Polar form of the Cauchy-Riemann equations is

Cauchy-Riemann equations:

Rectangular form:

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

f(z) to be analytic it needs to satisfy Cauchy Riemann equations

ux = vy, uy = -vx

\(\frac{{\partial u}}{{\partial x}} = \frac{{\partial v}}{{\partial y}},\frac{{\partial u}}{{\partial y}} = - \frac{{\partial v}}{{\partial x}}\)

Polar form:

f(z) = u(r, θ) + f v(r, θ)

\({u_r} = \frac{1}{r}{v_\theta }\) and uθ = -rvr

​​\(\dfrac{\partial u}{\partial r} = \dfrac{1}{r} \dfrac{\partial v}{\partial \theta} \ \text{and} \ \dfrac{\partial v}{\partial r}=-\dfrac{1}{r} \dfrac{\partial u}{\partial\theta }\)