Algebraic branch point ($w(z)={\sum }_{j=j_0}^{\infty }{c}_j(z-a{)}^{j/k}$)

At an algebraic branch point, a local solution becomes meromorphic after the finite substitution $z-a=t^k$. The displayed convergent expansion has only finitely many negative terms. A square-root branch point corresponds to $k=2$; an essential singularity or logarithmic singularity need not become meromorphic under any such finite substitution.