I started this by considering the cases about $Arg (z)$

$$ Arg(z) = \begin{cases} \arctan(y/x), & \text{if $x>0$ } \\ \pi+\arctan(y/x), & \text{if $x<0,y\ge 0$}\\ -\pi+\arctan(y/x), & \text{if $x<0,y< 0$}\\ \pi/2, & \text{if $x=0,y>0$}\\ -\pi/2, & \text{if $x=0,y<0$}\\ \undefind, & \text{if$x=0,y=0$} \end{cases}$$ Now I am not getting the way to go further. So please Help me to do further things.

Thank you in Advance.

  • $\begingroup$ Do you understand that $\arg(z)$ is continuous in the plane less the negative real axis? $\endgroup$ – Mark Viola Sep 23 '15 at 4:58
  • $\begingroup$ @Dr.MV I just know that $arg(z)$ is discontinuous in negative Real axis. $\endgroup$ – Chiranjeev_Kumar Sep 23 '15 at 6:52
  • $\begingroup$ Yes. Say we look on the principal branch, then for $y=0^+$ and $x<0$, the argument of $z$ is $\pi$, while for $y=0^-$, it is $-\pi$. The argument function is continous elsewhere. Now, upon squaring what happens? $\endgroup$ – Mark Viola Sep 23 '15 at 13:36
  • $\begingroup$ Dr. MV this become $\pi^2$ in both the cases, so it will become continuous in negative Real axis, now to finish the problem we have to show $\arg(z)$ is continuous elsewhere, can you tell me some initial lines to show this? $\endgroup$ – Chiranjeev_Kumar Sep 23 '15 at 14:53
  • $\begingroup$ @Chiranjeev,have a look at my answer. $\endgroup$ – Koro Sep 23 '15 at 23:49

Remember product of two continuous functions is continuous but product of two discontinuous function may be continuous.With this idea ,I proceed.
Let $f(z)=Arg(z)Arg(z)$ [$z\in C-\{0\}$]
$Arg(z)$ is continuous in the set: $C-\{z:Re(z)<0,Im(z)=0\}$....(A)
So, $f$ is continuous in (A).
Only problem is in the points of the set $\{z:Re(z)<0,Im(z)=0\}$....(B)
So let's check continuity only for set (B)
At negative $x- axis$,$\tan^{-1}(y/x)=0$;
$\lim_{y\to0{+}}Arg(z)=\pi \implies \lim_{y\to0^{+}}f(x+iy)=(Arg(z))^2=\pi^2$
$\lim_{y\to0{-}}Arg(z)=-\pi \implies \lim_{y\to0^{-}}f(x+iy)=(Arg(z))^2=(-\pi)^2=\pi^2$
$Arg(x+i0)=\pi \implies f(x)=\pi^2\implies f(x)=\lim_{y\to 0}f(x+iy)$
Hence proved!


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