Local and global finite branching of solutions of ODEs in the complex plane

Transcription

1 Local and global finite branching of solutions of ODEs in the complex plane Workshop on Singularities and Nonlinear ODEs Thomas Kecker University College London / University of Portsmouth Warszawa,

2 Outline To give a brief explanation about fixed and movable singularities

3 Outline To give a brief explanation about fixed and movable singularities To describe classes of (systems of) equations, with a simple local behaviour about the movable singularities of any of its solutions

4 Outline To give a brief explanation about fixed and movable singularities To describe classes of (systems of) equations, with a simple local behaviour about the movable singularities of any of its solutions To present the main ingredients for a method of proof that every movable singularity is of the type described

5 Outline To give a brief explanation about fixed and movable singularities To describe classes of (systems of) equations, with a simple local behaviour about the movable singularities of any of its solutions To present the main ingredients for a method of proof that every movable singularity is of the type described To discuss the local and global branching of the solutions

6 Outline To give a brief explanation about fixed and movable singularities To describe classes of (systems of) equations, with a simple local behaviour about the movable singularities of any of its solutions To present the main ingredients for a method of proof that every movable singularity is of the type described To discuss the local and global branching of the solutions To classify the admissible solutions of a certain 2 nd -order equation that are globally quadratic over the field of meromorphic functions

7 Fixed and movable singularities Consider the equation w = w 2 ( z 2 w 2 2 z + 1 ) z 2 w 2,

8 Fixed and movable singularities Consider the equation w = w 2 ( z 2 w 2 2 z + 1 ) z 2 w 2, which has the general solution w(z) = 1 z tan(z c), c C being an arbitrary integration constant.

9 Fixed and movable singularities Consider the equation w = w 2 ( z 2 w 2 2 z + 1 ) z 2 w 2, which has the general solution w(z) = 1 tan(z c), z c C being an arbitrary integration constant. The singularity located at z = 0 is fixed. All other singularities (located at z = c + πn 2, n Z) are movable square-root branch points.

10 Fixed and movable singularities Consider the equation w = w 2 ( z 2 w 2 2 z + 1 ) z 2 w 2, which has the general solution w(z) = 1 z tan(z c), c C being an arbitrary integration constant. The singularity located at z = 0 is fixed. All other singularities (located at z = c + πn 2, n Z) are movable square-root branch points. Definition A singularity z 0 is called an algebraic branch point if there is a rational number r > 0 such that the solution is represented, in a neighbourhood of z 0, by the sum of a Laurent series in (z z 0 ) r with finite principle part.

11 Analytic continuation and Painlevé s Lemma In the general situation we start from a local analytic solution about some point z 0 given by Cauchy s existence and uniqueness theorem and analytically continue the solution along some rectifiable path Γ.

12 Analytic continuation and Painlevé s Lemma In the general situation we start from a local analytic solution about some point z 0 given by Cauchy s existence and uniqueness theorem and analytically continue the solution along some rectifiable path Γ. Lemma (Painlevé) Let F k (z, y 1,..., y m ), k = 1,..., m, be analytic functions in a neighbourhood of a point (z, η 1,..., η m ) C m+1. Let Γ be a curve with end point z and suppose that (y 1,..., y m ) are analytic on Γ \ {z } and satisfy y k = F k (z, y 1,..., y m ), k = 1,..., m. Suppose there is a sequence (z n ) n N Γ such that z n z and y k (z n ) η k as n for all k = 1,..., n. Then the solution can be analytically continued to include the point z. z 0 z

13 First-order equations Theorem (Painlevé, 1888) Any movable singularity z of a solution of the first-order equation y (z) = P(z, y) Q(z, y), is an algebraic branch point, i.e. the solution can be represented by a series expansion y(z) = k=0 c k (z z ) k+k 0 n, k 0 Z, n N, convergent in some branched, punctured, neighbourhood of z.

14 First-order equations Theorem (Painlevé, 1888) Any movable singularity z of a solution of the first-order equation y (z) = P(z, y) Q(z, y), is an algebraic branch point, i.e. the solution can be represented by a series expansion y(z) = k=0 c k (z z ) k+k 0 n, k 0 Z, n N, convergent in some branched, punctured, neighbourhood of z. Proof. See E. L. Ince, Ordinary Differential Equations, Dover, New York, 1956; E. Hille, Ordinary Differential Equations in the Complex Domain, Wiley, New York, 1976.

15 First-order equations Theorem (Painlevé, 1888) Any movable singularity z of a solution of the first-order equation y (z) = P(z, y) Q(z, y), is an algebraic branch point, i.e. the solution can be represented by a series expansion y(z) = k=0 c k (z z ) k+k 0 n, k 0 Z, n N, convergent in some branched, punctured, neighbourhood of z. Proof. See E. L. Ince, Ordinary Differential Equations, Dover, New York, 1956; E. Hille, Ordinary Differential Equations in the Complex Domain, Wiley, New York, Question: For which classes of 2 nd -order, higher-order or systems of equations can we make a statement of this form?

16 Second-order and higher-order equations For differential equations of order 2 or higher other types of movable singularities do in general occur:

17 Second-order and higher-order equations For differential equations of order 2 or higher other types of movable singularities do in general occur: movable essential singularities ( y (y ) 2) ( ) y(y ) 3 = 0, y(z) = c 1 exp. z + c 2

18 Second-order and higher-order equations For differential equations of order 2 or higher other types of movable singularities do in general occur: movable essential singularities ( y (y ) 2) ( ) y(y ) 3 = 0, y(z) = c 1 exp. z + c 2 movable logarithmic branch points y + (y ) 2 = 0, y(z) = log(z + c 1 ) + c 2.

19 Second-order and higher-order equations For differential equations of order 2 or higher other types of movable singularities do in general occur: movable essential singularities ( y (y ) 2) ( ) y(y ) 3 = 0, y(z) = c 1 exp. z + c 2 movable logarithmic branch points y + (y ) 2 = 0, y(z) = log(z + c 1 ) + c 2. movable natural barriers, e.g. in Chazy s equation y = 2yy 3(y ) 2, any solution of which can only be defined on some disc, the radius of which depends on the initial conditions.

20 Painlevé property and Painlevé equations Restricting to the case where all movable singularities are poles, Painlevé, Gambier, Fuchs classified all second-order rational equations of the form y = R(z, y, y ) with this property, known as the Painlevé property.

21 Painlevé property and Painlevé equations Restricting to the case where all movable singularities are poles, Painlevé, Gambier, Fuchs classified all second-order rational equations of the form y = R(z, y, y ) with this property, known as the Painlevé property. For first-order equations y = R(z, y), the only type of equation with the Painlevé property is the Riccati equation y = a(z)y 2 + b(z)y + c(z).

22 Painlevé property and Painlevé equations Restricting to the case where all movable singularities are poles, Painlevé, Gambier, Fuchs classified all second-order rational equations of the form y = R(z, y, y ) with this property, known as the Painlevé property. For first-order equations y = R(z, y), the only type of equation with the Painlevé property is the Riccati equation y = a(z)y 2 + b(z)y + c(z). For second-order equations, up to transformations of the form z φ(z), y α(z)y + β(z) γ(z)y + δ(z), they found fifty canonical equations, all of which can be solved in terms of linear differential equations, classically known functions such as elliptic functions, and the solutions of the six nonlinear Painlevé equations.

23 The six Painlevé equations P I : P II : y = 6y 2 + z y = 2y 3 + zy + α P III : y = (y ) 2 y y z + 1 z (αy 2 + β) + γy 3 + δ y P IV : y = (y ) y 2 y 3 + 4zy 2 + 2(z 2 α)y + β y P V : y = 3y 1 2y(y 1) (y ) 2 y ( z + αy + β ) (y 1) 2 y z 2 P VI : y = 1 ( 1 2 y + 1 y ) (y ) 2 y z + y(y 1)(y z) z 2 (z 1) 2 ( α + β z + γy z ( 1 z + 1 z y z δy(y + 1) + (y 1) ) y y 2 + γ z 1 ) z(z 1) + δ (y 1) 2 (y z) 2

24 Painlevé test and Painlevé property About an arbitrary point z 0, the equation y = 2y 1 y (y ) 2 has a one-parameter family of Laurent series solutions y(z) = c k (z z 0 ) k, k= 1 i.e. the equation passes the Painlevé test.

25 Painlevé test and Painlevé property About an arbitrary point z 0, the equation y = 2y 1 y (y ) 2 has a one-parameter family of Laurent series solutions y(z) = c k (z z 0 ) k, k= 1 i.e. the equation passes the Painlevé test. However, it does NOT possess the Painlevé property, the general solution being of the form y(z) = log(tan(az + b)), which has a logarithmic branched singularity at z = b a.

26 Painlevé property of the Painlevé equations Painlevé s original proof for P I was somewhat incomplete

27 Painlevé property of the Painlevé equations Painlevé s original proof for P I was somewhat incomplete The first complete proof in the published literature (not using the iso-monodromy method) was given by Hinkkanen & Laine (1999).

28 Painlevé property of the Painlevé equations Painlevé s original proof for P I was somewhat incomplete The first complete proof in the published literature (not using the iso-monodromy method) was given by Hinkkanen & Laine (1999). A correct proof was circulating in lecture notes at the University of Tokyo by Hukuhara (1960), published by Okamoto & Takano in 2001

29 Painlevé property of the Painlevé equations Painlevé s original proof for P I was somewhat incomplete The first complete proof in the published literature (not using the iso-monodromy method) was given by Hinkkanen & Laine (1999). A correct proof was circulating in lecture notes at the University of Tokyo by Hukuhara (1960), published by Okamoto & Takano in 2001 Other direct proofs are by Steinmetz (2000), Shimomura (2001), Yan & Hu (2003)

30 Painlevé property for P I, y = 6y 2 + z The proof considers an auxiliary function W (z) = (y ) 2 4y 3 2zy + y y,

31 Painlevé property for P I, y = 6y 2 + z The proof considers an auxiliary function W (z) = (y ) 2 4y 3 2zy + y y, which satisfies a first-order linear differential equation W + 1 y 2 W = z y + y y 3.

32 Painlevé property for P I, y = 6y 2 + z The proof considers an auxiliary function W (z) = (y ) 2 4y 3 2zy + y y, which satisfies a first-order linear differential equation W + 1 y 2 W = z y + y y 3. Using the integrating factor ( I (z) = exp Γ ) dτ y(τ) 2, this can be converted into an integral representation for W, ( W (z) = I (z) 1 W (z) I (z) 2y y 2 Γ I (ζ) 2y(ζ) 4 (2ζy(ζ)3 1)dζ showing that W remains bounded as z z, provided that 1/y is bounded on the path Γ. ),

33 A curve deformation lemma To show that 1/y is bounded we employ a lemma by Shimomura (2001).

34 A curve deformation lemma To show that 1/y is bounded we employ a lemma by Shimomura (2001). Define u, v by y = u 2, ( y = u2 2 2u 3 + zu ) 2 u3 v.

35 A curve deformation lemma To show that 1/y is bounded we employ a lemma by Shimomura (2001). Define u, v by y = u 2, ( y = u2 2 2u 3 + zu ) 2 u3 v. Find a regular initial value problem in u and v ±u =1 + zu4 4 ± u5 4 u6 v 2 v = ± z2 u 8 + 3zu2 ± 8 ( 1 4 zv ) u 3 5u4 v 4 ± 3u5 v 2, 2 showing that u, v are analytic at z, corresponding to a pole of y.

36 Equations with algebraic singularities Theorem (R. A. Smith, 1953) 1) Let f and g be polynomials of degree n and m, respectively, where n > m, and let P be analytic at some point z. Then there is an infinite family of solutions of y + f (y)y + g(y) = P(z), which have an algebraic branch point at z. In a neighbourhood of z these solutions can be expressed in the form y(z) = a k (z z ) k/n, with a 1 0. k= 1

37 Equations with algebraic singularities Theorem (R. A. Smith, 1953) 1) Let f and g be polynomials of degree n and m, respectively, where n > m, and let P be analytic at some point z. Then there is an infinite family of solutions of y + f (y)y + g(y) = P(z), which have an algebraic branch point at z. In a neighbourhood of z these solutions can be expressed in the form y(z) = a k (z z ) k/n, with a 1 0. k= 1 2) Let Γ be contour of finite length in C having z as an end point. Let y(z) be a solution which can be continued analytically along Γ as far as z but not over it, then z is an algebraic branch point as described.

38 2) Let Γ be contour of finite length in C having z as an end point. Let y(z) be a solution which can be continued analytically along Γ as far as z but not over it, then z is an algebraic branch point as described. 3) If the singularity at z is not of the algebraic type described then Γ has infinite length and z is an accumulation point of algebraic branch points. Equations with algebraic singularities Theorem (R. A. Smith, 1953) 1) Let f and g be polynomials of degree n and m, respectively, where n > m, and let P be analytic at some point z. Then there is an infinite family of solutions of y + f (y)y + g(y) = P(z), which have an algebraic branch point at z. In a neighbourhood of z these solutions can be expressed in the form y(z) = a k (z z ) k/n, with a 1 0. k= 1

39 Equations with algebraic singularities In a series of papers, S. Shimomura (2007,2008) considered the equations y 2(2k + 1) = (2k 1) 2 y 2k + z, k N (1) y = k + 1 k 2 y 2k+1 + zy + α, k N \ {2}, (2) of P I -type and P II -type, respectively.

40 Equations with algebraic singularities In a series of papers, S. Shimomura (2007,2008) considered the equations y 2(2k + 1) = (2k 1) 2 y 2k + z, k N (1) y = k + 1 k 2 y 2k+1 + zy + α, k N \ {2}, (2) of P I -type and P II -type, respectively. Theorem Any singularity of a solution of one of the equations (1) or (2) that can be reached by analytic continuation along a finite length curve is algebraic.

41 Equations with algebraic singularities In a series of papers, S. Shimomura (2007,2008) considered the equations y 2(2k + 1) = (2k 1) 2 y 2k + z, k N (1) y = k + 1 k 2 y 2k+1 + zy + α, k N \ {2}, (2) of P I -type and P II -type, respectively. Theorem Any singularity of a solution of one of the equations (1) or (2) that can be reached by analytic continuation along a finite length curve is algebraic. When looking at more general equations (Filipuk & Halburd 2009) y = N 2 2(N + 1) (N 1) 2 y(z)n + a n (z)y(z) n, there are certain obstructions to the existence of algebraic singularities. n=0

42 Existence of formal series expansions Firstly, we look for leading-order behaviour of the form y(z) = c 0 (z ẑ) p + o((z ẑ) p ), as z ẑ.

43 Existence of formal series expansions Firstly, we look for leading-order behaviour of the form y(z) = c 0 (z ẑ) p + o((z ẑ) p ), as z ẑ. We find c N 1 0 = 1, p = 2 N 1.

44 Existence of formal series expansions Firstly, we look for leading-order behaviour of the form y(z) = c 0 (z ẑ) p + o((z ẑ) p ), as z ẑ. We find c N 1 0 = 1, p = 2 N 1. Substituting into the equation an expansion of the form y(z) = c n (z ẑ) (n 2)/(N 1), n=0

45 Existence of formal series expansions Firstly, we look for leading-order behaviour of the form y(z) = c 0 (z ẑ) p + o((z ẑ) p ), as z ẑ. We find c N 1 0 = 1, p = 2 N 1. Substituting into the equation an expansion of the form y(z) = c n (z ẑ) (n 2)/(N 1), n=0 leads to a recurrence relation for the coefficients of the form (n + N 1)(n 2N 2)c n = P n (c 0,..., c n 1 ).

46 Existence of formal series expansions Firstly, we look for leading-order behaviour of the form y(z) = c 0 (z ẑ) p + o((z ẑ) p ), as z ẑ. We find c N 1 0 = 1, p = 2 N 1. Substituting into the equation an expansion of the form y(z) = c n (z ẑ) (n 2)/(N 1), n=0 leads to a recurrence relation for the coefficients of the form (n + N 1)(n 2N 2)c n = P n (c 0,..., c n 1 ). Thus c 0,..., c 2N+1 are determined but for n = 2N + 2 we obtain the condition P 2N+2 (c 0,..., c 2N+1 ) = 0, leaving c 2N+2 arbitrary.

47 These conditions, for every c 0 satisfying c N 0 = 1, turn out to be a n 2(ẑ) = 0 (N even), + one additional condition (N odd).

48 These conditions, for every c 0 satisfying c N 0 = 1, turn out to be a n 2(ẑ) = 0 (N even), + one additional condition (N odd). Theorem (Filipuk & Halburd, 2009) Suppose a 0,..., a n 2 are analytic in a common domain Ω C and for each ẑ Ω there exists, for all c 0 satisfying c0 N = 1, a formal series solution of the form y(z) = c n (z ẑ) (n 2)/(N 1). (3) n=0 Let y be a solution that can be analytically continued along a finite length curve Γ up to but not including its endpoint z 0. Then y has a convergent series expansion in a punctured neighbourhood of ẑ = z 0 of the form (3).

49 These conditions, for every c 0 satisfying c N 0 = 1, turn out to be a n 2(ẑ) = 0 (N even), + one additional condition (N odd). Theorem (Filipuk & Halburd, 2009) Suppose a 0,..., a n 2 are analytic in a common domain Ω C and for each ẑ Ω there exists, for all c 0 satisfying c0 N = 1, a formal series solution of the form y(z) = c n (z ẑ) (n 2)/(N 1). (3) n=0 Let y be a solution that can be analytically continued along a finite length curve Γ up to but not including its endpoint z 0. Then y has a convergent series expansion in a punctured neighbourhood of ẑ = z 0 of the form (3). Remark The Theorem shows that for this class of equations, passing the (weak) Painlevé test is equivalent to having the (weak) Painlevé property.

50 The proof relies, like the proof for the Painlevé property of the Painlevé equations, on an auxiliary function W which remains bounded as z z.

51 The proof relies, like the proof for the Painlevé property of the Painlevé equations, on an auxiliary function W which remains bounded as z z. W is related to the Hamiltonian structure of the equation.

52 The proof relies, like the proof for the Painlevé property of the Painlevé equations, on an auxiliary function W which remains bounded as z z. W is related to the Hamiltonian structure of the equation. Consider the more general Hamiltonian H(z, p, q) = qn+1 N pm+1 M (i,j) I a ij (z)p i q j, where I = {(i, j) N 2 : i(n + 1) + j(m + 1) < (M + 1)(N + 1)} and the a ij, (i, j) I are analytic in some common domain Ω.

53 The proof relies, like the proof for the Painlevé property of the Painlevé equations, on an auxiliary function W which remains bounded as z z. W is related to the Hamiltonian structure of the equation. Consider the more general Hamiltonian H(z, p, q) = qn+1 N pm+1 M (i,j) I a ij (z)p i q j, where I = {(i, j) N 2 : i(n + 1) + j(m + 1) < (M + 1)(N + 1)} and the a ij, (i, j) I are analytic in some common domain Ω. dq dz = H p = pm + ia ij (z)p i 1 q j (i,j) I dp dz = H z = qn (4) ja ij (z)p i q j 1 (i,j) I

54 Theorem (TK, 2013) Suppose that for every pair of values (c 10, c 20 ) satisfying c MN 1 1,0 = 1, c 2,0 = c M 1,0, for all ẑ Ω the Hamiltonian system admits a formal series solution q(z) = c 1,k (z ẑ) k M 1 MN 1, p(z) = c 2,k (z ẑ) k N 1 MN 1. k=0 Then, every movable singularity z of a solution of the Hamiltonian system, obtained by analytic continuation along a rectifiable curve, is algebraic, and represented by series q(z) = k=0 k=0 k=0 C 1,k (z z ) kd N 1 MN 1, p(z) = C 2,k (z z ) kd M 1 MN 1, (5) where d = gcd{m + 1, N + 1, MN 1}, convergent in a branched, punctured, neighbourhood of z.

55 For the proof, the auxiliary function W is taken to be of the form W (z) = qn+1 N pm+1 M for some suitable set J N 2, (i,j) I a ij (z)p i q j + (k,l) J β kl p k q l,

56 For the proof, the auxiliary function W is taken to be of the form W (z) = qn+1 N pm+1 M (i,j) I a ij (z)p i q j + for some suitable set J N 2, such that W satisfies (k,l) J W = P(z, q, p)w + Q(z, q, p) + d R(z, q, p), dz β kl p k q l, where P, Q and R are certain rational functions in q and p. Also, the existence of the formal series solutions needs to be used here.

57 For the proof, the auxiliary function W is taken to be of the form W (z) = qn+1 N pm+1 M (i,j) I a ij (z)p i q j + for some suitable set J N 2, such that W satisfies (k,l) J W = P(z, q, p)w + Q(z, q, p) + d R(z, q, p), dz β kl p k q l, where P, Q and R are certain rational functions in q and p. Also, the existence of the formal series solutions needs to be used here. This first-order linear differential equation for W can be integrated ( ) W (z) = R(z) + I (z) C + (Q(ζ) + P(ζ)R(ζ))I (ζ) 1 dζ Γ ( ) I (z) = exp P(ζ)dζ, C = W (z 0 ) R(z 0 ), Γ Γ being a path of integration from z 0 to z.

58 By a certain modification of the curve Γ one can achieve that 1 q and 1 p, as well as certain terms of the form pk occuring in P, Q, R, are bounded q l in a neighbourhood of a movable singularity z.

59 By a certain modification of the curve Γ one can achieve that 1 q and 1 p, as well as certain terms of the form pk occuring in P, Q, R, are bounded q l in a neighbourhood of a movable singularity z. Using W, one can define variables u, v by p = u (N 1)/d, W (z, q, p) = v + F (z, u, v).

60 By a certain modification of the curve Γ one can achieve that 1 q and 1 p, as well as certain terms of the form pk occuring in P, Q, R, are bounded q l in a neighbourhood of a movable singularity z. Using W, one can define variables u, v by p = u (N 1)/d, W (z, q, p) = v + F (z, u, v). Changing the role of dependent and independent variable one finds a regular initial value problem dz du = u(mn 1)/d 1 A(z, u, v), dv = B(z, u, v), du showing that z is analytic in a neighbourhood of u = 0 with initial value z(0) = z and z z u (MN 1)/d. Transforming the variables back to p, q shows that z is an algebraic branch point of the solution.

61 An example M = N = 2: The Hamiltonian is given by H(z, q, p) = 1 3 q p3 + α 1,1 (z)qp + α 1,0 (z)q + α 0,1 (z)p The existence of formal series solutions q(z) = c 1,k (z ẑ) k, p(z) = k= 1 c 2,k (z ẑ) k k= 1 is equivalent to the resonance conditions α 1,1 0, α 1,0 0 and 0, i.e. one is essentially left with α 0,1 q = p 2 + zq + β, p = q 2 zp γ. (6) Theorem (A) shows that for system (6) all movable singularities of a solution (q, p) are poles, i.e. it has the Painlevé property. In fact, the solutions can be expressed in terms of the P IV transcendents.

62 Local vs. global branching We have presented a class of equations for which all solutions are locally finite-valued.

63 Local vs. global branching We have presented a class of equations for which all solutions are locally finite-valued. Apart from the meromorphic (single-valued) case, these equations are in general not integrable

64 Local vs. global branching We have presented a class of equations for which all solutions are locally finite-valued. Apart from the meromorphic (single-valued) case, these equations are in general not integrable The possibility of accumulation points of movable branch points, reached by analytic continuation about an inifinite length curve, cannot be excluded, see an example by R.A. Smith (1953): y + 4y 3 y + y = 0.

65 Local vs. global branching We have presented a class of equations for which all solutions are locally finite-valued. Apart from the meromorphic (single-valued) case, these equations are in general not integrable The possibility of accumulation points of movable branch points, reached by analytic continuation about an inifinite length curve, cannot be excluded, see an example by R.A. Smith (1953): y + 4y 3 y + y = 0. A condition for the equations to be integrable would be to demand that the solutions are also globally finite-valued. This leads to the notion of algebroid solutions, i.e. y satisfies an algebraic equation y n + s 1 (z)y n s n (z) = 0, (7) where s 1,..., s n are meromorphic functions.

66 Malmquist s Theorem In the case of a single first-order equation the following is known: Theorem (Malmquist 1913) Suppose there exists a transcendental algebroid solution for the equation y = P(z, y), P, Q C(z)[X ]. (8) Q(z, y) Then, by a rational transformation w = R(z, y), the equation can be reduced to a Riccati equation in w with rational coefficients, w = a(z)w 2 + b(z)w + c(z).

67 Malmquist s Theorem In the case of a single first-order equation the following is known: Theorem (Malmquist 1913) Suppose there exists a transcendental algebroid solution for the equation y = P(z, y), P, Q C(z)[X ]. (8) Q(z, y) Then, by a rational transformation w = R(z, y), the equation can be reduced to a Riccati equation in w with rational coefficients, w = a(z)w 2 + b(z)w + c(z). Remark: Usually Malmquist s Theorem is quoted as the following: If equation (8) has a transcendental meromorphic solution, then it must be already a Riccati equation.

68 Admissible solutions Employing Nevanlinna Theory, Malmquist s Theorem can be generalised to the notion of admissible solutions (K. Yosida, 1932). We adopt the usual notation for the Nevanlinna functions T (r, f ) = m(r, f ) + N(r, f ).

69 Admissible solutions Employing Nevanlinna Theory, Malmquist s Theorem can be generalised to the notion of admissible solutions (K. Yosida, 1932). We adopt the usual notation for the Nevanlinna functions T (r, f ) = m(r, f ) + N(r, f ). A meromorphic function that behaves like o(t (r, f )) as r, possibly outside some exceptional set of finite measure, is denoted by S(r, f ).

70 Admissible solutions Employing Nevanlinna Theory, Malmquist s Theorem can be generalised to the notion of admissible solutions (K. Yosida, 1932). We adopt the usual notation for the Nevanlinna functions T (r, f ) = m(r, f ) + N(r, f ). A meromorphic function that behaves like o(t (r, f )) as r, possibly outside some exceptional set of finite measure, is denoted by S(r, f ). We have the Lemma on the Logarithmic Derivative m (r, f ) = S(r, f ). f

71 Admissible solutions Employing Nevanlinna Theory, Malmquist s Theorem can be generalised to the notion of admissible solutions (K. Yosida, 1932). We adopt the usual notation for the Nevanlinna functions T (r, f ) = m(r, f ) + N(r, f ). A meromorphic function that behaves like o(t (r, f )) as r, possibly outside some exceptional set of finite measure, is denoted by S(r, f ). We have the Lemma on the Logarithmic Derivative m (r, f ) = S(r, f ). f Let y(z) be an algebroid solution of the differential equation F (z; y, y, y,..., y (k) ) = 0, with coefficients {a λ, λ I }. y is called admissible if for some j {1,..., n}, T (r, a λ ) = S(r, s j ) λ I.

72 Algebroid solutions of 2 nd -order equations Theorem (TK) Let y be an admissible solution of the equation y = 3 4 y 5 + a 4 (z)y 4 + a 3 (z)y 3 + a 2 (z)y 2 + a 1 (z)y + a 0 (z), (9) such that y also satisfies the irreducible equation y(z) 2 + s 1 (z)y(z) + s 2 (z) = 0. Then s 1 is proportional to a 4 and s 2 satisfies an equation which either reduces to a Riccati equation with coefficient functions that are rational expressions in a 0,..., a 4 and their derivatives, or to the equation s 2 = (s 2 )2 2s s (az + b)s ((az + b) 2 c)s 2.

73 At a singularity z 0 of y where a 0,..., a 4 are analytic, we have y 1, y 2 (z z 0 ) 1/2

74 At a singularity z 0 of y where a 0,..., a 4 are analytic, we have y 1, y 2 (z z 0 ) 1/2 From the differential equation, using the Lemma on the Logarithmic Derivative, one can obtain m(r, s j ) = S(r, y), j = 1, 2.

75 At a singularity z 0 of y where a 0,..., a 4 are analytic, we have y 1, y 2 (z z 0 ) 1/2 From the differential equation, using the Lemma on the Logarithmic Derivative, one can obtain m(r, s j ) = S(r, y), j = 1, 2. Since s 1 and s 2 are meromorphic and s 1 = (y 1 + y 2 ), s 2 = y 1 y 2, we see that s 1 has no pole at any such z 0, so N(r, s 1 ) = S(r, y), but therefore s 2 must have poles such that N(r, s 2 ) = O(T (r, y)).

76 Differentiating y 2 + s 1 y + s 2 = 0 yields 2yy + s 1y + s 1 y + s 2 = 0 = y = s 1 y + s 2 2y + s 1. (10)

77 Differentiating y 2 + s 1 y + s 2 = 0 yields 2yy + s 1y + s 1 y + s 2 = 0 = y = s 1 y + s 2 2y + s 1. (10) Differentiating again, using (9) to replace y and (10) to replace y and expanding by (2y + s 1 ) 2 yields an equation in y, s 1, s 1, s 1, s 2, s 2, s 2. Using y 2 = s 1 y s 2 repeatedly reduces the order of y until one ends up with F 1 (s 1, s 1, s 1, s 2, s 2, s 2 )y + F 0 (s 1, s 1, s 1, s 2, s 2, s 2 ) = 0.

78 Differentiating y 2 + s 1 y + s 2 = 0 yields 2yy + s 1y + s 1 y + s 2 = 0 = y = s 1 y + s 2 2y + s 1. (10) Differentiating again, using (9) to replace y and (10) to replace y and expanding by (2y + s 1 ) 2 yields an equation in y, s 1, s 1, s 1, s 2, s 2, s 2. Using y 2 = s 1 y s 2 repeatedly reduces the order of y until one ends up with F 1 (s 1, s 1, s 1, s 2, s 2, s 2 )y + F 0 (s 1, s 1, s 1, s 2, s 2, s 2 ) = 0. Since the equation y 2 + s 1 y + s 2 = 0 was assumed to be irreducible, we have F 0 = F 1 0. The relation F 1 0 yields the equation s 1 s a 4 s 4 1 a 3 s a 2 s 2 1 a 1 s 1 + 2a 0 =s 2 (5s 3 1 4a 4 s a 3 s 1 + 2a 2 ) + s 2 2 (2a 4 5s 1 )

79 Differentiating y 2 + s 1 y + s 2 = 0 yields 2yy + s 1y + s 1 y + s 2 = 0 = y = s 1 y + s 2 2y + s 1. (10) Differentiating again, using (9) to replace y and (10) to replace y and expanding by (2y + s 1 ) 2 yields an equation in y, s 1, s 1, s 1, s 2, s 2, s 2. Using y 2 = s 1 y s 2 repeatedly reduces the order of y until one ends up with F 1 (s 1, s 1, s 1, s 2, s 2, s 2 )y + F 0 (s 1, s 1, s 1, s 2, s 2, s 2 ) = 0. Since the equation y 2 + s 1 y + s 2 = 0 was assumed to be irreducible, we have F 0 = F 1 0. The relation F 1 0 yields the equation s 1 s a 4 s 4 1 a 3 s a 2 s 2 1 a 1 s 1 + 2a 0 =s 2 (5s 3 1 4a 4 s a 3 s 1 + 2a 2 ) + s 2 2 (2a 4 5s 1 ) Since T (r, s 1 ) = S(r, s 2 ), the left and right hand sides must both vanish identically, showing s 1 = 2 5 a 4.

80 Letting a 4 = s 1 = 0, one obtains from the relation F 0 0 the equation s 2 = (s 2 )2 2s s3 2 2a 3 (z)s a 1 (z)s 2, for which s 2 is a meromorphic solution.

81 Letting a 4 = s 1 = 0, one obtains from the relation F 0 0 the equation s 2 = (s 2 )2 2s s3 2 2a 3 (z)s a 1 (z)s 2, for which s 2 is a meromorphic solution. At any pole z 0, we find s 2 (z) = α z z 0 + c k (z z 0 ) k, α = ±1. k=0

82 Letting a 4 = s 1 = 0, one obtains from the relation F 0 0 the equation s 2 = (s 2 )2 2s s3 2 2a 3 (z)s a 1 (z)s 2, for which s 2 is a meromorphic solution. At any pole z 0, we find s 2 (z) = α z z 0 + c k (z z 0 ) k, α = ±1. k=0 Computing the coefficients c k, k = 0, 1,... recursively, one finds, at k = 2, the condition αa 3 (z 0 ) + a 3 (z 0 )a 3(z 0 ) 2a 1(z 0 ) = 0.

83 We have two cases: Case 1: Only one type of pole, s 2 α z z 0 occurs with frequency of order O(T (r, s 2 )), the other, s 2 α z z 0, with frequency S(r, s 2 ). Let w(z) = s 2(z) + αs 2 (z) 2 αa 3 (z)s 2 (z), (11) which has no pole if at z 0 if s 2 α z z 0, i.e. N(r, w) = S(r, s 2 ). Since also m(r, w) = S(r, s 2 ), we have T (r, w) = S(r, s 2 ), i.e. s 2 satisfies the Riccati equation (11) admissibly.

84 We have two cases: Case 1: Only one type of pole, s 2 α z z 0 occurs with frequency of order O(T (r, s 2 )), the other, s 2 α z z 0, with frequency S(r, s 2 ). Let w(z) = s 2(z) + αs 2 (z) 2 αa 3 (z)s 2 (z), (11) which has no pole if at z 0 if s 2 α z z 0, i.e. N(r, w) = S(r, s 2 ). Since also m(r, w) = S(r, s 2 ), we have T (r, w) = S(r, s 2 ), i.e. s 2 satisfies the Riccati equation (11) admissibly. Case 2: Both types of poles, α = ±1, occur with frequency of order O(T (r, s 2 )). Then one finds from the resonance conditions a 3 0, (a 2 3 4a 1 ) 0. Letting a 3 (z) = 2(az + b) and a 1 (z) = (az + b) 2 c one obtains s 2 = (s 2 )2 2s s (az + b)s ((az + b) 2 c)s 2.