Laplace Contour Integrals and Linear Differential Equations

The purpose of this paper is to determine the main properties of Laplace contour integrals Λ(z)=12πi∫Cϕ(t)e-ztdt\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Lambda (z)=\frac{1}{2\pi i}\int _\mathfrak {C}\phi (t)e^{-zt}\,dt \end{aligned}$$\end{document}that solve linear differential equations L[w](z):=w(n)+∑j=0n-1(aj+bjz)w(j)=0.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} L[w](z):=w^{(n)}+\sum _{j=0}^{n-1}(a_j+b_jz)w^{(j)}=0. \end{aligned}$$\end{document}This concerns, in particular, the order of growth, asymptotic expansions, the Phragmén–Lindelöf indicator, the distribution of zeros, the existence of sub-normal and polynomial solutions, and the corresponding Nevanlinna functions.


Introduction
Special functions usually admit several quite different definitions and representations.For example, Airy's equation w − zw = 0 has a particular solution, known as Airy function, that may be written as Laplace contour integral (1) Ai(z) = 1 2πi C e −zt+t 3 /3 dt; the contour C consists of the straight line from +∞ e iπ/3 to the origin followed by the straight line from 0 to +∞ e −iπ/3 .In [1] the authors G. Gundersen, J. Heittokangas and Z-T.Wen investigated two special families of linear differential equations, namely the three-term equations with the intention "to make a contribution to this topic which includes a generalization of the Airy integral [. ..] to have more examples of solutions of complex differential equations that have concrete properties".To this end they determined several contour integral solutions 1 2πi C F (z, w) dw with varying kernels F and contours C.Although [1] does not contain any hint how to find appropriate kernels and contours, not to mention appropriate (families of) differential equations like (2) and (3), the examples in [1] reveal the nature of the contour integrals: in any case they are equal or may be transformed into Laplace contour integrals of particular analytic functions φ over canonical contours or paths of integration C. We will prove that to each linear differential equations ( 5) (a j + b j z)w (j) = 0 (a 0 + b 0 z ≡ 0) there exists some distinguished Laplace contour integral solution (4) with kernel φ L that is uniquely determined by the operator L and itself determines canonical contours C. The main properties of these solutions -denoted Λ L -, which strongly remind on the Airy integral, will be revealed.This concerns, in particular, the order of growth, the deficiency of the value zero, asymptotic expansions in particular sectors, the distribution of zeros, the Phragmén-Lindelöf indicator, the Nevanlinna functions T (r, Λ L ) and N (r, 1/Λ L ), and the existence of sub-normal solutions.

Kernels and contours
The following reflections on linear differential equations (6) L[w] := n j=0 P j (z)w (j) = 0, are more or less part of mathematical folklore, but unfortunately not as well-known as they should be.One can find few remarks in Wasow [9], p. 123 ff., where the method of Laplace contour integrals and the saddle-point method is applied to Airy's equation.Hille [4], p. 216 ff., and Ince [5], chapter XVIII, deal with linear differential equations (6) under the constraint deg P n ≥ deg P j .This, however, is by no means necessary and even prevents the discussion of the most important case P n (z) ≡ 1, where the solutions are entire functions of finite order of growth.
We start with linear differential equations (6) with polynomial coefficients (7) and are looking for solutions that may be written as Laplace contour integrals (4); the function φ and thereby possible paths of integration C have to be determined.Formally we have and ( 8) denotes the variation of Ψ along C, when C starts at a and ends at b; of course, C may be closed.We demand ( 9) These calculations are justified if (9) holds and the integrals converge absolutely with respect to arc-length, and locally uniformly with respect to z.With ) may be written as and we have the following theorem, which forms the basis of our considerations.
2.1.Theorem.Let φ be any non-trivial solution to (12) and C any contour such that the integrals (8) converge absolutely with respect to arc-length of C and locally uniformly with respect to z, and also (9) holds.Then the Laplace contour integral (4) solves the differential equation (6).
Of course one has to check in each particular case that w is non-trivial.In most cases this is a corollary of the inverse formula and/or uniqueness theorems for the Laplace transformation.

Equations with coefficients of degree one
3.1.The kernel.The method of the previous section turns out to be most useful in the study of equation ( 5), in which case where b q = 0 and b j = 0 for j > q is assumed.By a linear change of the independent variable z we may achieve (13) b q = (−1) n−q+1 (and a q = 0 if desired); this will be assumed henceforth.Then up to an arbitrary constant factor.More precisely, (15) holds.Sometimes it suffices to know that φ has the form ( 16) where ψ is holomorphic on {t : |t| > R 0 , | arg t| < π} and satisfies Occasionally, any function (14) will denoted φ L .

Appropriate contours.
There is some freedom in the choice of the contour C, which is restricted only by condition (9).Closed simple contours work in particular cases only, namely when λ ν is an integer, and yield less interesting examples.More interesting are the contours C R,α,β that consist of three arcs as follows: 1. the line t = re iα , where r runs from +∞ to R ≥ 0; 2. the circular arc on |t| = R from Re iα via t = Re i(α+β)/2 to Re iβ ; 3. the line t = re iβ , where r runs from R to +∞.
In place of C 0,α,β , C R,α,−α , and C 0,α,−α we will also write C α,β , C R,α , and C α , respectively.Note that C R,α,−α passes through R, while C R,α,2π−α passes through −R.To ensure convergence of the contour integral and condition (9), the real part of t n−q+1 has to be negative on arg t = α and arg t = β; this will tacitely be assumed.A canonical and our preferred choice is α = π n−q+1 and β = −α.It is, however, almost obvious by Cauchy's theorem that the contour integral is independent of α, β, and R within their natural limitations.This will be proved later on.
3.3.The protagonist.By w = Λ L (z) we will denote any solution to equation (5) given by with associated function (14).Of course, Λ L is (up to a constant factor) uniquely determined by the operator L and differs from operator to operator, but its essential properties depend only on n − q.Hence the Airy function is a typical representant of the solutions Λ L with q = n − 2, where n may be arbitrarily large.
Since φ L has an essential singularity at t = 0, one may choose C : |t| = 1; it is, however, more interesting to choose C = C π n−k+1 if n−q is even; if n−q is odd, the differential equation does match our normalisation only after the change of variable z → −z.We note that the substitution (the conformal map) t = −1/u transforms the Laplace contour integral with kernel φ L up to sign into some integral in accordance with formula (5.2) in [1].
4. Asymptotic expansions and the order of growth which justifiably may be called characteristic equation.Remember that we assume b q = (−1) n−q+1 and b j = 0 for j > q.Also let p ≤ q denote the smallest index such that b p = 0.As z → ∞, equation ( 19) has solutions in any case (n − q odd or even), one By z 1/r and log z we mean the branches on | arg z| < π that are real on the positive real axis.If p = 1, polynomial solutions may exist, but not otherwise.The central index method for solutions to linear differential equations ( 6) is based on the relation where z r is any point on |z| = r such that |f (z r )| = M (r, f ).This holds with j (r) → 0 as r → ∞ with the possible exception of some set E j of finite logarithmic measure; for a proof see Wittich [10], chapter I, or Hayman [2].
hence any such v(r) is asymptotically correlated with some solution to the characteristic equation ( 19) as follows: as r → ∞ holds even without exceptional set by the regular behaviour of z r y j (z r ); since ν(r) is positive, z r y j (z r ) is asymptotically positive, this giving additional information on the possible values of arg z r .Thus the transcendental solutions to equation ( 5) have possible orders of growth = 1 + 1 n−q : solutions of this order always exist; = 1: necessary for solutions of this order to exist is p < q; = 1 − 1 p : necessary for solutions this order to exist is p > 1; = 0: necessary for polynomial solutions is p = 1.4.3.Asymptotic expansions.The differential equation ( 5) may be rewritten in the usual way as a first-order system (20) ) with n × n-matrices A and B. The following details can be found in Wasow's fundamental monograph [9].The system (20) has rank one, but the matrix B has vanishing eigenvalues.Thus the theory of asymptotic integration yields only a local result (Theorem 19.1.in [9]), which makes its applicability unpleasant and involved: to every angle θ there exists some sector S θ : | arg(ze −iθ )| < δ such that the system (20) has a distinguished fundamental matrix A is holomorphic on |z| > 0, the number δ, the constant n × n-matrix G, and the matrix Π = diag (Π 1 , . . ., Π n ), whose entries are polynomials in z 1/r for some positive integer r, are universal, that is, they do not depend on θ; V (z|θ) has an asymptotic expansion the latter means as z → ∞ on S θ , uniformly on every closed sub-sector; again the matrices V j are independent of θ.The fundamental matrices V, that is, the matrices V (z|θ) may vary from sector to sector.
[ 1 ] Returning to equation (5) this means that given θ, there exists a distinguished fundamental system • w 1 (z|θ), . . ., • w j (z|θ) = e Πj (z 1/r ) z ρj Ω j (z|θ) on S θ ; Π j is a polynomial in z 1/r , ρ j is some complex number, and Ω j is a polynomial in log z with coefficients that have asymptotic expansions in z 1/r on S θ .The triples (Π j , ρ j , Ω j ) are mutually distinct.Again this system may vary from sector to sector, but only in the coefficients of Ω j (z|θ)!The leading terms of the Π j can be found among those of the y ν dz.

4.4.
The Phragmén-Lindelöf indicator.The following facts can be found in Lewin's/Levin's monographs [6,7].Let f be an entire function of positive finite order such that log is called Phragmén-Lindelöf indicator or just indicator of f of order ; it is continuous and always assumed to be extended to the real axis as a 2π-periodic function.
1 Independence of the matrices V j and dependence of V (z|θ) on θ yields no contradiction.
Asymptotic series may represent different analytic functions on sectors.
Having h(ϑ) at hand, the Nevanlinna functions may be computed explicitly (for definitions and results we refer to Hayman [3] and Nevanlinna [8]): have to be considered.

Results on normal and sub-normal solutions
The Airy integral has an asymptotic representation 2. has Phragmén-Lindelöf indicator h(ϑ) = −2 3 cos 3 2 ϑ on |ϑ| < π; 3. has infinitely many zeros, all on the negative real axis.4. T (r, Ai) ∼ 8 9π r 3/2 and N (r, 1/Ai) ∼ 4 9π r 3/2 .In the general case one cannot expect results of this high precision, but the following theorem seems to be a good approximation.
5.1a.Suppose q = n − 1.Then either Λ L (z) = e −z 2 /2 P (z), P some non-trivial polynomial, or else the following is true: In the second case, Λ L again has 'many' zeros; they are distributed over arbitrarily small sectors about arg z = ± 3 4 π.The proof of Theorem 5.1 and 5.1a will be given in section 9. 5.2.Subnormal solutions.Solutions of maximal order = 1 + 1/(n − q) always exist, while the existence of so-called sub-normal solutions having order = 1, = 1 − 1/p, and = 0 (polynomials), respectively, remains open; necessary but by far not sufficient conditions are q > p, p > 1, and p = 1, respectively.Sufficient condition are coupled with the poles of Q 0 /Q 1 and their residues.We note that at t = 0, Q 0 /Q 1 either has a pole of order p ≥ 1 or is regular (p = 0).We have to distinguish two cases.
5.2a Theorem.Let t = 0 be a pole of Q 0 /Q 1 of order p ≥ 1 with integer residue λ 0 .Then the solution w(z) = res has order of growth = 1 − 1/p if t = 0 is an essential singularity of φ L , that is, if p > 1, and otherwise (p = 1) is either a non-trivial polynomial of degree λ 0 ≥ 0 or else vanishes identically (λ 0 < 0).

5.2b
Theorem.Let t = t 0 = 0 be a pole of Q 0 /Q 1 of order m with integer residue λ 0 .Then the solution where W is a transcendental entire function of order of growth at most 1 − 1/m if m > 1, and otherwise is a polynomial of degree λ 0 ≥ 0 or vanishes identically (λ 0 < 0).
Proof of 5.2a.First assume that t = 0 is a simple pole of holds, where H is holomorphic at t = 0 and H(0) = 0. Then (assuming λ 0 ≥ 0) is a polynomial of degree λ 0 .If, however t = 0 is essential for φ L , then p > 1 and holds as z → ∞.This yields (w) ≤ 1 − 1/p.Since w is not a polynomial, the only possibility is = 1 − 1/p.This proves part a.
To deal with 5.2b write Then by part a of the proof, either W is a polynomial of degree λ 0 ≥ 0 or vanishes identically (λ 0 < 0) or is a transcendental entire function of order of growth at most 1 − 1/m; the value zero is a Borel or even Picard exceptional value of w.
Remark.By w = e −t0z W (t 0 = 0), equation ( 5) is transformed into (26) (a j +b j z); in particular, B q = b q and A 0 +B 0 z = Q 0 (t 0 )+Q 1 (t 0 )z hold, hence q, b q , and the maximal order of growth 1+1/(n−q) are invariant under this transformation, but not necessarily the index p.If in Theorem 5.2b, t 0 is not a zero of Q 0 (t) and m > 1, then in (26) we have B m = 0, but B j = 0 for j < m, hence W has the order 1 − 1/m by part a.

Rotational symmetries
The functions Ai(ze −2νπi/3 ) also solve Airy's equation and coincide up to some non-zero factor with the Laplace contour integrals obviously the sum over the three integrals vanishes identically, while any two of these functions are linearly independent; this is well-known, and will be proved below in a more general context.In the general case there are two obvious obstacles: 1. Λ L (ze −2νπi/(n−q+1) ) is not necessarily a solution to L[w] = 0. Nevertheless one may consider the contour integral solutions and their Phragmén-Lindelöf indicators h ν for 0 ≤ ν ≤ n − q, but has to take care since φ L may be many-valued.The next theorem is just a reformulation of Theorem 5.1 and 5.1a.We assume the indicator h 0 of Λ 0 = Λ L be extended to the real line as a 2π-periodic function.Although Λ ν (z) is different from Λ L (ze −2νπi/(n−q) ) in general, both functions share their main properties as are stated in Theorem 5.1 and 5.1a for Λ L .
If φ L has no singularities, the sum vanishes identically.On the other hand, poles t k of Q 0 /Q 1 are poles or essential singularities of φ L ; if these singularities are non-critical, that is, if the residues are integers, the following holds.
6.2.Theorem.Suppose the residues λ k are integers.Then the sum Λ either vanishes identically or may be written as a linear combination of sub-normal solutions W k has order of growth less than one.
2. φ L may be many-valued on C \ { poles of Q 0 /Q 1 }.This is the case if some of the residues λ k are not integers.Nevertheless φ L may be single-valued on |t| > R 0 : if t goes around once the positively oriented circle k λ k is an integer.Assuming this, Λ is a sub-normal solution of order at most one.This follows from (28) and We have thus proved 6.3.Theorem.If the sum of residues k λ k is an integer, the sum Λ either vanishes identically or is a sub-normal solution to L[w] = 0.
The next theorem is concerned with the linear space spanned by the functions Λ ν , 0 ≤ ν ≤ n − q.
and  Remark.Note that φ L has no singularities at all only in case of a j w (j) + (−1) n+1 zw = 0 (with our normalisation).The most simple three-term example is equation ( 2).The Laplace solutions to w (n) + (−1) n+1 zw = 0 seem to be qualified candidates to be named special functions.We do not know what happens if k λ k is not an integer.Can one guarantee the existence of subnormal solutions?Is the sum Λ itself sub-normal?
3) Section 6 in [1] is devoted to the differential equation (unfortunately not with our normalisation).The authors found three contour integral solutions denoted H(z), H(−z), and U (z) = H(z) + H(−z) of order 3/2; any two of them are linearly independent.The transformation w(z) = v(iz) transforms this equation into have order of growth 2, and the sum is sub-normal with order of growth 2/3 (n = 4, q = p = 3); it corresponds to the solution U in [1] and has power series expansion k runs over every non-negative integer such that k + m − 1 ≥ 0.
4) To every n ≥ 2 there exist a unique family of differential equations (a j + b j z)w (j) = 0 depending on n − 2 parameters with solution w = e −z 2 /2 .5) The differential equation w (4) Λ 0 + Λ 1 is a non-trivial linear combination of w 2 , w 3 , and w 4 .
Remark.Actually Proposition 8.1 was stated and proved in [1] (Theorem 3) for without reference to the Phragmén-Lindelöf indicator.Examining the proof shows that it works for any φ given by (29) and m ≥ 2, but not for m = 1, which was out of sight in [1].For the proof of Theorem 5.1 we only need m ≥ 2 (m = n − q in our notation), but for the addendum to this theorem we need the case m = 1.
Proof of Proposition 8.1 for m = 1.Our object now is First of all we notice that R > R 0 may take any value since for R 1 ≥ R 2 > R 0 , the simple closed curve C R2, π 2 C R1, π 2 is contained in the domain of φ.We choose R = |z|, where > 0 depending on θ = arg z will be determined during the proof.Secondly we choose λ > 1 such that λθ < π/4 and show that for 0 < θ < π/4, say, the contour C |z|, π 2 may be replaced with C |z|, π to be done if θ = 0).To this end we have to show that the integral over the arc Γ r : t = ire −iϑ , 0 ≤ ϑ ≤ λθ vanishes in the limit r → ∞.This follows from  Note that every > 0 works and λ doesn't appear.

4. 1 .
The characteristic equation.The information on possible orders of growth and asymptotic expansions of transcendental solutions to(5) is encoded in the algebraic equation(19)
holds on I n−p−k for every j = k, k + 1 mod (n − p + 1).Now let n−p ν=0 c ν Λ ν be any non-trivial linear combination of the trivial solution, if any, and assume c k = 0 for some k.Then h k (ϑ) ≤ max{h ν (ϑ) : c ν = 0, ν = k} holds on the other hand, and h k (ϑ) = h(ϑ)on I n−p−k ∪ I n−p−k+1 on the other.This implies c k±1 = 0, hence c ν = 0 for every ν, and proves that any collection of n − q functions Λ ν is linearly independent.