Question

Write the cauchy riemann equation in coordinates in polar form

Answer 1

What I have so far:

Cauchy-Riemann Equations: Let f(z)f(z) = u(x,y)u(x,y) +iv(x,y)iv(x,y) be a function on an open domain with continuous partial derivatives in the underlying real variables. Then f is differentiable at z=x+iyz=x+iy if and only if ∂u∂x(x,y)∂u∂x(x,y) = ∂v∂y(x,y)∂v∂y(x,y) and ∂u∂y(x,y)∂u∂y(x,y) = −∂v∂x(x,y)∂v∂x(x,y). So we have f′(z)=∂u∂x(z)+i∂v∂x(z)f′(z)=∂u∂x(z)+i∂v∂x(z). Let f(z)f(z) = f(reiθ)f(reiθ)= u(r,θ)u(r,θ) +iv(r,θ)iv(r,θ) be a function on an open domain that does not contain zero and with continuous partial derivatives in the underlying real variables. Then f is differentiable at zz = reiθreiθ if and only if r∂u∂r=∂v∂θr∂u∂r=∂v∂θ and ∂u∂θ∂u∂θ = −r∂v∂r−r∂v∂r.

Sorry, if this is not very good. I just decided to start learning complex analysis today...