Cauchy Riemann Equations For Analytic Function
Cauchy - Riemann Equations :- Cauchy-Riemann Equation Necessary Conditions For A Function To Be Analytic : Cauchy-Riemann Equation Statement :- Cauchy-Riemann Equation The necessary conditions for w = f(z) = u(x,y) + i v(x,y) to be analytic (differentiable) at any point z = x + i y of its domain D is that the four patial derivatives ∂u/∂x , ∂u/∂y ,∂v/∂x ,∂v/∂y should exists and satisfy the C-R partial differential equations . ∂u/∂x = ∂v/∂y and ∂u/∂y = - ∂v/∂x Proof :- Cauchy-Riemann Equation Let f(z) = u(x,y) + i v(x,y) be analytic at any point z of its domain , then f'(z) = lim f(z+δz) - f(z) / δz δx-->0 exists and is unique . i.e it is independent of the path along which δz -->0 Let z = x+ iy ∴ δz = δx + i δy and as δz-->0 then δx , δy --> 0 ∴ f'(z) = lim [u(x+δx ,y+δy) + iv(x+δx,y+δy)]/δx+iδy