Using Cauchy theorem to compute $\int_{\gamma} z\overline{z}$

Question

Compute $\int_{\gamma} z\overline{z}$ where $\gamma=\{z||z|=1\}$.

I thought I could apply Cauchy Theorem and conclude the integral is zero since $\gamma$ is the unit ball hence connected and closed curve.

But there is still the condition that $z\overline{z}$ is holomorphic or analytic left to prove. I do not know if Icould use the Cauchy Riemann equations or the derivative definition:

$\lim_{z\to z_0}\frac{z\overline{z}-z_0\overline{z_0}}{z-z_0}=\lim_{z\to z_0}\frac{|z|^2-|z_0|^2}{z-z_0}$

But I do not see how I should continue.

Questions:

How should I end the computation?

Is it the same to use Cauchy- Riemann equations or the derivative definition in order to determine if a function is holomorphic or analytic?

Thanks in advance!

Answer 1

HINT: $z\bar z = |z|^2 = 1$ on the circle $|z|=1$. What is the integral of $1$ over a circle?

You can use the derivative definition to find out if a function is holomorphic. Also, a holomorphic function must satisfy the Cauchy-Riemann equations. But the C-R equations are not sufficient to conclude that a function is holomorphic. However, if the C-R equations hold and the real and imaginary part of the function are differentiable as well, then you can say that the function is holomorphic.

The function $z\bar z =|z|^2$ is not holomorphic inside $|z|=1$ (or anywhere alse) because it doesn't satisfy the Cauchy-Riemann equations.