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I want to determine the domain of analyticity of this function: $$f(z) =\sqrt{\coth^2(a\ z) + \coth^2(b\ z) - c}$$ And $$c \in ]0,1]$$

Where $$a,b \in \mathbb{Z} - \mathbb{Z}^+$$ and $a , b$ finite say $a,b \in [-1000 , 0]$


Suppose instead i took that $$f(z)=\sqrt{\coth^2(- \ z) + \coth^2(-3\ z) - 1}$$

Letting $$w={\coth^2(- \ z) + \coth^2(-3\ z) - 1}$$

And finding when $w=0$ is that engough to find the branch points .

Is that enugh or we must see how is $Arg({\coth^2(- \ z) + \coth^2(-3\ z) - 1})$ behave ?

If so , How can i see that ?

Ok , I Don't see where is the problem ?

If for example we have $$f(z)=\sqrt{\coth(z)}=\sqrt{\frac{\cosh(z)}{\sinh(z)}}=\sqrt{1+\frac{2}{e^{2z}-1}}$$

We get Roots $$ z= \frac{1}{2} i(2\pi n + \pi) \ \ , z \in \mathbb{Z}$$

Does that mean that we have infinite branch points ? including 0

So Where is the $\sqrt{\coth(z)} $ is Analytic ?

Than you !

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    $\begingroup$ Is Coth the hyperbolic cotangent? (maybe dumb question...) $\endgroup$ – coffeemath Mar 21 '19 at 8:00
  • $\begingroup$ Yes it is the hyperbolic cotangent . $\endgroup$ – Mahmoud Hassan Mar 21 '19 at 8:11
  • $\begingroup$ Square roots of negatives are OK in the complex numbers, provided one takes care about branches... $\endgroup$ – coffeemath Mar 21 '19 at 9:10
  • $\begingroup$ So , Is this function analytic in the first Quadrant ? $\endgroup$ – Mahmoud Hassan Mar 21 '19 at 9:19
  • $\begingroup$ I don't know... what branch of square root chosen would make a difference I guess, but I'm no expert on that. $\endgroup$ – coffeemath Mar 21 '19 at 9:39

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