Verifying CR Equations for $z\cos z$

[Dustinsfl]

$z\cos z$

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?

 

[Chris L T521]

$z\cos z$

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 $\cos (x + yi) = \cos x\cosh y\,{\color{red}-}\,i\sin x\sinh y$.
So
$$
f(z) = z\cos z = x\cos x\cosh y\,{\color{red}+}\, y\sin x\sinh y + i({\color{red}-}\,x\sin x\sinh y + y\cos x\cosh y).
$$
Then
$$
u(x,y) = x\cos x\cosh y\,{\color{red}+}\, y\sin x\sinh y\quad\text{and}\quad
v(x,y) = y\cos x\cosh y\,{\color{red}-}\, x\sin x\sinh y,
$$

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?

You had a sign error. See all my changes in red.

 

[Dustinsfl]

You had a sign error. See all my changes in red.

Bad admin for missing the solved tag in the title for before you posted.

 

[Chris L T521]

Bad admin for missing the solved tag in the title for before you posted.

I clicked on the link from the home page, and it didn't show a solved tag. :P