To find the imaginary part of an analytic function, we need to use the Cauchy-Riemann equations. These equations relate the partial derivatives of the real and imaginary parts of the function.



The Cauchy-Riemann equations state that for a function f(z) = u(x,y) + iv(x,y) to be analytic, the following conditions must be satisfied:

• ∂u/∂x = ∂v/∂y


• ∂u/∂y = -∂v/∂x

We are given the real part of the analytic function as u(x,y) = x² - 3xy² + 3x² - 3y². To find the imaginary part v(x,y), we first need to check if the Cauchy-Riemann equations are satisfied.

• ∂u/∂x = 2x - 3y² + 6x = 8x - 3y²


• ∂u/∂y = -6xy - 6y = -6y(1 + x)

Now, we can use the first equation to find ∂v/∂y:

∂v/∂y = ∂u/∂x = 8x - 3y²

And we can use the second equation to find ∂v/∂x:

∂v/∂x = -∂u/∂y = 6y(1+x)



Now we have the partial derivatives of the imaginary part v(x,y) with respect to x and y. We can integrate these to find v(x,y):

v(x,y) = ∫(6y(1+x))dx = 3x²y + 2xy² + f(y)

v(x,y) = ∫(8x - 3y²)dy = 4xy - y³ + g(x)

where f(y) and g(x) are integration constants.

Therefore, the imaginary part of the analytic function is:

v(x,y) = 3x²y + 2xy² + C1

+ 4xy - y³ + C2

where C1 and C2 are constants of integration.