AMAT 415 — Week 2 Problems

(You will be asked to submit a subset of these questions as part of Assignment 1.)

P7) Find \sqrt{7+24i} and \sqrt{24+7i}.  (Try to be clever and avoid duplication of effort.)

P8) For a complex number z=x+iy, define the matrix m_z=\begin{pmatrix}x&-y\\y&x\end{pmatrix}.  Show that m_{z+w}=m_z+m_w, m_{zw}=m_zm_w, and m_z^{-1}=m_{1/z}.

P9) Express e^{\log(2)+7\pi i/4} and i^i in the form x+iy.

P10) For which z=x+iy does \overline{e^z}=e^{\bar{z}} hold?  How about \overline{e^{iz}}=e^{i\bar{z}}?

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