# 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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