Question

c. r equation in polar coordinate

Answer 1

Let f(z) = u(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+iy if and only if
∂u
∂x

(x,y) =
∂v
∂y

(x,y) and
∂u
∂y

(x,y) = −
∂v
∂x

(x,y). So we have f′(z)=
∂u
∂x

(z)+i
∂v
∂x

(z). Let f(z) = f(reiθ)= u(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 z = reiθ if and only if r
∂u
∂r

=
∂v
∂θ

and
∂u
∂θ

= −r
∂v
∂r

.

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



this is your answer