# Thread: Differentiation using Cauchy Riemann Equations

1. ## Differentiation using Cauchy Riemann Equations

FOlks,

Use the C-R equations to determine whether f(z)=u(x,y)+iv(x,y) is complex analytic for the following, if so calculate f'(z).

a) $\frac{x-iy}{x^2+y^2}$

1) Is complex analytic the same as complex differentiable?

2) I determine f(z) = $\frac{x}{x^2+y^2}$ +i $\frac{-y}{x^2+y^2}$ and $f'(z) =u_x+i v_x$

Using the quotient rule I determine $u_x=v_y= \frac{-x^2+y^2}{(x^2+y^2)^2}$ and $u_y=-v_x= \frac{-2xy}{(x^2+y^2)^2}$

Therefore $f'(z)= \frac{-x^2+y^2}{(x^2+y^2)^2}+ i \frac{2xy}{(x^2+y^2)^2}$...?

2. ## Re: Differentiation using Cauchy Riemann Equations

1) On Wikipedia: "a function that is complex-differentiable in a whole domain (holomorphic) is the same as an analytic function."
2)
du/dx = dv/dy => True
-dv/dx => True

Thus, f'(z) = (y^2-x^2)/(x^2+y^2)^2 + (2 i x y)/(x^2+y^2)^2

Theorem 3.3.4 If verifies the Cauchy-Riemann Formulas at and if the partial derivatives of and are continuous at , then is derivable at and . [source]