- #1

dyn

- 773

- 62

- Homework Statement
- Is f = u(x,y) + iv(x,y) an analytic function of z=x+iy where ## u(x,y) = x^3 - 3x(y^2) ## and ##v(x,y) = -y^3-3(x^2)y ##

- Relevant Equations
- The Cauchy-Riemann equations ,

## u_x = 3x^2 -3y^2 ## and ## v_y = -3y^2-3x^2 ##

## u_y = -6xy## and ## v_x = -6xy##

To be analytic a function must satisfy ##u_x = v_y## and ##u_y = -v_x##

Both these conditions are met by x=0 and y taking any value so I think the functions is analytic anywhere on the line x=0

However the answer is that the function is not analytic ; I don't understand that. Any help would be appreciated

## u_y = -6xy## and ## v_x = -6xy##

To be analytic a function must satisfy ##u_x = v_y## and ##u_y = -v_x##

Both these conditions are met by x=0 and y taking any value so I think the functions is analytic anywhere on the line x=0

However the answer is that the function is not analytic ; I don't understand that. Any help would be appreciated