# Analyticity of this function $\sqrt{\coth^2(a\ z) + \coth^2(b\ z) - c}$

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 !

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