Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 12 (Complex Differential Equations)

Choose $0<R<R_1$, $0<S<S_1$ and $0<T<T_1$. On this smaller closed polydisc, the holomorphic function $A$ and its $w$-derivative are bounded; write $|A|\le M$ and $|{\partial }_{w}A|\le L$. Choose $0<r<R$ with $rM<S/2$ and $rL<1$, replacing a zero bound by any positive upper bound if necessary, and choose $0<\epsilon <T$.

Consider functions $u(z,\alpha )$ continuous on $\overline{D(0,r)}\times \overline{D(0,\epsilon )}$, jointly holomorphic in the interior, with $u(0,\alpha )=0$ and $\Vert u{\Vert }_{\infty }\le S/2$. This is a closed subset of a Banach space in the supremum norm. The integral operatoris well defined on the straight segment and is jointly holomorphic. Indeed, it equals $z{\int }_0^{1}A(tz,u(tz,\alpha ),\alpha )dt$, an integral of jointly holomorphic functions locally uniformly bounded in both variables. Convexity of the $w$-disc and the derivative bound giveThus the Banach fixed-point theorem gives a fixed point $f(z,\alpha )$. More explicitly, the successive integral iterates starting from zero converge uniformly, and locally uniform convergence of holomorphic functions makes their limit jointly holomorphic. Differentiating the integral equation gives the required complex ordinary differential equation and initial value. This proves local existence, uniqueness and joint holomorphic dependence on $(z,\alpha )$. Uniqueness among arbitrary local solutions follows first on a small disc where their values stay within $|w|<S$, by the same contraction mapping estimate, and then throughout their common connected domain by the identity theorem.

For holomorphic dependence of ordinary differential equations on parameters in the initial value, fix an admissible initial value $w_{\ast }$ and put $u=f-w_0$. The equation becomesIts right-hand side is jointly holomorphic in $(z,u,w_0)$ near $(0,0,w_{\ast })$. The preceding construction supplies a common disc and a jointly holomorphic solution for $w_0$ near $w_{\ast }$. Consequently $f=u+w_0$ is jointly holomorphic in $z$ and its initial value $w_0$.

The assertion is true. The assumed finite multiplicity means that $f_0-w_1$ is not identically zero and has an isolated zero of order $k$ at $z_1$. Choose $\rho >0$ so that $\overline{D(z_1,\rho )}\subset D(0,r)$ and $f_0-w_1$ has no other zeros in this closed disc. On its boundary,The joint holomorphy established above gives $f_{\alpha }\to f_0$ uniformly on that circle as $\alpha \to 0$. For sufficiently small $|\alpha |$, therefore, $|f_{\alpha }-f_0|<\eta$. By Rouché's theorem, $f_{\alpha }-w_1$ has exactly $k$ zeros in $D(z_1,\rho )$, counted with multiplicity. In particular it has at least one zero in $D(0,r)$. This preserves the total local zero count, without asserting that the zeros remain together.

The assertion is false for $k\ge 2$, but true for $k=1$. For $k\ge 2$, take the jointly holomorphic right-hand side $A_{\alpha }(z,w)=k{z}^{k-1}+\alpha$. Its solution with zero initial value isAt $\alpha =0$, the value zero has multiplicity $k$ at the origin. For $\alpha \ne 0$, however, every zero is a simple zero: $f_{\alpha }^{\prime }(0)=\alpha$, and at a nonzero zero $z^{k-1}=-\alpha$,Thus there is no zero of multiplicity $k$ anywhere, even though the $k$ simple zeros stay close to zero for small $\alpha$. When $k=1$, part (a) gives exactly one zero counted with multiplicity in a small disc, so that zero is simple. Alternatively, the holomorphic implicit function theorem gives its holomorphic location as a function of $\alpha$.

For a first-order system, it is important to specify the frame in which its coefficients are written. In the simple-pole convention intended here, a Fuchsian linear differential system has, at a finite singular point $a$,The residue is the matrix $R_a={\lim }_{z\to a}(z-a)A(z)$, computed entry by entry. Such a point is a regular singular point: on a sector, the coefficient bound $\Vert A(z)\Vert \le C/|z-a|$ and integration along rays give at most power growth of solutions. Intrinsically, a regular singular point means moderate power growth of all solutions on sectors, allowing logarithmic factors. This intrinsic definition is more general than requiring a simple coefficient pole in an arbitrary meromorphic frame.

For $A(z)=R/z$, a fundamental matrix on a chosen complex logarithm branch isDifferentiating the matrix exponential verifies ${\Phi }^{\prime }=R\Phi /z$. A counterclockwise circuit replaces $\log z$ by $\log z+2\pi i$, soThis is the residue and monodromy of a constant Fuchsian system. A change of solution basis conjugates $M$. Its eigenvalues are $e^{2\pi i\lambda }$ for the residue eigenvalues $\lambda$; a nontrivial Jordan block supplies the corresponding logarithmic terms in $z^R$. The monodromy does not uniquely recover $R$, since integer shifts of residue eigenvalues can leave their exponentials unchanged.

The coordinate-independent coefficient is the matrix-valued one-form $A(z)dz$. With $t=1/z$, its coefficient isBeing Fuchsian at infinity means $A(z)=O(z^{-1})$, while being ordinary there means $A(z)=O(z^{-2})$. If the finite simple poles are $a_1,\dots ,a_s$, subtract their principal parts:Each entry of $H$ is entire and tends to zero at infinity. Liouville's theorem therefore gives $H=0$. Expanding this partial fraction decomposition at infinity yieldsThus the sum of all residues, including the residue at infinity of the one-form, is zero. If infinity is ordinary, $R_{\infty }=0$ and the finite residues alone sum to zero. Treating $A$ merely as a scalar function of $z$, without transforming $dz$, would give the wrong infinity convention.

Put $\omega =e^{2\pi i/3}$. The possible finite singular points of $(B+zC)/(1-z^3)$ are $1,\omega ,{\omega }^2$, andA point $\zeta$ is removable exactly when $B+\zeta C=0$ as a matrix. At infinity,which is holomorphic at $t=0$. Consequently the actual set is $S=\{\zeta :{\zeta }^3=1,B+\zeta C\ne 0\}$, with $\{1,\omega ,{\omega }^2\}$ the generic set. The finite residues sum to zero, as follows also from ${\sum }_{{\zeta }^3=1}{\zeta }^{-1}={\sum }_{{\zeta }^3=1}{\zeta }^{-2}=0$.

Conversely, suppose a Fuchsian linear differential system has its finite singular points among these three roots and has no singularity at infinity. Then $Q(z)=(1-z^3)A(z)$ extends to an entire matrix function, and ordinariness at infinity gives $Q(z)=O(z)$. By the Cauchy estimate, each entry has zero derivatives of order two or higher, so $Q(z)=B+zC$. The same argument applies to any subset $S$, with additional cancellations at the omitted roots. This proves the requested normal form.

There is a genuine convention limitation: the normal form is not true under intrinsic regular singularity alone in an unrestricted frame. For example, if $N\ne 0$ and $N^2=0$, then $A(z)=N/(z-1{)}^2$ has fundamental matrix $I-N/(z-1)$. Its solutions have only power growth, so $1$ is intrinsically a regular singular point, and infinity is ordinary. Nevertheless its coefficient has a double pole and cannot equal $(B+zC)/(1-z^3)$. The simple-pole convention makes the preceding residue and normal-form assertions correct.

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.

For the unrestricted zero sequence, introduce the Weierstrass elementary factorsFor $|u|\le 1/2$, cancellation of the first $p$ terms of the logarithm power series gives a holomorphic logarithm of the factor:Define the canonical productFix any compact disc $|z|\le R$. Since $|z_n|\to \infty$, eventually $|z/z_n|\le 1/2$ uniformly on this disc. The tail logarithms are then bounded by $2(1/2{)}^{n+1}$, a summable bound independent of $z$. Their sum is holomorphic by locally uniform convergence of holomorphic functions, and its exponential is nowhere zero. Multiplication by the finite initial product therefore gives an entire function. Each factor has a simple zero exactly at its specified $z_n$, the exponential factor has none, and a tail beginning beyond any given compact disc is nonvanishing there. Thus $f$ has precisely the prescribed zeros and no others, each simple, and $f(0)=1$. This proves the required existence without an assumption on ${\sum }_n|z_n{|}^{-1}$.

For the final, unheaded product request, suppose now that ${\sum }_n|z_n{|}^{-1}<\infty$. On $|z|\le R$, all sufficiently late factors have $|z/z_n|\le 1/2$, so part (a) bounds their principal complex logarithms by $2R/|z_n|$. The logarithms converge absolutely and uniformly; infinite product convergence from logarithmic tails proves thatis entire and again has exactly the prescribed simple zeros. For its finite partial products,Passing to the limit provesIn particular $P$ is an entire function of exponential type. This argument uses a valid modulus estimate and does not require the false logarithm assertion in part (c).

For $|w|\le 1/2$, the power series of the principal complex logarithm is absolutely convergent and givesThus $C_1=2$ works. The value $1-w$ stays in the right half-plane, so there is no logarithm branch ambiguity. This is the bounds for the principal logarithm near one estimate used to control product tails.

For every real $r\ge 0$,Taking $r=|w|$ proves the required logarithm inequality, even without the lower bound $|w|\ge 1/2$. Hence $C_2=1$ works.

The printed assertion is false. Take real $w=1-\epsilon$ with $0<\epsilon <1$. The principal complex logarithm is unambiguous, butNo finite $C_3$ can satisfy the proposed estimate, even on these points within the logarithm's domain. At $w=1$ the logarithm is undefined, and the usual principal analytic branch also excludes its cut. These are additional reasons the literal all-$w$ assertion cannot hold.

The valid estimate for the final product proof is insteador equivalently $|1-w|\le e^{|w|}$ for every $w$, including $w=1$. The upper bound on the real part of the logarithm is enough for growth; a bound on the full modulus of the logarithm is unnecessary. The preceding root Solution supplies the complete convergence and exponential-growth proof using this correction.

Write ${\log }^{+}x=\max (0,\log x)$. The Nevanlinna proximity function and Nevanlinna integrated counting function areThe pole sum counts multiplicity; it is finite for each $r$ because $f(0)$ is finite and nonzero and poles of a meromorphic function are isolated. Equivalently $N(r,f)={\int }_0^{r}n(t,f)dt/t$, where $n(t,f)$ counts poles in the disc. The Nevanlinna characteristic isFor a general nonzero meromorphic function $g$ with signed order $\nu$ at zero, use $g(z)=c{z}^{\nu }(1+O(z))$, $c\ne 0$. The general counting convention includes a pole at zero:where $n(0,g)=\max (0,-\nu )$. This convention will also cover $g=f-a$ when the target $a=f(0)$.

Here is a direct proof of the exact reciprocal identity underlying the Nevanlinna first main theorem. Choose a circle with no zero or pole of $g$ on it, and a slightly larger disc with the same interior zeros and poles. Divide $g$ by $z^{\nu }$ and its nonzero zero factors and multiply by its nonzero pole factors, obtaining a holomorphic, nowhere-zero function $H$ on that disc. For any factor $z-a$ with $|a|<r$,Indeed, factor out $r{e}^{i\theta }$; the remaining logarithm has real part equal to that of $-{\sum }_{k\ge 1}(a/r{)}^k{e}^{-ik\theta }/k$, whose angular mean is zero. Also, $\log |H|$ is harmonic, so the mean value property for harmonic functions equates its circle mean to $\log |H(0)|$. Putting back the zero and pole factors provesThe identity ${\log }^{+}x-{\log }^{+}(1/x)=\log x$ now yieldsZeros or poles on the circle cause only integrable logarithmic singularities; the same identities hold at those radii by limits. A boundary zero or pole contributes $\log (r/r)=0$ to its counting term.

For a finite target $a$, set $g=f-a$. Its finite poles and their orders equal those of $f$. The elementary estimatesgive $|T(r,f-a)-T(r,f)|\le \log (1+|a|)$. Combining this with the reciprocal identity proves the first main theorem:The error is bounded independently of $r$, with an explicit bound $\log (1+|a|)+|\log |c_a||$, where $c_a$ is the first nonzero local coefficient of $f-a$ at zero. The count on the right records the occurrences of the value $a$ with multiplicity. The target infinity is the defining identity $T=m+N$. If $f$ is a constant equal to $a$, the reciprocal expression is undefined; the value-distribution assertion assumes $f-a\not\equiv 0$, as it automatically does for nonconstant $f$. The exact reciprocal identity remains valid for every nonzero constant $f$.

Taking $g=f$, whose signed order is zero and whose local coefficient is $f(0)$, rearranges that exact identity into Jensen's formula:Thus the zero and pole terms, their signs and the initial-value constant follow from the same proved identity, rather than from a theorem citation without proof.

Finally let $f$ be a nonzero bounded holomorphic function on the unit disc, with $|f|\le M$. If its order at zero is $m$, write $f(z)=z^{m}h(z)$, $h(0)\ne 0$. The factorization argument above works on every compact subdisc and gives, for its nonzero zeros,For $1/2\le r<1$ this is a uniform finite upper bound. As $r\uparrow 1$, monotone convergence givesSince $1-u\le -\log u$ for $0<u<1$, and there are only $m$ zeros at zero, we obtainThis is precisely the Blaschke condition, so the zeros, counted with multiplicity, form a Blaschke sequence.

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 $w+z$.