Solution
= Solution
On the cut plane choose $0<\theta<2\pi$. For $z=x+iy$,
$$
u(x,y)=\frac12\log(x^2+y^2),
\qquad v(x,y)=\arg(x+iy).
$$
Their derivatives are
$$
u_x=\frac{x}{x^2+y^2}=v_y,
\qquad
u_y=\frac{y}{x^2+y^2}=-v_x.
$$
The <Cauchy-Riemann equations> hold, so this branch is holomorphic. Its derivative is
$$
\boxed{\frac{dw}{dz}=u_x+iv_x
=\frac{x-iy}{x^2+y^2}=\frac1z.}
$$