Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2002 / Paper 1 / 11

Complex conjugation reverses the argument, reciprocation also inverts the modulus, and multiplication by $-1$ adds $\pi$ to the argument. With all arguments chosen in $[0,2\pi )$ and $0<\varphi <\pi /2$,Thus $z$ is in the first quadrant, $z^{\ast }$ and $z^{-1}$ lie on the same fourth-quadrant ray at radii two and one-half, and $-z$ is diametrically opposite $z$.

Euler's formula and its version with $-\theta$ giveSquaring $e^{i\theta }=\cos \theta +i\sin \theta$ and comparing real and imaginary parts with $e^{2i\theta }$ yields the double-angle formulas

Factor $e^{i\varphi /2}$ from the numerator and denominator:For $0<\varphi <\pi$, the tangent ranges over all positive real values. Thus $\boxed{\mathrm{Re}w=0,\mathrm{Im}w>0}$. Using the principal complex logarithm,The real part ranges over the entire real line, while the imaginary part is fixed: the locus is the horizontal line $\mathrm{Im}\log w=\pi /2$. Any other fixed logarithm branch gives the parallel line $\mathrm{Im}\log w=\pi /2+2\pi k$ for one integer $k$; allowing every branch gives the union of those lines.