Hey I'm new to complex analysis and I'm a bit lost with Holomorphic functions and what to do for questions like these:
Let $$ u(x, y) = x^2 + y^2 $$ Is $u$ the real part of a holomorphic function $f$? If so, find $f$. If not, explain why not
I just made a basic function but what does it mean find $x$ and what would a function that's not holomorphic look like? Thanks
EDIT: $u(x,y)$ is the real part, since $f$ is a holomorphic function, this means that it satisfies the Cauchy-Riemann equations $$ \frac {\partial u} {\partial x} = \frac {\partial v} {\partial y} $$ $$ \frac {\partial u} {\partial y} = - \frac {\partial v} {\partial x} $$
therefore $$ \frac {\partial u} {\partial x} = 2x \quad\text{meaning}\quad \frac {\partial v} {\partial y} = 2x $$
and $$ \frac {\partial v} {\partial y} = 2y \quad\text{meaning}\quad \frac {\partial v} {\partial x} = -2y $$
$$ \int \frac {\partial v} {\partial y}\,dy = i \int 2x\,dy = 2ixy $$
$$ \int \frac {\partial v} {\partial x}\,dy = -i \int 2y\,dx = -2ixy $$
Then I'm not too sure where to go with it
An example of a function that is not holomorphic (i.e. not analytic in the complex plane) is $f(z)=\bar{z}$.
Suppose $f(z)=\bar{z}$ is holomorphic, then $f$ must satisfy the CR equations (Recall that if $z=x+iy$, then $\bar{z}=x-iy$)
Writing $z=x+iy$, then $f(z)=f(x,y)=x-iy=u(x,y)+iv(x,y)$, where $u(x,y)=x$ and $v(x,y)=-y$. Then, $u_x=1$, $u_y=0$, $v_x=0$, and $v_y=-1$.
The CR equations below, $$u_x=v_y\text{ and }u_y=-v_x,$$ must be satisfied.
Substituting our values for $f$ we obtain straight away
$$1=-1,$$ which is a contradiction. Therefore, $f(z)$ is not holomorphic.