If the complex-value function  f = u + iv  is analytic, then the Cauchy-Riemann equations are held

     =       (1)

and

     = -     (2)


(as the reference, see this Wikipedia article https://en.wikipedia.org/wiki/Cauchy%E2%80%93Riemann_equations ).


    I am very sorry, but in these equations the symbol "d" must be a Greek letter ("d rounded"), not a "d" Latin,
    but in this editor I can not reproduce it, unfortunately.  So, concider this "d" as "d rounded" in all my 
    formulas in this post).



From equation (1), after differentiating  , we have

     = cos(x)*cosh(y)      (3)


From equation (2), after differentiating  , we have

     = -sin(x)*sinh(y)     (4)


So, we need to find function v  as antiderivative from equations  (3) and (4).


It is easy to do:  the function  v(x,y) = cos(x)*sinh(y)  satisfies both equations  (3) and (4).


THEREFORE,  the answer to the problem's question is  v = cos(x)*sinh(y).