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Let $z = x + yi$.

Then $f(z) = (x + yi)\cos (x + yi)$.

By the addition rule for cosine and the identities $\cos yi = \cosh y$ and $-i\sin yi = \sinh y\Leftrightarrow \sin yi = i\sinh y$, we have that $\cos (x + yi) = \cos x\cosh y + i\sin x\sinh y$.

So

$$

f(z) = z\cos z = x\cos x\cosh y - y\sin x\sinh y + i(x\sin x\sinh y + y\cos x\cosh y).

$$

Then

$$

u(x,y) = x\cos x\cosh y - y\sin x\sinh y\quad\text{and}\quad

v(x,y) = y\cos x\cosh y + x\sin x\sinh y,

$$

I am trying to verify the CR equations but there is a negative sign difference. There has to be an error in my algebra but I can't find it. What is wrong with the above?