Chapter 2.1 PG 80

[Min Gyu Woo]

There's this little portion of text before the defining Laplace's Equation that I am confused about.

It reads:

"A word of warning is merited here: Not every function $u(x,y)$ is the real part of an analytic function".

If we always define $f = u +iv$, doesn't this guarentee that $u(x,y)=\text{Re}(f)$?

[Meiqun Lu]

I think what the textbook is trying to say is that even though you could find a function f(z) whose real part can be written as u(x,y), but f(z) does not have to be analytic.