TITLE: $f=u+iv$ holomorphic, $xu+yv = (x^2+y^2)e^x \cos y$, what is $f$?
QUESTION [5 upvotes]: $f(z)=u(x,y)+iv(x,y)$ holomorphic, $xu+yv = (x^2+y^2)e^x \cos y$, what is $f$?
I tried to interprete $xu+yv$ as some part of a new function, for example, as the real part of $\overline{z}f$，but this function is no more holomorphic, so I don't know how to continue. (Maybe $\dfrac{\partial}{\partial\overline{z}}（\overline{z}f)=1$? But how to make use of it?)

REPLY [6 votes]: What you need to notice is that
$$ \frac{f(z)}{z} = \frac{u+iv}{x+iy} = \frac{(u+iv)(x-iy)}{x^2+y^2}, $$
of which the real part is
$$ \frac{xu+yv}{x^2+y^2}, $$
which is equal to $e^x \cos{y}$ according to the problem. Now find the harmonic conjugate of $e^x \cos{y}$, which is well-known to be $e^x \sin{y}$ for obvious Euler's formula-related reasons, and rewrite the equation to include the imaginary part:
$$ \frac{f(z)}{z} = e^x(\cos{y}+i\sin{y}) = e^{x+iy}=e^z. $$