$\displaystyle u=\ r^2\cos{2\theta{}}-r\cos{\theta{}}+2 \\[2ex] \displaystyle u=r^2\left( {\cos}^2{\theta{}}- {\sin}^2{\theta{}}\right)-\left(r\cos{\theta{}}\right)+2\\[2ex] u=x^2-y^2-x+2 \\[3ex] \displaystyle \left\{x=r\cos{\theta{}},\ y=r\sin{\theta{}}\right\} \\[2ex] $

$\displaystyle Since\ f\left(z\right)\ is\ analytic, \\[2ex] \displaystyle u_x=2x-1=v_y \\[2ex] \displaystyle u_y=\ -2y=\ -v_x \\[2ex] \displaystyle \ By\ Milne\ Thompson^{'}sMethod,\ \\[2ex] \displaystyle f^{'}\left(z\right)={u_x}_{\left(z,0\right)}-i{u_y}_{\left(z,0\right)}\\[2ex] f^{'}(z)=2z-1-i\left(0\right)\\[2ex] f^{'}(z)=2z-1 \\[2ex] \displaystyle f\left(z\right)=\int{}f^{'}\left(z\right)dz\ =\int{}2z-1\ dz \\[2ex] \displaystyle f(z)=z^2-z+c\ \\[2ex] $