Question

et {z ∈ ℂ: im $\frac{1}{z}=\frac{1}{4}$}

Explanation:

Answer:

A circle in the complex - plane. Let \(z=x + iy\), then \(\frac{1}{z}=\frac{1}{x + iy}=\frac{x-iy}{x^{2}+y^{2}}\). The imaginary part of \(\frac{1}{z}\) is \(-\frac{y}{x^{2}+y^{2}}\). Setting \(-\frac{y}{x^{2}+y^{2}}=\frac{1}{4}\), we get \(x^{2}+y^{2}+4y = 0\), which can be rewritten as \(x^{2}+(y + 2)^{2}=4\) (a circle with center \((0,-2)\) and radius \(2\)).

Discipline: Mathematics
Sub - field: Complex Analysis (a part of Algebra in a broader sense)