Question

State and prove cauchy riemann equations in polar coordinates

Answer 1

Cauchy-Riemann Equations: 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...