Complex ordinary differential equation ($w^{\prime }=A(z,w)$)

A first-order ordinary differential equation with complex independent and dependent variables is solved by a holomorphic function on an ordinary domain, and its analytic continuation can have branch points. A holomorphic right-hand side gives local existence and uniqueness by the integral contraction mapping used in the Picard-Lindelöf theorem.

The Painlevé property excludes movable critical singularities, such as branch points, of the analytically continued general solution of a complex differential equation. A common stronger formulation requires every movable singularity to be a pole, so local solutions are meromorphic there. Fixed singularities determined by the coefficients are distinguished from movable ones whose locations depend on initial data. Riccati equations with meromorphic coefficients have only movable poles at ordinary points: linearization expresses their solutions as logarithmic derivatives of linear-system solutions. More generally, set $w=f/g$ and solve the holomorphic linear system $f^{\prime }=bf+cg$, $g^{\prime }=-af$. Its quotient satisfies $w^{\prime }=a{w}^2+bw+c$ directly. Initial data $g(z_0)=1$, $f(z_0)=w(z_0)$ recover every local solution, and holomorphic $f,g$ make the quotient a meromorphic function, even at zeros of $a$. At a zero of $g$ with $a\ne 0$, uniqueness prevents $f=0$ there, so $g^{\prime }=-af\ne 0$ and the pole is simple. This proves the absence of movable branching without assuming that the quadratic coefficient never vanishes.

For a first-order ordinary differential equation rational in the dependent variable, with coefficients meromorphic on the chosen independent-variable surface, analytic continuation along an arc ending outside the fixed exceptional set has a definite limit in the Riemann sphere. Its movable singularities are poles or algebraic branch points. To see the limiting assertion, the cluster set of a continuous lifted arc is connected and compact. Any cluster value where the vector field is holomorphic permits uniform local continuation by the Picard-Lindelöf theorem. Otherwise the cluster set lies in finitely many coefficient poles, so it is a singleton. At a pole of the vector field, the inverse equation is holomorphic; its solution has a finite-order critical point, giving a convergent root expansion. The reciprocal dependent variable handles infinity. Points where both reduced numerator and denominator vanish belong to the fixed exceptional set and are excluded from this argument.

A solution has a movable singularity if its location varies with the initial data. For example $w^{\prime }=w^2$ has solutions $w=-1/(z-c)$, whose pole moves with $c$. The Painlevé determinateness theorem restricts movable singularities of first-order equations rational in the dependent variable to poles and algebraic branch points.

A singular location is fixed if its position is determined by the coefficients, independently of the initial data. Not every solution must actually be singular there. For a rational first-order equation, coefficient poles and exceptional fibres where the reduced numerator and denominator have common zeros are included among the possible fixed locations.

If the right-hand side and initial data are jointly holomorphic, the local solution is jointly holomorphic in its independent variable and parameters. On a smaller closed polydisc, bound $|A|$ by $M$ and $|{\partial }_{w}A|$ by $L$. Choose $r$ so that $rM$ stays inside the dependent-variable disc and $rL<1$. The integral contraction mapping has jointly holomorphic iterates converging uniformly, so locally uniform convergence of holomorphic functions proves the assertion. Shifting the dependent variable turns initial values into parameters.