def forward_diff_z(beta, x, y, z, dt):
    """

    Update equation for coordinate :math:`z`: 
        
        :math:`z[n+1] = (x[n]y[n] - \\beta z[n])t_{\\delta} + z[n]`

    INPUT::

        beta : float
            Free parameter.
        x : float
            Current value of coordinate x (x[n]).
        y : float
            Current value of coordinate y (y[n]).
        z : float
            Current value of coordinate z (z[n]).
        dt : float
            Time step (s).

    OUTPUT::
        
        float
            z at the next time step (z[n+1]).


    """

    return (x * y - beta * z) * dt + z