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 !