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I'm asked to plot $|z+1-3i|\leq1$ and $\text{Im}(z) \geq3$, I've plotted both the inequalities, PS see the attachment. enter image description here Now, I'm unable to determine the difference between the greatest and least values of $\arg z$ for points lying in this region.

PS assist,

Also any resources that will be me get better to solve such questions will be of great help.

Thanks Arif

Teddy38
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Arif
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  • Geometrically, it should be clear that the least value of $\operatorname{arg}z$ will be that of the complex number $z=3i$ (far right of the half-disc), which is $\pi/2$, and that the greatest value will be that of the complex number $z=-2+3i$ (far left of the half-disc), which is... – Guest Dec 20 '16 at 08:20
  • answer given is 0.588 radians. Any resource that can help me get better at this topic will be really helpful. – Arif Dec 20 '16 at 08:29

1 Answers1

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By taking a look at your plot, it is clear that:

  • The least value of $\arg z$ is attained at $$ z_1:=3i $$ (far right of the half-disc). This value is clearly $\pi/2$ radians.
  • The greatest value of $\operatorname{arg}z$ is attained at $$z_2:=-2+3i$$ (far left of the half-disc). This value is clearly $\pi/2+\alpha$ radians, where $\alpha$ is the angle (in radians) at $(0,0)$ of the triangle with vertices $(0,0)$, $(-2,3)$ and $(0,3)$ in the complex plane. We have $\alpha=\arctan(2/3)$ by simple trigonometry.

Hence, the difference you seek is, in radians, \begin{align} \arg z_2-\arg z_1&=\left(\frac{\pi}{2}+\arctan\frac{2}{3}\right)-\frac{\pi}{2}\\ &=\arctan\frac{2}{3}\\ &\approx0.588 \end{align}

Guest
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  • need some resources(online/books) that will me get better at such concepts. – Arif Dec 21 '16 at 03:28
  • @Arif Maybe you could just take a look at the questions tagged (complex-numbers) on this site. Try to solve them and if you don't succeed at least you will have solutions to guide you. If you want to get better at this I think you should solve as many similar problems as you can. – Guest Dec 21 '16 at 03:35
  • @Arif If you let $z:=a+ib$ (cartesian form for $z$) then $$|z+1-3i|\leq1\iff\sqrt{(a+1)^2+(b-3)^2}\leq1$$ and $$\operatorname{Im} z\geq3\iff b\geq3.$$ Using these two equations, WolframAlpha can plot the half-disc. – Guest Jan 04 '17 at 05:33