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A branch point is a point around which a sufficiently small loop can change a germ of a holomorphic function under analytic continuation. This is a local condition: a large loop around a regular point might also enclose a distant branch point, so that large loop alone cannot classify the regular point. For the complex logarithm and square root, every nonzero point has a local holomorphic branch, but a loop once around $0$ changes the chosen branch. ^[ref]

For $z=r{e}^{it}$ with fixed $r>0$, start at $t=0$ with values $\sqrt{r}$ and $\log r$. Their continuations areOne counterclockwise turn gives $w(2\pi )=-\sqrt{r}$ and $L(2\pi )=\log r+2\pi i$. At $r=4$, the square root therefore changes from $2$ to $-2$, while the logarithm changes from $\log 4$ to $\log 4+2\pi i$. Two turns restore the square root to $2$; no nonzero number of turns restores the logarithm. This distinguishes a two-sheeted algebraic branch point from a logarithmic singularity.

A pole need not be a branch point: $1/z$ is unbounded near $0$, yet $1/(r{e}^{it})=r^{-1}{e}^{-it}$ returns to its starting value after one turn. Conversely, the square-root values tend to $0$ at their branch point. The issue is whether a single holomorphic choice survives going around the point, rather than whether the values become large. On the Riemann sphere, infinity is also a branch point of both $\sqrt{z}$ and $\log z$: in the local coordinate $\zeta =1/z$, these are ${\zeta }^{-1/2}$ and $-\log \zeta$, whose continuations around $\zeta =0$ change sign and subtract $2\pi i$, respectively. ^[ref]

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.

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.

Write the rational function $A$ in reduced form. We first exclude a finite pole in its dependent variable at a generic point $(a,b)$. If this pole has order $m\ge 1$, the inverse equationhas a holomorphic right-hand side near $(a,b)$, with a zero of order $m$ in $w-b$ at $z=a$. Choosing a generic point avoids collisions of pole branches and zeros of the reduced numerator. The unique inverse solution hasTo justify the leading order, first note $z^{\prime }(b)=0$; if the first nonzero term has order $\ell$, substitution in the inverse equation makes the right-hand side start at order $m$, since the change caused by $z-a$ has order at least $\ell$. Thus $\ell -1=m$. Inverting produces an algebraic branch point of order $m+1$. Away from the fixed coefficient exceptions, $a$ can be chosen throughout an open set on the pole curve; inverse solutions based at these different $a$ give different initial values at a nearby ordinary point. These are movable singularities of a complex differential equation, contradicting the hypothesis. Hence the denominator has no genuine $w$-dependent factor: $A$ is a polynomial in $w$, with rational functions of $z$ as coefficients.

Let its degree in $w$ be $d$. Use the dependent coordinate $v=1/w$ on the Riemann sphere. Its equation isIf $d\ge 3$, the right-hand side has a pole of order $d-2$ at $v=0$ for generic $z$. The same inverse-equation argument gives a movable algebraic branch point of order $d-1$. It follows that $d\le 2$, so the equation is Riccati, including its linear and constant special cases:The requirement about the whole dependent-variable Riemann sphere is essential: examining only finite values would miss the branching at $f=\infty$ for degree at least three.

The linearization of a Riccati equation is particularly symmetric in the systemOn a simply connected domain where the coefficients are holomorphic, its solutions are holomorphic. Direct differentiation givesA nonzero initial vector never becomes the zero vector, by uniqueness for the linear differential equation, so the quotient is meromorphic. Choosing initial vector $(f(z_0),1)$ represents every finite initial value, and $(1,0)$ represents infinity. At fixed coefficient singularities, a single globally holomorphic vector solution need not exist; the assertion is local on ordinary domains, or on their appropriate continuation surface.

Finally, for any two finite solutions $u,v$ of the Riccati equation,Use this identity on the four differences inThe logarithmic derivative of this cross-ratio isThus $K$ is constant wherever the expression is initially defined, and the identity extends as an identity of meromorphic functions. Distinctness of $g,h,j$ is preserved at ordinary points by uniqueness, interpreted on the Riemann sphere. Solving the constant cross-ratio equation gives the Riccati cross-ratio superpositionFor $K=\infty$ this means $f=j$; $K=0$ gives $g$, and $K=1$ gives $h$. The constant is selected by the initial value of $f$, so the formula represents every solution, with its denominator zeros interpreted as poles.

A fixed singularity of a complex differential equation has a location determined by the coefficients, independently of the initial value. A movable singularity of a complex differential equation belongs to an individual continued solution and changes location when the initial value changes. For example, $w^{\prime }=w^2$ has the movable pole $z=c$ in $w=-1/(z-c)$. A fixed exceptional location need not be singular for every solution.

The precise first-order form of the Painlevé determinateness theorem needed here is as follows. Let $w^{\prime }=A(z,w)$ be rational in $w$, with coefficients meromorphic in $z$ on the chosen independent-variable surface. Algebraic coefficients can be made single valued on their own surface. Exclude the fixed coefficient singularities and exceptional fibres where a reduced numerator and denominator vanish together, working also in the dependent coordinate $v=1/w$. If a solution is continued along an arc $\gamma :[0,1)\to U$ with endpoint $z_{\ast }={\lim }_{t\to 1}\gamma (t)$ outside that exceptional set, then $w(\gamma (t))$ has a definite finite or infinite limit. Any singularity there is a pole or an algebraic branch point; there is no movable essential or logarithmic singularity. The standard rectifiable-arc formulation is included in this statement.

Here is a proof. First suppose a sequence along the arc has $(z_n,w_n)\to (z_{\ast },w_{\ast })$ at a point where the vector field is holomorphic. Choose a smaller closed bidisc about this point. Uniform bounds on the vector field and its dependent-variable derivative give, by the integral contraction mapping proof, one radius $r>0$ on which every solution starting sufficiently close to $(z_{\ast },w_{\ast })$ exists and remains bounded. For large $n$, the solution through $(z_n,w_n)$ is therefore defined on a fixed disc about $z_{\ast }$. The entire sufficiently late tail of $\gamma$ lies in that disc. Uniqueness identifies the original continued solution with this local solution throughout that tail, so the solution extends through $z_{\ast }$ and has a limit. The same argument applies to an ordinary point in the reciprocal coordinate $v=1/w$.

If the limit did not exist, consider the cluster set $K$ in the dependent-variable Riemann sphere:Each closed tail is compact and connected, and these sets are nested; hence $K$ is nonempty, compact and connected. The preceding paragraph excludes every ordinary point from $K$. At $z_{\ast }$ the rational vector field has only finitely many nonordinary dependent values, in both coordinate charts: an entire fibre of singular points was excluded among the fixed exceptions. Thus $K$ is a connected subset of a finite set, so it is a singleton. Compactness then forces convergence to its unique point, contradicting the assumed lack of a limit. This proves determinateness.

It remains to determine the local type rather than merely the limit. At a finite limiting value $w_{\ast }$ which is a pole of the vector field, write the reduced equation as $w^{\prime }=P/Q$. Since $z_{\ast }$ is not exceptional, $P(z_{\ast },w_{\ast })\ne 0$, and the inverse equationis holomorphic near $(z_{\ast },w_{\ast })$. The holomorphic dependence of ordinary differential equations on parameters makes its local solution curves jointly holomorphic in $w$ and their initial $z$ values. In particular, the value at the transverse line $w=w_{\ast }$ is a holomorphic first integral labelling these curves. The original graph has constant label; taking its endpoint limit shows that it lies on the inverse solution through $(z_{\ast },w_{\ast })$. That inverse solution cannot be the vertical curve $z\equiv z_{\ast }$: this would require $Q(z_{\ast },w)$ to vanish identically near $w_{\ast }$, an excluded fixed singular fibre. Consequently its power series has a first nonzero term,Here $k\ge 2$ at a vector-field pole. Factor the right side as $(w-w_{\ast }{)}^k$ times a nonvanishing holomorphic function and take its local $k$th root. This gives a new dependent coordinate with nonzero derivative, and the holomorphic inverse function theorem expresses $w$ as a convergent power series in $(z-z_{\ast }{)}^{1/k}$. Thus the singularity is an algebraic branch point. At an infinite limit, repeat the argument for $v=1/w$. An ordinary zero of $v$ gives a pole of $w$, while a finite ramification of $v$ gives an algebraic pole branch. This proves the asserted local types and the theorem.

For the given equation, the finite-coordinate rational field isThe whole fibre $z=-1$ is singular in the displayed coefficient, and at $(z,w)=(0,0)$ the reduced numerator and denominator vanish together. Hence the finite fixed exceptional set is $\{-1,0\}$. Away from these two locations, the only possible finite dependent limits at a singularity are $w_{\ast }=\sigma z_{\ast }$, with $\sigma =\pm 1$.

At such a point the inverse equation isIt is holomorphic when $z_{\ast }\ne 0,-1$ and $w_{\ast }=\sigma z_{\ast }$. Its inverse solution satisfies $z_w(w_{\ast })=0$, and differentiation givesThereforeandThese are genuine square-root algebraic branch points. The inverse solutions can be based at arbitrary $z_{\ast }\notin \{0,-1\}$ and then continued to nearby ordinary initial points. Their distinct branch locations require different initial data, so these branch points are movable. The signs specify the two inverse branches near each of the two values $w_{\ast }=\pm z_{\ast }$.

There are no movable poles. Indeed the reciprocal equation isFor every finite $z\ne -1$ it is holomorphic at $v=0$, and $v\equiv 0$ is its unique solution with zero initial value. A nonzero reciprocal solution therefore cannot reach zero there. This rules out both an ordinary pole and an infinite algebraic branch at a movable finite location. Together with the theorem, it exhausts the finite movable singularities.

The fixed point zero must not be mistaken for a generic movable collision. It is ordinary for solutions with $w(0)\ne 0$. But solutions reaching $w=0$ can have a fixed square-root singularity. To construct them rather than just balance powers, set $z(w)=w^{2}h(w)$ in the inverse equation. The equation for $h$ isEquivalently,On a small disc this is a contraction mapping of the closed ball $\Vert h-1/2{\Vert }_{\infty }\le 1/4$: both its departure from $1/2$ and its Lipschitz constant are $O(r^2)$. Its unique holomorphic fixed point gives $z=w^2/2+O(w^4)$. Thusis a genuine branch at the fixed location zero. The zero solution itself extends through zero as a function, although the original displayed coefficient is undefined at $(0,0)$.

The point $-1$ can produce a fixed nonalgebraic singularity. For an explicit example take $z=-1+x$ real, $0<x<x_0<1$, and $w=iy$ with initial $y>2$. Put $q=y^2$ and $L=-\log x$. ThenAs $L$ increases, $q$ increases, its derivative is bounded above by $2$ and bounded below by a positive constant, and therefore the solution exists for all large $L$ and $q\to \infty$. The derivative then tends to $2$, so $q/L\to 2$. ConsequentlyThis is unbounded but grows more slowly than every negative power of $z+1$. It cannot be a meromorphic pole even after a finite ramified substitution, demonstrating an actual fixed singularity with logarithmic growth. The determinateness theorem does not exclude such behavior at fixed exceptions.

Finally, on the independent-variable Riemann sphere use $t=1/z$. For finite $w$ the transformed equation iswhich is ordinary at $t=0$. At simultaneous $z=\infty$, $w=\infty$, however, the reciprocal field iswith an indeterminate point at $(t,v)=(0,0)$. Thus infinity is an additional fixed exceptional location on the sphere, rather than an unavoidable singularity of solutions with finite limiting value. There are solutions escaping to infinity there: for real initial data $w(z_0)>z_0>0$, the difference $d=w-z$ stays positive, since its derivative becomes positive as $d\downarrow 0$. Moreover $d^{\prime }\le 0$ when $d\ge 1$, so $0<d\le \max (1,d(z_0))$. Such a solution continues for all positive $z$, with $w=z+O(1)\to \infty$.

In the finite plane, the fixed exceptional locations are $0,-1$; all other solution singularities are movable square-root branches at $w=\pm z$, and there are no movable poles. On the sphere, infinity is also a fixed exception at the infinite dependent value. Being fixed or movable describes the base location, not merely whether the coefficient contains the factors $w-z$ and $w+z$.