## Coursera: analysis complex kind Week 3.2 Cauchy Riemann equations

### Jul 18, 2016

Definition complex function

F(z) = u(x,y) + i v(x,y), z= x +iy,

Where u , v are functions with two real inputs , x and y, and one real output.

E.g. f(z)=z^2, u(x,y) = x^2 – y^2, v(x,y) = 2xy

Derivative of z, f'(z) = 2z

Partial derivatives w.r.t. x =u_x , treat y as a constant = 2x, because y^2 treated as a constant

Similarly u_y, treat x as a constant, = -2y

From v(x,y) = 2xy

V_x (y constant) = 2y

V_y (x Constant) = 2x

Cauchy-riemann : u_x = v_y, u_y = – v_x

Further f'(z) = f_x(z) = u_x + i v_x = – i f_y(z) = -i(u_y + I v_y)