(a) Write down the definition of the complex derivative of the function $f(z)$ of a single complex variable.

(b) Derive the Cauchy-Riemann equations for the real and imaginary parts $u(x, y)$ and $v(x, y)$ of $f(z)$, where $z=x+i y$ and

$$f(z)=u(x, y)+i v(x, y)$$

(c) State necessary and sufficient conditions on $u(x, y)$ and $v(x, y)$ for the function $f(z)$ to be complex differentiable.