Year: 1997
Paper: 2
Question Number: 5
Course: LFM Stats And Pure
Section: Complex Numbers (L8th)
No solution available for this problem.
Difficulty Rating: 1600.0
Difficulty Comparisons: 0
Banger Rating: 1484.0
Banger Comparisons: 1
The complex numbers $w=u+\mathrm{i}v$ and $z=x+\mathrm{i}y$ are related by the
equation
$$z=
(\cos v+\mathrm{i}\sin v)\mathrm{e}^u.$$
Find all $w$ which correspond to $z=\mathrm{i\,e}$.
Find the loci in the $x$--$y$ plane corresponding to the lines $u=$ constant
in the $u$--$v$ plane. Find also the loci corresponding to the lines $v=$
constant. Illustrate your answers with clearly labelled sketches.
Identify two subsets $W_1$ and $W_2$ of the $u$--$v$ plane each of
which is in one-to-one correspondence with the first quadrant
$\{(x,\,y):\,x>0,\,y>0\}$ of the $x$--$y$ plane.
Identify also two subsets $W_3$ and $W_4$ each of which
is in one-to-one correspondence with the set $\{z\,:0<\,\vert z\vert\,<1\}$.
\noindent[{\bf NB} `one-to-one' means here that to each value of
$w$ there is only one corresponding value of $z$, and vice-versa.]