Complex Analysis: What is the magnitude of e^(2-3i)?

[lastlydreaming]

What would the magnitude of e^(2-3i) be? Since it is in the 3rd quadrant, what is its argument?

 

[pka]

\(\displaystyle \L
\begin{array}{rcl}
e^{a + bi} & = & e^a (\cos (b) + i\sin (b)) \\
\left| {e^{a + bi} } \right| & = & \left| {e^a (\cos (b) + i\sin (b))} \right| \\
& = & \left| {e^a } \right|\left| {\cos (b) + i\sin (b)} \right| \\
& = & e^a \\
\end{array}\)

As a general rule
\(\displaystyle \L\left| {e^z } \right| = e^{Re(z)}\)

 

[lastlydreaming]

So with respect ot all of its arguments, does it become?


e^ 2 [(cos 3+ 2pin) + i sin(3+ 2pi n)] where n = 0, +/- , +/- 2.... ?

Thanks.