Introduction:
In fluid mechanics, the stream function and the velocity potential are two important concepts used to describe the motion of a fluid flow. The stream function is a scalar field that gives a measure of the flow's circulation, while the velocity potential is a scalar field that gives a measure of the flow's irrotationality. In this question, we are given a function \psi(x,y) and we need to determine if it can represent the stream function of an analytic function f(z) and find the complex potential and velocity potential if it does.

Stream Function:
The stream function \psi(x,y) represents the flow in a two-dimensional fluid flow. It satisfies the continuity equation in polar coordinates, given by:
∂ψ/∂x = -∂ψ/∂y

Analytic Function:
An analytic function f(z) is a complex-valued function that is differentiable at every point within its domain. In terms of the stream function, an analytic function f(z) can be written as:
f(z) = u(x,y) + iv(x,y)

Complex Potential:
The complex potential, denoted by Φ(z), is related to the stream function \psi(x,y) through the Cauchy-Riemann equations. In particular, we have:
Φ(z) = ψ(x,y) + iϕ(x,y)

Velocity Potential:
The velocity potential, denoted by φ(x,y), is related to the stream function \psi(x,y) through the following equations:
u(x,y) = ∂φ/∂x
v(x,y) = -∂φ/∂y

Solution:
To determine if the given function \psi(x,y) can represent the stream function of an analytic function f(z), we need to check if it satisfies the continuity equation. Taking the partial derivatives of \psi(x,y) with respect to x and y, we have:
∂ψ/∂x = 2x - 2y
∂ψ/∂y = -2y - 2x

Comparing these equations, we can see that the continuity equation is not satisfied, as ∂ψ/∂x is not equal to -∂ψ/∂y. Therefore, the function \psi(x,y) cannot represent the stream function of an analytic function f(z).

Since the function \psi(x,y) does not represent the stream function, we cannot find the complex potential and velocity potential associated with it.

Conclusion:
In conclusion, the given function \psi(x,y) = x^2 - y^2 - 2xy 2x - 3y does not represent the stream function of an analytic function f(z). Hence, we cannot find the complex potential and velocity potential associated with it.