Day: February 14, 2023

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202302140910 Exercises 2.1

What do the following equations represent geometrically? Give sketches.

i. |z+2|=6,
ii. |z-3\,\mathrm{i}|=|z+\mathrm{i}|,
iii. |\mathrm{i}z-1|=|\mathrm{i}z+1|,
iv. |z-\omega |=|z-1| (\omega=e^{2\pi\,\mathrm{i}/3}).

Extracted from H. A. Priestley. (2003). Introduction to Complex Analysis.

Roughwork.

i.

\begin{aligned} |z+2|&=6 \\ |(x+\mathrm{i}y)+2| & = 6 \\ |(x+2)+\mathrm{i}y| & = 6 \\ \sqrt{(x+2)^2+y^2} & = 6 \\ (x+2)^2 + y^2 & = 6^2 \\ \end{aligned}

i.e., a circle centred at (-2,0) with 6 units in radius.

ii.

\begin{aligned} |z-3\,\mathrm{i}| & = |z+\mathrm{i}| \\ |(x+\mathrm{i}y)-3\,\mathrm{i}| & = |(x+\mathrm{i}y)+\mathrm{i}| \\ |x+\mathrm{i}(y-3)| & = |x+\mathrm{i}(y+1)| \\ \sqrt{x^2+(y-3)^2} & = \sqrt{x^2+(y+1)^2} \\ x^2+(y-3)^2 & = x^2 + (y+1)^2 \\ y^2-6y+9 & = y^2+2y+1 \\ y & = 1 \\ \end{aligned}

i.e., a horizontal line with y-intercept 1 unit.

iii.

\begin{aligned} |\mathrm{i}z-1| & = |\mathrm{i}z+1| \\ |\mathrm{i}(x+\mathrm{i}y)-1| & = |\mathrm{i}(x+\mathrm{i}y)+1| \\ |(-y+\mathrm{i}x)-1| & = |(-y+\mathrm{i}x+1|\\ |(-y-1)+\mathrm{i}x| & = |(-y+1)+\mathrm{i}x| \\ \sqrt{(-y-1)^2+x^2} & = \sqrt{(-y+1)^2+x^2} \\ y^2+2y+1 & = y^2-2y+1 \\ y & = 0 \\ \end{aligned}

i.e., the x-axis.

iv.

\begin{aligned} |z-\omega | & = |z-1| \\ |(x+\mathrm{i}y)-\mathrm{cis}(2\pi/3)|& = |(x+\mathrm{i}y)-1| \\ \bigg|\bigg(x+\frac{1}{2}\bigg)+\mathrm{i}\bigg(y-\frac{\sqrt{3}}{2}\bigg)\bigg| & = |(x-1)+\mathrm{i}y| \\ \end{aligned}

Not to be completed.