6.4: Exercises- Complex Numbers, Vectors, and Functions
Express \(|e^z|\) in terms of \(x\) and/or \(y\).
Confirm that \(e^{ln(z)} = z\) and \(ln(e^z) = z\)
Find the real and imaginary parts of \(\cos (z)\) and \(\sin (z)\)
Show that \(\cos^{2}(z)+\sin^{2}(z) = 1\)
With \(z^{w} \equiv e^{w ln(z)}\) for complex \(z\) and \(w\) compute \(\sqrt{i}\)
Verify that \(\cos (z)\) and \(\sin (z)\) satisfy the Cauchy-Riemann equations and use the proposition to evaluate their derivatives.
Submit a Matlab diary documenting your use of residue in the partial fraction expansion of the transfer function of
\[B = \begin{pmatrix} {2}&{0}&{0}\\ {-1}&{4}&{0}\\ {0}&{-1}&{2} \end{pmatrix} \nonumber\]