Consider two complex numbers $z=(1+i)^p$ and $w=(1-i)^q$ where $p$ and $q$ are positive inte

Question

Consider two complex numbers $z=(1+i)^p$ and $w=(1-i)^q$ where $p$ and $q$ are positive integers. Express each of $z$ and $w$ in polar form, giving your answers in terms of $p$ and $q$. Find the smallest values of $p$ and $q$ such that $z=w$.

Accepted Answer

(a) $z=2^{p/2}(\cos\alpha_p+i\sin\alpha_p),\ \alpha_p\equiv\frac{p\pi}{4}\pmod{2\pi};\quad w=2^{q/2}(\cos\beta_q+i\sin\beta_q),\ \beta_q\equiv-\frac{q\pi}{4}\pmod{2\pi},\ \alpha_p,\beta_q\in(-\pi,\pi]$ (b) $p=q=4$

Equating powers of conjugate complex numbers

Equating powers of conjugate complex numbers

Solution:

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