Suppose that a function f (z) is analytic at a point z0 = z(t0) lying on a smooth arc z = z(t) (a ≤ t ≤ b). Show that if w(t) = f [z(t)], then
w'(t) = f'[z(t)]z'(t)
when t = t0.
Suggestion: Write f (z) = u(x, y) + iv(x, y) and z(t) = x(t) + iy(t), so that
w(t) = u[x(t), y(t)] + iv[x(t), y(t)].
Then apply the chain rule in calculus for functions of two real variables to write
W' = (uxx'+ uyy') + i(vxx'+ vyy'),
and use the Cauchy-Riemann equations.