Past exam of the mathematics course of the University of Cambridge / 2019 / ia / Paper 1 / 1C (Vectors and Matrices)

The powers of the imaginary unit have period four. HencewhileThus $x+iy=1-i$, so

Since $(1+i)/(1-i)=i$ and $(1+i{)}^2=2i$,

Use the principal complex logarithm, for which $\mathrm{Log}i=i\pi /2$, and setThen the definition of complex exponentiation givesand thereforeOther choices of branch of the complex logarithm produce further valid answers.

Write $z=r{e}^{i\theta }$ with $r>0$. On a fixed branch of the complex logarithm on which $\log z=\log r+i\theta$ and $\log \bar{z}=\log r-i\theta$, the equation becomesIts real and imaginary parts give the same condition $\log r=\theta$, orThis is a logarithmic spiral. For the principal complex logarithm, it is the portion parametrized by $-\pi <\theta <\pi$ that avoids the branch cut.