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The real part function is $$u(x,y) = \sin(x^2-y^2)\cosh(2xy)$$ and my task is to find all functions $f(z) = u(x,y) + iv(x,y)$ such that $f(z)$ is analytic over all $\mathbb{C}$. Normally what I would do is to write $u$ in terms of the real-valued part of certain functions or try to solve Cauchy-Riemann equations. But here I am lost. Any ideas?

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Hint

$$u(x,y)+i\sinh(2xy)\cos(x^2-y^2)=\sin{z^2}$$

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