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Mathematics LibreTexts

6.4: Exercises- Complex Numbers, Vectors, and Functions

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    21836

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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

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