Example 4.3-1 Potential Flow around a cylinder
The rest of the example to find the pressure 
>  restart;
>  w:=z->-vinf*R*(z/R+R/z); Eq. 4.3-16

>  phi:=(x,y)->evalc(Re(w(x+I*y)));psi:=(x,y)->evalc(Im(w(x+I*y))); w(z)=phi(x,y)+i*psi(x,y). note that Maple requires i to be I.
>  assume(x,real,y,real,vinf,real,R,real);
>  phi(x,y);

>  psi(x,y); eq. 4.3-18
>  dw:=diff(w(z),z);
>  dw:=unapply(dw,z);
>  vx:=(x,y)->evalc(Re(-dw(x+I*y)));
>  vy:=(x,y)->evalc(Im(dw(x+I*y)));
>  vx(x,y);

>  vy(x,y);

>  vx2:=(r,theta)->simplify(vx(r*cos(theta),r*sin(theta)));
>  vx2(r,theta);

>  vxbook:=(r,theta)->vinf*(1-(R/r)^2*cos(2*theta)); eq. 4.3-21
>  simplify(vx2(r,theta)-vxbook(r,theta));
>  expand(%);

>  vy2:=(r,theta)->simplify(vy(r*cos(theta),r*sin(theta)));
>  vy2(r,theta); compare to eq. 4.3-22

>  vsq:=theta->simplify((vx2(R,theta)^2+vy2(R,theta)^2));
>  vsq(theta); eq. 4.3-23

>  dP:=rho/2*(vinf^2-vsq(theta)); eq.4.3-24

>  dPbook:=rho/2*vinf^2*(1-4*sin(theta)^2); eq. 4.3-25
>  simplify(dP-dPbook);