I have the following question:

Given that

$\displaystyle u(x,y) = x~cosx~coshy + y~sinx~sinhy$

Find a funvtion $\displaystyle v(x,y)$ such that

$\displaystyle w(z) = u(x,y) + iv(x,y) $

is an analytic function of $\displaystyle z$. Find $\displaystyle w(z)$ explicitly in terms of $\displaystyle z$.

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I know how to do these type of questions by using Cauchy-Riemann equations but I was wondering if there was a way of making this one simpler as it becomes very tedious. Is there an identity I can use to make the process much simpler?