIndex{3}

Exercise 7.4.4\PageIndex{4}

Exercise 7.4.5\PageIndex{5}

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Exercise $\PageIndex{1}$

Exercise $\PageIndex{2}$

Exercise $\PageIndex{3}$

Exercise $\PageIndex{4}$

Exercise $\PageIndex{5}$