Limits in differential fields of holomorphic germs

Limits in differential fields of holomorphic germs D. Gokhman Division of Mathematics, Computer Science and Statistics University of Texas at San Antonio

Complex Variables 28:27–36 (1995) c 1995 Gordon and Breach Science Publishers S.A.

Abstract Differential fields of germs of continuous real valued functions of one real variable (Hardy fields) have the property that all elements have limits in the extended real numbers and thus have a canonical valuation. For differential fields of holomorphic germs this is not generally the case. We provide a criterion for differential fields of holomorphic germs for its elements to have uniform limits in a partial neighborhood of infinity as an extended complex number. We apply the criterion to the specific case of a differential field of germs generated by the solutions of the Riccati equation W 0 +W 2 = e2z and extend the asymptotic validity of the usual series for the solutions from the positive real axis to a region in the complex plane.

1

Introduction

A proper differential algebraic framework for the asymptotic analysis of functions of a real variable is provided by the theory of Hardy fields — differential fields of germs. We refer to [7] and [2] for an introduction to the subject. One of the principal applications of Hardy field theory is the asymptotic theory of algebraic differential and difference equations. For example, given an algebraic differential equation whose coefficients belong to a Hardy field, one tries to extend the Hardy field to include the solutions as well. Singer’s theorem (see [7]) gives a positive answer to this problem for first order algebraic differential equations with coefficients in a Hardy field. There are theorems of this type for second order algebraic differential equations and some difference equations as well. As a consequence of being in the framework of a Hardy field, the usual formal operations (e.g. with power series) acquire an asymptotic meaning. 1

Limits in differential fields of holomorphic germs

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In [7] the reader can find an explicit illustration of this technique for the Riccati equation W 0 + W 2 = F 2 where F → ∞ and its germ belongs to a Hardy field. Part of the trick lies in the fact that elements of a Hardy field H can be characterized by the following non-oscillation property (also known as regular growth): [f ] ∈ H if and only if for any differential polynomial P , the function P (f, f 0 , f 00 , ...) ultimately has definite sign. As a consequence, for each germ [f ] ∈ H, lim f exists as an extended real number, so we have a canonical differential valuation (see [7]). The situation with complex functions is far less clear. In particular, we cannot even always make the last assertion about the existence of limits, which is essential to asymptotic analysis. However, there are indications that some of the Hardy field theory may be applicable (see also [6]). In particular, in some cases, it is possible to establish an analogue of regular growth for solutions of differential equations (see [4]), thus completing the crucial step of extending the differential field of germs spanned by the coefficients of the equation to include the solutions. With this motivation in mind, we shall introduce the notion of a complex Hardy field as a differential field of complex germs on a suitable domain. The natural question then arises, whether a complex Hardy field has a canonical valuation. Some light on the existence of limits is shed by the theory of value distribution near an essential singularity. In particular, E. Lindel¨of’s theorem [5] provides a criterion for the existence of uniform limits in sectors. Specifically, if a holomorphic function omits at least two values in an open sector and has a limit along a ray of that sector, then it has a uniform limit in any closed subsector. Conformal mappings can be used to generalize the type of region that the theorem applies to. We shall see that functions representing elements of a complex Hardy field will satisfy the first hypothesis. If the domain of definition of the complex Hardy field ultimately contains the real axis, then we can apply real Hardy field theory to the real and imaginary parts of the germ and, thus, satisfy the second hypothesis along the real axis. We illustrate this approach by applying it to the Riccati equation W 0 + W 2 = e2z .

(1)

We refer to [3] for the existence of solutions to such an equation in regions (horizontal strips) of the form D = {x + iy: x > x0 , y1 < y < y2 }, where x0 is some large real number and − π2 + nπ < y1 < y2 < π2 + (n + 1)π, n ∈ Z. We

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refer to [4] for the proof that solutions of (1) have regular growth in partial neighborhoods of infinity D defined as follows. Definition 1.1 Let D be a region in the complex plane bounded above and x below by curves of the form ±KD e−x−2D e , where KD and D are positive constants, and bounded on the left by a straight line x = x0 , where x0 is a large real number. Note that such regions are contained in any horizontal strip symmetric about the real axis, if x0 is chosen sufficiently large. The differential field C (ez ) generated by the coefficients of (1) is a complex Hardy field with a canonical valuation in certain horizontal strips. The regular growth of solutions W to (1) in regions D as in Definition 1.1 implies that the complex Hardy field C (ez ) can be extended to include the solutions to (1). Specifically, the differential field C (ez , W ) is represented by germs in regions where W has regular growth. We shall show that we can extend the canonical valuation on C (ez ) to the complex Hardy field extension C (ez , W ) in a region D as in Definition 1.1. The existence of the canonical valuation for C (ez , W ) gives asymptotic meaning to the usual formal calculations in a manner similar to sec. 3 of [7]. Thus, we extend the asymptotic validity of the formal asymptotic expansions W ∼ ±ez +

∞ X

ak e−kz

(2)

k=0

from the positive real axis to any partial neighborhood of infinity as in Definition 1.1.

2

About the domains

In [3] we considered domains near infinity with orthogonal curvilinear coordinates (p, q) D = {z ∈ C: p(z) ∈ [p0 , ∞), q(z) ∈ [q1 , q2 ]} Such domains can be thought of as partial neighborhoods of ∞. Typical examples are (i) a sector: (p, q) = (ρ, θ), ρ > ρ0 , θ ∈ (θ1 , θ2 ); (ii) a strip: (p, q) = (x, y), x > x0 , y ∈ (y1 , y2 );

Limits in differential fields of holomorphic germs r

(iii) a funnel shaped region with p =



y 2 ln y −

1 2



4

+ x2 , q = − lnxy .

In [4] we considered domains as in Definition 1.1. These regions are “smaller” than those of type (iii) and will be of particular importance here. Curvilinear coordinates (p, q) determine a directed neighborhood system n o 1 Vδ = z ∈ C: |p(z)| < δ and with it the following notions: Definition 2.1 Limit of a function as z approaches ∞ in a domain D: limD f (z) = L ∈ C means that for any  > 0 there is a δ > 0 such that |f (z) − L| <  if z ∈ Vδ ; limD f (z) = ∞ means that for any  > 0 there is a 1 δ > 0 such that |f (z)| <  if z ∈ Vδ . Definition 2.2 A germ of a continuous function on a domain D is an equivalence class of functions each defined on some Vδ , where two functions are considered equivalent if they agree on some Vδ . A function belongs to its own equivalence class (germ). Here, as in most literature, the abuse of notation, by which a function is identified with its germ, will be perpetuated. Definition 2.3 A property is said to hold ultimately in D, when in holds on some Vδ . The above notions are sufficient to develop a general asymptotic theory, so we shall consider domains with neighborhood systems when dealing with differential fields of germs. An important special case are regions as in Definition 1.1 with cartesian coordinates. In this case limD is just uniform limit with respect to y as x tends to infinity and a property holds ultimately when it holds for all sufficiently large x.

3

Complex Hardy fields

We start by introducing the notion of a complex Hardy field. This notion is somewhat tentative, since not all of the usual properties of real Hardy fields carry over. We provide some examples and deal with the relationship of complex Hardy fields to the notion of regular growth introduced in [4]. Definition 3.1 Let D be a domain in the complex plane with a directed neighborhood system at ∞. A complex Hardy field is a differential field H, whose elements are germs with respect to the neighborhood system,

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represented by complex valued functions defined on D such that each such function is ultimately continuous on D, the field operations in H correspond to pointwise operations on functions and the derivation on H is ordinary complex differentiation. Clearly a complex Hardy field on a partial neighborhood of ∞ is still a complex Hardy field on an ultimately smaller partial neighborhood of infinity. If H is a complex Hardy field on a domain D ⊂ C and f is a complex valued function on D that represents an element of H (we loosely write f ∈ H), then f is ultimately differentiable (holomorphic) on the interior of D. If f is not identically zero, then since every nonzero element of a field is a unit, f1 ∈ H. This means that ultimately f has no zeros or poles in D. Conversely we can state a self-evident criterion for a set of complex valued functions to be contained in a complex Hardy field. Lemma 3.1 A differential ring R of holomorphic germs on a partial neighborhood D of infinity is an integral domain whose field of quotients is a complex Hardy field on D, if each element of R is ultimately zero free on D or identically zero. The following is a complex analogue of a characterization of Hardy field extensions (cf. Definition 6.2 in [1]). A function has regular growth in the sense of [4] over some complex Hardy field H in some partial neighborhood D of infinity if, and only if, it is contained in a complex Hardy field extension of H on D. Proposition 3.1 Suppose D is domain, H is a complex Hardy field on D and f represents a germ on D. Then there exists a complex Hardy field extension H1 of H such that [f ] ∈ H1 if, and only if, for any differential polynomial P with coefficients in H, the function P (f, f 0 , f 00 , ...) is ultimately zero free in D. Proof: If f ∈ H1 , then 1/P (f, f 0 , f 00 , ...) ∈ H1 , so P (f, f 0 , f 00 , ...) is ultimately zero free. Conversely if P (f, f 0 , f 00 , ...) is ultimately zero free, by the lemma H [f, f 0 , f 00 , ...] is an integral domain and we let H1 = H (f, f 0 , f 00 , ...). Motivated by the fact that existence of definite limits (finite or infinite) for elements of a real Hardy field is crucial to asymptotic analysis, we introduce the following definition.

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Definition 3.2 A valued complex Hardy field is a complex Hardy field H, such that each element of H has a definite uniform limit (as an element of the Riemann sphere) as we approach infinity in D. The criteria for a complex Hardy field to be a valued complex Hardy field are discussed in the next section. The result below sheds some light on the origin of the definition. Proposition 3.2 Suppose H is a valued complex Hardy field on a partial neighborhood D of infinity. Then H admits a canonical valuation corresponding to the valuation subring R of germs [f ] on D for which the limit of f is finite. Proof: The well known theorems on limits imply that limD (f g) = limD f limD g and limD (f + g) = limD f + limD g so R is indeed a subring. Now let f ∈ H. If f 6∈ R, i.e. f → ∞, then 1/f → 0 ∈ C, so 1/f ∈ R. Thus R is a valuation subring of K. This is the usual correspondence between places and valuations. We define what it means for two functions to be asymptotic by saying that f ∼ g exactly when ν(f − g) > ν(g). It is easily seen that in a complex Hardy field we have f ∼ g ⇔ limD f /g = 1. We pause for two important examples. The first is a classical example where the valuation is discrete and, in fact, directly related to the familiar notion of degree of a rational function. The second deals with the field generated by the coefficient of the Riccati equation that we are considering. Example 3.1 The field of rational functions C(z) is a complex Hardy field in the entire punctured neighborhood of ∞, i.e. we may take D = C. Here the valuation is just minus the degree map on the rational functions. Note that l’Hospital’s rule is satisfied with respect to formal differentiation, so this is actually a differential valuation. Indeed, let a, b ∈ C(z) such that ν(a) ≥ 0 and ν(b) > 0, i.e. deg a ≤ 0 and deg b < 0. If b = p/q, where p, q ∈ C[z], then b0 /b = (p0 q − qp0 )/(pq). Since deg b 6= 0, we have deg (b0 /b) = −1, so deg (a0 b/b0 ) = deg a. If deg (a0 b/b0 ) < 0, we are done. If deg a = 0, then a = c+r/s, where c is a constant and r, s are polynomials with deg r < deg s. If we replace a with a − c, the derivative stays the same, but we can apply the previous case. Example 3.2 The field C (ez ) is a complex Hardy field on a horizontal strip 

π π D = (x, y): x0 < x < ∞, − < y < 2 2



.

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Indeed, since the derivative of ez is itself, we have a differential field. Each element f of this field can be represented by a rational function. For any k ∈ Z, the uniform limit of ekz exists. Depending on whether k is positive, negative, or zero, the limit is infinite, zero, or 1 respectively. For example if k is positive, the set where the real part of kz is positive consists of sectors of angle πk one of which contains the positive real axis and is symmetric with respect to it. Such a sector contains the strip for sufficiently large x. The above is still true if we take a slightly smaller closed subsector. This shows that the real part of kz is ultimately bounded away from zero in D, so the limit is uniform. We can extend this idea to a rational function by applying the division algorithm to it or its multiplicative inverse and extracting the leading term. The function is asymptotic to the leading term, so is ultimately zero free in D, and has a uniform limit. Thus our field is a complex Hardy field. In the next section we shall see that it is actually a valued complex Hardy field.

4

Existence of uniform limits

In this section we establish a criterion for the existence of valuation on a complex Hardy field. We use a result from complex analysis, a theorem of E. Lindel¨of, which is a refinement of Picard’s Big Theorem for sectors. A statement of the theorem with a proof by P. Montel can be found in sec. 15 of [5] (Theorem 15.4.4). The theorem is quoted here for convenience. Theorem (E. Lindel¨of) Let f (z) be holomorphic in the sector S : α < arg z < β, 0 < |z| < R, and suppose that f (z) omits two values when z is restricted to S. Suppose further that there is a γ such that α < γ < β and limr→0 f reiγ ≡ c exists where c may be finite or infinite. Then 



limr→0 f reiθ = c for α < θ < β, uniformly in any fixed interior sector. The only property of an essential singularity used in P. Montel’s proof of Picard’s Big Theorem using normal families is that limits, as one approaches the singularity, are not uniform. In the case of a sector we see that unless we do have uniform limits, then all but two values are attained in any neighborhood of the singularity (intersected with the sector). In particular, we may find a value that is attained infinitely often for a sequence of points converging to the vertex in the sector. Theorem 4.1 Suppose R > 0 and n

o

D = z = ρeiθ : ρ ∈ (0, R), θ ∈ [α, β]

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is a sector and F is a function holomorphic on D. If F represents an element of a complex Hardy field on D and the limit of F at zero exists along a single ray in the sector, then the uniform (with respect to θ) limit of F at zero exists as an element of C ∪ {∞}. Proof: Suppose the contrary. Then by Lindel¨ of’s theorem there is a value ω that is attained infinitely often for a sequence of points converging to the vertex in the sector. Then the germ F − ω is not invertible, so cannot belong to a differential field of germs. While Lindel¨of’s theorem is stated for a sector, we can generalize to other regions by the use of conformal maps. In particular, the application of − log(z) transforms a sector to a horizontal strip in the right half plane.


Lemma 4.1 Let α ∈ 0, π2 , r0 ∈ (0, 1) and n o

D = ζ = reiθ : 0 < r < r0 , −α < θ < α . Then the conformal map z = − log ζ takes D into D1 = {z = x + iy: x > − log r0 , −α < y < α}. Furthermore, under this mapping the neighborhood system at zero in D given by polar coordinates is carried to the neighborhood system at infinity in D1 given by cartesian coordinates.

Proof: Let ζ = reiθ ∈ D, then z = − log ζ = − log r − iθ. This implies the first conclusion of the lemma. By varying r0 we get the same result for each neighborhood. A further application of log(z) gives a funnel shaped region.


Lemma 4.2 Let C ∈ 0, π2 , ξ0 > 0 and D1 = ζ = ξ + iηξ > ξ0 , −C < η < C. Then the conformal map z = log ζ takes D1 into a funnel shaped region D2 bounded by the curves 

y = ± arctan √



C , e2x − C 2

Furthermore, under this mapping D1 given by cartesian coordinates infinity in D2 given by curvilinear by ! q y = arctan p 2x , e − q2

x = log ξ0 +

  1 log 1 + tan2 y . 2

the neighborhood system at infinity in is carried to a neighborhood system at coordinates (p, q) implicitly determined x = log p +

  1 log 1 + tan2 y . 2

9

Limits in differential fields of holomorphic germs

Proof: Let z = ξ + iη ∈ D1 , then z = log ζ = log

p

ξ 2 + η 2 − i arctan p

ξ2

 

C 2,

η ξ

.

First, we keep + we √ η = C constant and solve for ξ. Since x = log 2x 2 have ξ = ± e − C . Substitution of this expression into the formula for y gives the equations of the top and bottom boundary curves. Now we keep ξ = ξ0 constant and solve for η = ξ0 tan y. This gives the equation of the boundary curve on the left. Varying ξ0 and C gives the conclusion about the neighborhood systems. Note that since C < π2 , the range of x = x(p, q) on D2 is bounded for each fixed p and the bound is independent of p. Furthermore, y = arctan

p

q

e2x − q 2

!

∼ qe−x .

Note that, since tangent is asymptotic to the identity function for small values of the argument, the same is true for arctangent, so the boundary curves for the above funnel shaped region have asymptotically exponential decay. We now state the following corollary for a funnel shaped region with exactly exponential boundaries. This is used in turn in the next section to show that the complex Hardy field generated over C by ez and a solution W to the Riccati equation (1) is a valued complex Hardy field in regions as in Definition 1.1, whose boundaries decay faster than exponentials. Corollary 4.1.1 Let DC be a funnel shaped region around the positive real axis bounded by curves y = ±Ce−x and x = x0 , where C > 0. Suppose F is a function holomorphic on DC . If F represents an element of a complex Hardy field on D and the limit of F at zero exists along a the positive real axis, then the uniform (with respect to y) limit of F (z) exists as an element of C ∪ {∞} as z → ∞ in DC1 for any C1 < C. Proof: Let C1 < C. Let DC denote the funnel shaped region corresponding to C in the preceding lemma. Letting C2 ∈ (C1 , C) we obtain DC2 ⊂ DC , if in the lemma we choose ξ0 sufficiently large as we well may. By the preceding lemma we may apply Lindel¨of’s theorem in DC2 and, assuming the hypotheses for F , obtain a uniform limit for F in DC3 for any C3 ∈ (C1 , C2 ). However, if we now take x0 sufficiently large, we have DC1 ⊂ DC3 , so the uniform limit holds in DC1 . Note that since the range of y becomes small as x gets large, we have x ∼ log p uniformly with respect to y. Furthermore, the neighborhood system determined by the coordinates (p, q) as in the above lemma is equivalent to the one given by cartesian coordinates in the sense that each neighborhood of infinity in one system contains one in the other

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Limits in differential fields of holomorphic germs

system and vice versa. Thus, uniformity with respect to q is equivalent to uniformity with respect to y. We conclude this section with an important example dealing with the field of functions generated by ez , the coefficient in the Riccati equation under consideration. Example 4.1 As we have seen in Example 3.2, the field C (ez ) is a complex Hardy field on a horizontal strip 

π π D = (x, y): x0 < x < ∞, − < y < 2 2



.

In fact, C (ez ) is a valued complex Hardy field. We can see this by showing that each element of C (ez ) has a limit as z → ∞ along the positive real (ez ) z axis and applying Lindel¨ of’s theorem. Indeed, let PQ(e z ) ∈ C (e ), where P and Q are polynomials in one variable. Multiplying the numerator and the denominator by the complex conjugate of the denominator and taking real and imaginary parts we obtain on the real axis 







Im P (ex )Q(ex ) Re P (ex )Q(ex ) P (ez ) +i . = Q(ez ) |Q(ex )|2 |Q(ex )|2 The real and imaginary parts are rational functions of ex , so they possess limits (finite or infinite) by the classical Hardy field theory. Thus, the limit (ez ) of PQ(e z ) exists as an element of the Riemann sphere as z → ∞ along the positive real axis and coincides with the uniform limit of substrip of D.

5

P (ez ) Q(ez )

in any closed

Field extension and the validity of asymptotic formulas

In this section we extend the validity of an asymptotic expansion for a solution of the Riccati equation W 0 + W 2 = F 2 , where F = ez , from the positive real axis to regions as in Definition 1.1. In general, asymptotic analysis of a differential equation consists of finding approximate solutions, showing the existence of solutions asymptotic to the approximate solutions and refining the approximation by showing the asymptotic validity of formal series representations for the solutions. This is a well traveled path, especially for meromorphic differential equations and equations with coefficients asymptotic to products of iterated logarithms. In [3, 4] and in this paper we have considered the Riccati equation

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W 0 + W 2 = F 2 in the complex domain, where the coefficient F grows faster than rational functions of z. In this paper we take the case F = ez to its logical conclusion. The Riccati equation W 0 + W 2 = F 2 has F and −F as approximate solutions. The existence of solutions asymptotic to ±F has been shown in [3] for a class of functions F in various regions of the complex plane satisfying certain criteria. In one case (F= ez ) the solution is asymptotic to F uniformly in any closed substrip of (x, y): x > x0 , − π2 < y < π2 (see [3]). We can obtain a formal series expansion in powers of F less than or equal to 1, for example by substitution of a general formal power series into the differential equation and comparing terms to obtain recursion relations for the coefficients (Frobenius method). In [7] it is shown that the formal expansion is actually an asymptotic expansion on the positive real axis for all F → ∞ in a real Hardy field. This can be seen by using Singer’s theorem to extend the Hardy field generated by F to include the solution of the Riccati equation and by performing the formal operations in the framework of the larger Hardy field, thereby endowing them with asymptotic validity. In fact, these operations are performed explicitly in [7]. In this section we will apply the same strategy in a region in complex plane, as in Definition 1.1, which contains the positive real axis, for F = ez . This will allow us to extend the asymptotic validity of the series from the positive real axis into the complex plane, albeit not very far (not even as far as horizontal strips, where the solution is asymptotic to F = ez two paragraphs above). The result follows from the fact that C (ez , W ), where W is our solution, is a valued complex Hardy field in regions as in Definition 1.1. The remainder of the section is devoted to establishing this result. We start with the valued complex Hardy field C (ez ) on a horizontal strip   π π D = z = x + iy: x0 < x < ∞, − < y < . 2 2 endowed with cartesian coordinates (see Example 3.2). The coordinates provide a directed neighborhood system at ∞, so it is clear what we mean by a uniform approach to ∞ in D. Each element of C (ez ) is represented by a holomorphic function on D which possesses a uniform limit (finite or infinite) in D. We have shown in [4] that a solution W ∼ F satisfies a certain nonoscillation property, namely that given a differential polynomial P over C (ez ), the function P (W ) is ultimately zero free. This is shown for regions as in Definition 1.1.

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The non-oscillation criterion is precisely what is needed for C (ez , W ), where W is a solution as above, to be a complex Hardy field in the given region. Since we need the existence of limits to prove asymptotic validity of the formal expansion, we show that for a region of type (1.1) C (ez , W ) is actually a valued complex Hardy field. Thus, we have the following theorem (a complex analogue to Theorem 5 in [7]). Theorem 5.1 Let F = ez . Any solution of the differential equation W 0 + W 2 = F 2 is asymptotic to F or −F and for solutions W ∼ F , C (ez , W ) is a valued complex Hardy field on regions D as in Definition 1.1. Furthermore, formal substitution of a series in F −k , where k = −1, 0, 1, 2, ... and the coefficient corresponding to k = −1 is assumed to be 1, yields an asymptotic series representation for W valid uniformly with respect to y in D. Proof: Since W 0 is a polynomial in W and F , we see that C (ez , W ) is closed under differentiation. Every element of C (ez , W ) is a polynomial in W and the non-oscillation property proved in [4] shows that it is ultimately zero free in D. Thus, we have a complex Hardy field. The existence of limits on the positive real axis is a consequence of the usual real variable Hardy field theory. The function F = ez is real valued on the positive real axis and we can choose a solution W with the same property. Each element of C (ez , W ) can be written as a rational function in ez and W . There may be complex constant coefficients in the polynomials, but we can rationalize (multiply the numerator and the denominator by the complex conjugate of the denominator), take real and imaginary parts and deal with them separately similarly to what was done in Example 3.2. Corollary 4.1.1 implies the existence of uniform limits in funnel shaped regions which ultimately contain D, so C (ez , W ) is a valued complex Hardy field in D and in light of the discussion preceding the theorem the validity of asymptotic expansions is thereby extended from the positive real axis to D.

6

Acknowledgments

I am indebted to Maxwell Rosenlicht for introducing me to this subject, for his guidance and assistance. I am very grateful to all the people who have helped me with this project, especially William C. Jagy.

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References [1] M. Boshernitzan. An Extension of Hardy’s Class L of “Orders of Infinity”. J. d’Analyse Math., 39:235–255, 1981. [2] M. Boshernitzan. New “Orders of Infinity”. J. d’Analyse Math., 41:130– 167, 1982. [3] D. Gokhman. An asymptotic existence theorem in C for the Riccati equation. Complex Variables, 24:145–159, 1994. [4] D. Gokhman. Regular growth of solutions of the Riccati equation W 0 + W 2 = e2z in the complex plane. Complex Variables, 27:365–382, 1995. [5] E. Hille. Analytic function theory, volume II. Chelsea, New York, 1973. [6] M. Rosenlicht. Differential valuations. Pac. J. Math., 86(1), 1980. [7] M. Rosenlicht. Hardy fields. J. Math. Anal. Appl., 93:297–311, 1983.