6.4: Exercises- Complex Numbers, Vectors, and Functions

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Sep 17, 2022
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- 6.3: Complex Differentiation
- 7: Complex Analysis II

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Exercise $\PageIndex{1}$

Express $|e^z|$ in terms of $x$ and/or $y$ .

Exercise $\PageIndex{2}$

Confirm that $e^{ln(z)} = z$ and $ln(e^z) = z$

Exercise $\PageIndex{3}$

Find the real and imaginary parts of $\cos (z)$ and $\sin (z)$

Exercise $\PageIndex{4}$

Show that $\cos^{2}(z)+\sin^{2}(z) = 1$

Exercise $\PageIndex{5}$

With $z^{w} \equiv e^{w ln(z)}$ for complex $z$ and $w$ compute $\sqrt{i}$

Exercise $\PageIndex{6}$

Verify that $\cos (z)$ and $\sin (z)$ satisfy the Cauchy-Riemann equations and use the proposition to evaluate their derivatives.

Exercise $\PageIndex{7}$

Submit a Matlab diary documenting your use of residue in the partial fraction expansion of the transfer function of

$B = \begin{pmatrix} {2}&{0}&{0}\\ {-1}&{4}&{0}\\ {0}&{-1}&{2} \end{pmatrix} \nonumber$

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Exercise 6.4.1\PageIndex{1}

Exercise 6.4.2\PageIndex{2}

Exercise 6.4.3\PageIndex{3}

Exercise 6.4.4\PageIndex{4}

Exercise 6.4.5\PageIndex{5}

Exercise 6.4.6\PageIndex{6}

Exercise 6.4.7\PageIndex{7}

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