Blaschke factor (${\psi }_{w,R}(z)=R(z-w)/(R^2-\overline{w}z)$)

For $|w|<R$, this Möbius transformation maps the radius-$R$ disc to the unit disc, vanishes at $w$ and has modulus one on the boundary. Expanding gives $|R^2-\overline{w}z{|}^2=R^2|z-w{|}^2$ when $|z|=R$; its pole lies outside the disc. Dividing a holomorphic function by factors corresponding to its interior zeros removes those zeros without changing its boundary modulus. This gives a direct factorization proof of Jensen's formula.