Q:

Find a value of a simple closed curve.

Suppose that $f(z,\bar{z})$ is a $C^1$ , complex valued function defined $D_1(0)$ that satisfies
the partial differential equation
$$
\partial_zf=0
$$
Let $\Gamma$ be a  $C^1$, simple closed in $D_1(0)$. What can you say about the value of
$$
\oint_\Gamma f(z,\bar{z})d\bar{z}?
$$
Give an example of non-constant function that satisfies $\partial_zf=0$.
I don't know how to deal with problems with $f(z,\bar{z})$ because I don't think it's a 2 variable function, since $z$ is not independent with $\bar{z}$. So please tive me how to understand $f(z,\bar{z})$ and then give me some ideas about this problem. THANK YOU.

A:

Let $w=\bar{z}$ so $f(z,\bar{z})=f(z,w)$. Since $\partial_z f=0$, $f$ is independent from $z$, so $f(w)$ is a function of the complex variable $w$. You may apply complex variable theory, such as the residue theorem to express the integral.
An example of a function where $\partial_z f=0$ may be a Laurent series of $\bar{z}$, $f(z,\tilde{z})=\sum_{n=-\infty}^\infty a_n\bar{z}^n$. The are known as antiholomorphic functions.
A formal definition of an holomorphic function comes from, $\frac{1}{2}(\frac{d}{dx}+i\frac{d}{dy})f=\frac{df}{d\bar{z}}=0$ an equivalent form of the Cauchy-Riemann conditions.
An anti-holomorphic function will satisfy $\frac{1}{2}(\frac{d}{dx}-i\frac{d}{dy})f=\frac{df}{dz}=0$.

