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# 6.4: Exercises- Complex Numbers, Vectors, and Functions

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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

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