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Let $h(z)=\sqrt{z^2+1}$ (principal square root).

$a)$ Where is $h$ continuous?

$b)$ Find $h(A)$ where $A=\{z=x+iy\in\Bbb C: x\ge 0\ \text{and}\ |y|\le x\}$

$c)$ Find the image of the axes.

MY ATTEMPT:

$a)$ We have to $h(z)=g(f(z))$ where $g(z)=\sqrt{z}$ and $f(z)=z^2+1$. $g(z)$ is continuous on $D=\Bbb C-(-\infty,0]$, then $h(z)$ will be continuous on $D-B$, where $B=\{z\in D:f(z)\in (-\infty,0]\}$.
After doing some calculations, the set $B$ is the following, $B=\{z=x+iy\in\Bbb C:|y|\ge 0\ \text{and}\ x=0\}$.

Could you tell me if my exercise is right, and could you help me with the other two that I have no idea on how to do or how to even start.
TY

arnav_de
  • 709
James A.
  • 910
  • "Find the image of the axis." Which axis? You say $f(z)=z^3+1$. Shouldn't it be $z^2+1$? Also $|y|\geq 0$ is always true. – Gary May 09 '22 at 05:57
  • I made a mistake in the topography, I already corrected it. And by the axes I mean the real axis and the imaginary axis.@Gary – James A. May 09 '22 at 06:05
  • This may help: https://math.stackexchange.com/questions/988828/how-to-find-the-branch-points-and-cut – Gary May 09 '22 at 06:07

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