A First Course in Complex Analysis · Problem 2.5.25

Differentiation — Beck, Marchesi, Pixton & Sabalka

Mathematics — Complex Analysis · cauchy-riemann-polar

Problem

Derive the Cauchy--Riemann equations in polar form. That is, if

f(re^{i\theta}) = u(r,\theta) + iv(r,\theta)

, show that if

f

is holomorphic then

u_r = \frac{1}{r} v_\theta \quad \text{and} \quad v_r = -\frac{1}{r} u_\theta

and

f'(z) = e^{-i\theta}(u_r + i v_r).