We can write .
Now that is zero only if .
Now I may have misread your question. But does this help?
I must find the domain of analyticity of the function .
Since is analytic over the whole complex plane, f is analytic on the whole complex plane except when .
This is equivalent to say that if , with and . I've searched in google the zeros of the complex function cosh and I didn't find anything (not even in wikipedia) that showed them. Of course likely I've missed something but still. I'm out of ideas about how to go further in the expression . What does this represent... Any idea?