Laplace Contour Integrals and Linear Differential Equations

Abstract

The purpose of this paper is to determine the main properties of Laplace contour integrals

$$\begin{aligned} \Lambda (z)=\frac{1}{2\pi i}\int _\mathfrak {C}\phi (t)e^{-zt}\,dt \end{aligned}$$

that solve linear differential equations

$$\begin{aligned} L[w](z):=w^{(n)}+\sum _{j=0}^{n-1}(a_j+b_jz)w^{(j)}=0. \end{aligned}$$

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.

1 Introduction

Special functions usually admit several quite different definitions and representations. For example, Airy’s differential equation

$$\begin{aligned} w''-zw=0 \end{aligned}$$

has a particular solution, known as the Airy function, that may be written as a Laplace contour integral

$$\begin{aligned} \mathrm{Ai}(z)=\frac{1}{2\pi i}\int _\mathfrak {C}e^{-zt+t^3/3}\,dt;\end{aligned}$$

(1)

the contour $\mathfrak {C}$ consists of the straight line from $+\infty \,e^{i\pi /3}$ to the origin followed by the straight line from 0 to $+\infty \,e^{-i\pi /3}$. 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

$$\begin{aligned} L[w]:=w^{(n)}+(-1)^{n+1}bw^{(k)}+(-1)^{n+1}zw=0,\quad 0<k<n,~b\in \mathbb {C}, \end{aligned}$$

(2)

and

$$\begin{aligned} L[w]:=w^{(n)}-zw^{(k)}-w=0,\quad 1<k<n, \end{aligned}$$

(3)

with the intention “to make a contribution to this topic which includes a generalization of the Airy integral [$\ldots$]  to have more examples of solutions of complex differential equations that have concrete properties”. To this end they determined several contour integral solutions

$$\begin{aligned} \frac{1}{2\pi i}\int _\mathfrak {C}F(z,w)\,dw \end{aligned}$$

with varying kernels F and contours $\mathfrak {C}$. Although [1] does not contain any hint of 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 to or may be transformed into Laplace contour integrals

$$\begin{aligned} \frac{1}{2\pi i}\int _\mathfrak {C}\phi (t)e^{-zt}\,dt \end{aligned}$$

(4)

of particular analytic functions $\phi$ over canonical contours or paths of integration $\mathfrak {C}$. We will prove that to each linear differential equation

$$\begin{aligned} L[w]:=w^{(n)}+\sum _{j=0}^{n-1}(a_j+b_jz)w^{(j)}=0,\quad a_0+b_0z\not \equiv 0, \end{aligned}$$

(5)

there exists some distinguished Laplace contour integral solution (4) with kernel $\phi _L$ that is uniquely determined by the operator L and itself determines canonical contours $\mathfrak {C}$. The main properties of these solutions—denoted $\Lambda _L$—, which strongly resemble 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 $T(r,\Lambda _L)$ and $N(r,1/\Lambda _L)$, and the existence of sub-normal solutions.

2 Kernels and Contours

The following reflections on linear differential equations

$$\begin{aligned} L[w]:=\sum _{j=0}^nP_j(z)w^{(j)}=0, \end{aligned}$$

(6)

are more or less part of mathematical folklore, but unfortunately not as well-known as they should be. One can find a few remarks in Wasow [10, 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, Ch. XVIII] deal with linear differential equations (6) under the constraint $\deg P_n\ge \deg P_j$. This, however, is by no means necessary and even prevents the discussion of the most important case $P_n(z)\equiv 1$, where the solutions are entire functions of finite order of growth.

We start with linear differential equations (6) with polynomial coefficients

$$\begin{aligned} P_j(z)=\sum _\alpha c_{j\alpha }z^\alpha ,\quad P_n(z)P_0(z)\not \equiv 0,\quad d=\max \deg P_j\ge 1, \end{aligned}$$

(7)

and are looking for solutions that may be written as Laplace contour integrals (4); the function $\phi$ and also possible paths of integration $\mathfrak {C}$ have to be determined. Formally we have

$$\begin{aligned} w^{(j)}(z)=\frac{1}{2\pi i}\int _\mathfrak {C}\phi _j(t)e^{-zt}\,dt\quad \mathrm{with}\quad \phi _j(t)=(-t)^j\phi (t) \end{aligned}$$

and

$$\begin{aligned} \begin{aligned} (-1)^\alpha \int _\mathfrak {C}z^\alpha \phi _j(t)e^{-zt}\,dt&=\int _\mathfrak {C}\phi _j(t)D^\alpha [e^{-zt}]\,dt\quad (\mathrm{where~}D= d/{dt})\\&=\big [ \phi _j(t)D^{\alpha -1} [e^{-zt}]\big ]_\mathfrak {C}-\int _\mathfrak {C}\phi _j'(t)D^{\alpha -1}[e^{-zt}]\,dt \end{aligned} \end{aligned}$$

(8)

if $\alpha \ge 1$; here

$$\begin{aligned} \big [\Psi (t)\big ]_\mathfrak {C}=\lim _{t\rightarrow b}\Psi (t)-\lim _{t\rightarrow a}\Psi (t) \end{aligned}$$

denotes the variation of $\Psi$ along $\mathfrak {C}$, when $\mathfrak {C}$ starts at a and ends at b; of course, $\mathfrak {C}$ may be closed. We demand

$$\begin{aligned} \big [\phi ^{(\alpha -1)}_j(t) e^{-zt}\big ]_\mathfrak {C}=0,\quad 0\le j\le n,~1\le \alpha \le \max \limits _{0\le j\le n}\deg P_j, \end{aligned}$$

(9)

to obtain

$$\begin{aligned} L[w](z)=\frac{1}{2\pi i}\int _\mathfrak {C}\Bigg [\sum _{j,\alpha }c_{j\alpha }D^\alpha [(-t)^j\phi (t)]\Bigg ] e^{-zt}\,dt, \end{aligned}$$

hence $L[w](z)\equiv 0$ if

$$\begin{aligned} \sum _{j,\alpha }c_{j\alpha }D^\alpha [(-t)^j\phi (t)]\equiv 0. \end{aligned}$$

(10)

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

$$\begin{aligned} Q_\alpha (t)=\sum _{j=0}^nc_{j\alpha }(-t)^j, \end{aligned}$$

(11)

condition (10) may be written as

$$\begin{aligned} \sum _{\alpha =0}^d \big [Q_\alpha \phi \big ]^{(\alpha )}(t)\equiv 0,\quad d=\max \deg P_j, \end{aligned}$$

(12)

and we have the following theorem, which forms the basis of our considerations.

2.1 Theorem

Let $\phi$ be any non-trivial solution to (12) and $\mathfrak {C}$ any contour such that the integrals (8) converge absolutely with respect to arc-length on $\mathfrak {C}$ 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.

3 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

$$\begin{aligned} Q_0(t)=(-t)^n+\sum _{j=0}^{n-1}a_j(-t)^j\quad \mathrm{and}\quad Q_1(t)=b_q(-t)^q+\sum _{j=0}^{q-1}b_j(-t)^j, \end{aligned}$$

where $b_q\ne 0$ and $b_j=0$ for $j>q$ is assumed. By a linear change of the independent variable z we may achieve

$$\begin{aligned} b_q=(-1)^{n-q+1}\quad (\mathrm{and}~ a_q=0~\mathrm{if~desired}); \end{aligned}$$

(13)

this will be assumed henceforth. Then $Q_0\phi +(Q_1\phi )'=0$ gives

$$\begin{aligned} \phi (t)=\frac{1}{Q_1(t)}\exp \Bigg [-\int \frac{Q_0(t)}{Q_1(t)}\,dt\Bigg ] \end{aligned}$$

(14)

up to an arbitrary constant factor. More precisely,

$$\begin{aligned} \begin{array}{c} \phi (t)=\displaystyle \prod _{\nu =1}^m(t-t_\nu )^{-m_\nu -\lambda _\nu }\exp \left[ R_0(t)+\sum _{\nu =1}^m R_\nu \left( \frac{1}{t-t_\nu }\right) \right] \quad \mathrm{with}\\ \displaystyle \lambda _\nu =\mathop {{\text {res}}}_{t_\nu }\left[ \frac{Q_0}{Q_1}\right] ,\quad \deg R_\nu \le m_\nu -1, \quad R_0(t)=\frac{t^{n-q+1}}{n-q+1}+\cdots \end{array} \end{aligned}$$

(15)

holds. Sometimes it suffices to know that $\phi$ has the form

$$\begin{aligned} \phi (t)=\exp \Bigg [\frac{t^{n-q+1}}{n-q+1}+\psi (t)\Bigg ], \end{aligned}$$

(16)

where $\psi$ is holomorphic on $\{t:|t|>R_0,~ |\arg t|<\pi \}$ and satisfies

$$\begin{aligned} |\psi (t)|\le C|t|^{n-q}\quad \mathrm{as}~t\rightarrow \infty . \end{aligned}$$

(17)

Occasionally, any function (14) will denoted $\phi _L$.

3.2 Appropriate Contours

There is some freedom in the choice of the contour $\mathfrak {C}$, which is restricted only by condition (9). Closed simple contours work in particular cases only, namely when $\lambda _\nu$ is an integer, and yield less interesting examples. More interesting are the contours $\mathfrak {C}_{R,\alpha ,\beta }$ that consist of three arcs as follows:

• 1. the line $t=re^{i\alpha }$, where r runs from $+\infty$ to $R\ge 0$;

• 2. the circular arc on $|t|=R$ from $Re^{i\alpha }$ via $t=Re^{i(\alpha +\beta )/2}$ to $Re^{i\beta }$;

• 3. the line $t=re^{i\beta }$, where r runs from R to $+\infty$.

In place of $\mathfrak {C}_{0,\alpha ,\beta }$ , $\mathfrak {C}_{R,\alpha ,-\alpha }$ , and $\mathfrak {C}_{0,\alpha ,-\alpha }$ we will also write $\mathfrak {C}_{\alpha ,\beta }$ , $\mathfrak {C}_{R,\alpha }$ , and $\mathfrak {C}_{\alpha }$ , respectively. Note that $\mathfrak {C}_{R,\alpha ,-\alpha }$ passes through R, while $\mathfrak {C}_{R,\alpha ,2\pi -\alpha }$ 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 $\arg t=\alpha$ and $\arg t=\beta$; this will tacitly be assumed. A canonical and our preferred choice is $\alpha =\pi /(n-q+1)$ and $\beta =-\alpha$. It is, however, almost obvious by Cauchy’s theorem that the contour integral is independent of $\alpha ,$ $\beta ,$ and R within their natural limitations. This will be proved later on.

3.3 The Protagonist

By $w=\Lambda _{L}(z)$ we will denote any solution to Eq. (5) given by

$$\begin{aligned} \Lambda _L(z)=\frac{1}{2\pi i}\int _{\mathfrak {C}_{R,\pi /(n-q+1)}}\phi _L(t)e^{-zt}\,dt \end{aligned}$$

(18)

with associated function (14). Of course, $\Lambda _L$ 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 representative of the solutions $\Lambda _L$ with $q=n-2$, where n may be arbitrarily large.

3.4 Elementary Examples

• 1. Airy’s Equation $w''-zw=0$ and $\Lambda _L=\mathrm{Ai}$: $Q_0(t)=t^2,$ $Q_1(t)=-1$, $\phi _L(t)=e^{t^3/3}$, and $\mathfrak {C}=\mathfrak {C}_{\pi /3}$. By Cauchy’s theorem, any contour $\mathfrak {C}_{R,\alpha ,\beta }$ with $R\ge 0$, $\alpha \in (\pi /2,\pi /6)$, and $\beta \in (-\pi /6,-\pi /2)$, is admissible.

• 2. $w^{(n)}+(-1)^{n+1}bw^{(k)}+(-1)^{n+1}zw=0$, see (2): $Q_0(t)=(-1)^nt^n-(-1)^{n+1+k}bt^k$, $Q_1(t)=(-1)^n$, hence

  $$\begin{aligned} \phi _L(t)=\exp \Bigg [\frac{t^{n+1}}{n+1}+\frac{(-1)^{k+1}bt^{k+1}}{k+1}\Bigg ] \end{aligned}$$

  (see [1, (3.2)]). The contour $\mathfrak {C}=\mathfrak {C}_{\pi /(n+1)}$ is canonical.

• 3. $w^{(n)}-zw^{(k)}-w=0$, $1<k<n$, see (3): $Q_0(t)=(-1)^nt^n-1$, $Q_1(t)=(-1)^{k+1}t^k$, hence

  $$\begin{aligned} \phi _L(t)=\frac{1}{t^k}\exp \Bigg [\frac{(-1)^{n-k}t^{n-k+1}}{n-k+1}+\frac{(-1)^kt^{-k+1}}{k-1}\Bigg ]. \end{aligned}$$

  Since $\phi _L$ has an essential singularity at $t=0$, one may choose $\mathfrak {C}:|t|=1$; it is, however, more interesting to choose $\mathfrak {C}=\mathfrak {C}_{1,\pi /(n-k+1)}$ if $n-q$ is even; if $n-q$ is odd, the differential equation does match our normalisation only after the change of variable $z\mapsto -z$. We note that the substitution (the conformal map) $t=-1/u$ transforms the Laplace contour integral with kernel $\phi _L$ up to sign into some integral

  $$\begin{aligned} \frac{1}{2\pi i}\int _{\tilde{\mathfrak {C}}}u^{k-2}\exp \left[ \frac{z}{u}-\frac{u^{-n+k-1}}{n-k+1}-\frac{u^{k-1}}{k-1}\right] \,du \end{aligned}$$

  in accordance with [1, (5.2)].

4 Asymptotic Expansions and the Order of Growth

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

$$\begin{aligned} y^n +\sum _{j=0}^{n-1}(a_j+b_jz)y^j=0, \end{aligned}$$

(19)

which justifiably may be called the characteristic equation. Remember that we assume $b_q=(-1)^{n-q+1}$ and $b_j=0$ for $j>q$. Also let $p\le q$ denote the smallest index such that $b_p\ne 0$. As $z\rightarrow \infty$, Eq. (19) has solutions

• 1. $y_j(z)\sim \gamma _j z^{1/(n-q)}$, $\gamma _j^{n-q}=(-1)^{n-q}$, $1\le j\le n-q$, hence

  $$\begin{aligned} \int y_j(z)\,dz\sim \frac{\gamma _jz^{1+1/(n-q)}}{1+\frac{1}{n-q}}. \end{aligned}$$

  in any case ($n-q$ odd or even), one $\gamma _j$ equals $-1$.

• 2. $y_{n-q+j}\sim t_j$, $1\le j\le q-p$, where $t_j\ne 0$ is a root of $Q_1(-t)=0$ (counting multiplicities), hence

  $$\begin{aligned} \int y_{n-q+j}(z)\,dz\sim t_j z; \end{aligned}$$

• 3. $y_{n-p+j}(z)\sim \tau _jz^{-1/p}$, $\tau _j^p=-a_0/b_p$, $1\le j\le p$, hence

  $$\begin{aligned} \qquad \quad \int y_{n-p+j}(z)\,dz\sim \frac{\tau _jz^{1-1/p}}{1-\frac{1}{p}}~\mathrm{if}~ p>1 \mathrm{~and~}\int y_n(z)\,dz\sim \tau _1\log z~\mathrm{if~} p=1. \end{aligned}$$

By $z^{1/r}$ and $\log z$ we mean the branches on $|\arg z|<\pi$ that are real on the positive real axis. If $p=1$, polynomial solutions may exist, but not otherwise.

4.2 The Order of Growth

Let

$$\begin{aligned} f(z)=\sum _{n=0}^\infty c_nz^n \end{aligned}$$

be any transcendental entire function with maximum modulus, maximum term, and central index

$$\begin{aligned} \begin{aligned} M(r,f)=&~\max \{|f(z)|:|z|=r\},\\ \mu (r)=&~\max \{|c_n|r^n:n\ge 0\},\mathrm{~and~}\\ \nu (r)=&~\max \{n:|c_n|r^n=\mu (r)\}, \end{aligned} \end{aligned}$$

respectively. Then f has order of growth

$$\begin{aligned} \varrho (f)=\limsup _{r\rightarrow \infty }\frac{\log \log M(r,f)}{\log r}=\limsup _{r\rightarrow \infty }\frac{\log \log \mu (r)}{\log r}=\limsup _{r\rightarrow \infty }\frac{\log \nu (r)}{\log r}. \end{aligned}$$

The central index method for solutions to linear differential equations (6) is based on the relation

$$\begin{aligned} \frac{f^{(j)}(z_r)}{f(z_r)}=\Bigg (\frac{\nu (r)}{z_r}\Bigg )^j(1+\epsilon _j(r)),\quad j=1,2,\ldots , \end{aligned}$$

where $z_r$ is any point on $|z|=r$ such that $|f(z_r)|=M(r,f)$. This holds with $\epsilon _j(r)\rightarrow 0$ as $r\rightarrow \infty$ with the possible exception of some set $E_j$ of finite logarithmic measure; for a proof see Wittich [11, Ch. I] or Hayman [2]. If $w=f(z)$ solves (5), $\nu (r)/z_r=v(r)$ satisfies

$$\begin{aligned} (1+\epsilon _n(r)v(r)^n+\sum _{j=0}^{n-1}(a_j+b_jz_r)(1+\epsilon _j(r))v(r)^j=0, \end{aligned}$$

hence any such v(r) is asymptotically correlated with some solution to the characteristic equation (19) as follows:

$$\begin{aligned} \nu (r)\sim z_r y_j(z_r) \end{aligned}$$

as $r\rightarrow \infty$ holds even without an exceptional set by the regular behaviour of $z_ry_j(z_r)$; since $\nu (r)$ is positive, $z_ry_j(z_r)$ is asymptotically positive, this giving additional information on the possible values of $\arg z_r$. Thus the transcendental solutions to Eq. (5) have possible orders of growth

$$\begin{aligned} \varrho= & {} 1+\frac{1}{n-q}: \text { solutions of this order always exist;}\\ \varrho= & {} 1: \text { necessary for solutions of this order to exist is } p<q;\\ \varrho= & {} 1-\frac{1}{p}: \text { necessary for solutions of this order to exist is } p>1;\\ \varrho= & {} 0: \text { necessary for polynomial solutions is } p=1.\\ \end{aligned}$$

4.3 Asymptotic Expansions

The differential equation (5) may be rewritten in the usual way as a first-order system

$$\begin{aligned} z^{-1}\mathfrak {v}'=\big (B+z^{-1}A\big )\mathfrak {v}\end{aligned}$$

(20)

for $\mathfrak {v}=(w,w',\ldots ,w^{(n-1)})^\top$ with $n\times n$-matrices A and B. The following details can be found in Wasow’s fundamental monograph [10]. The system (20) has rank one, but the matrix B has vanishing eigenvalues. Thus the theory of asymptotic integration yields only a local result [10, Thm. 19.1], which makes its applicability unpleasant and involved: to every angle $\theta$ there exists some sector $S_\theta :|\arg (ze^{-i\theta })|<\delta$ such that the system (20) has a distinguished fundamental matrix

$$\begin{aligned} \mathfrak {V}(z|\theta )=V(z|\theta )z^Ge^{\Pi (z)}; \end{aligned}$$

since $B+z^{-1}A$ is holomorphic on $|z|>0$, the number $\delta$, the constant $n\times n$-matrix G, and the matrix $\Pi =\mathrm{diag\,}(\Pi _1,\ldots ,\Pi _n)$ are universal, that is, they do not depend on $\theta$. The entries of $\Pi$ are polynomials in $z^{1/r}$ for some positive integer r, and $V(z|\theta )$ has an asymptotic expansion

$$\begin{aligned} V(z|\theta )\sim \sum _{j=0}^\infty V_j\,z^{-1/r}\quad \mathrm{on~} S_\theta ; \end{aligned}$$

(21)

the latter means

$$\begin{aligned} \Vert V(z|\theta )-\sum _{j=0}^m V_j\,z^{-1/r}\Vert =o(|z|^{-m/r}),\qquad m=0,1,2,\ldots , \end{aligned}$$

as $z\rightarrow \infty$ on $S_\theta$, uniformly on every closed sub-sector; again the matrices $V_j$ are independent of $\theta$. The fundamental matrices $\mathfrak {V}$, that is, the matrices $V(z|\theta )$ may vary from sector to sector.¹ Returning to Eq. (5) this means that given $\theta$, there exists a distinguished fundamental system

$$\begin{aligned} \buildrel \circ \over w_{1}(z|\theta ),\ldots , \buildrel \circ \over w_{n}(z|\theta ) \end{aligned}$$

(22)

such that

$$\begin{aligned} \buildrel \circ \over w_{j}(z|\theta )=e^{\Pi _j(z^{1/r})}z^{\rho _j} \Omega _j(z|\theta )\quad \mathrm{on~}S_\theta ; \end{aligned}$$

(23)

$\Pi _j$ is a polynomial in $z^{1/r}$, $\rho _j$ is some complex number, and $\Omega _j$ is a polynomial in $\log z$ with coefficients that have asymptotic expansions in $z^{1/r}$ on $S_\theta$. The triples $(\Pi _j,\rho _j,\Omega _j)$ are mutually distinct. Again this system may vary from sector to sector, but only in the coefficients of $\Omega _j(z|\theta )$! The leading terms of the $\Pi _j$ can be found among those of the algebraic functions $\int y_\nu \,dz$ at $z=\infty$.

4.4 The Phragmén–Lindelöf Indicator

The following facts can be found in Lewin’s/Levin’s monographs [6,7,8]. Let f be an entire function of positive finite order $\varrho$ such that $\log M(r,f)=O(r^\varrho )$ as $r\rightarrow \infty$. Then

$$\begin{aligned} h(\vartheta )=\limsup _{r\rightarrow \infty }\frac{\log |f(re^{i\vartheta })|}{r^{\varrho }},\quad \vartheta \in \mathbb {R}\end{aligned}$$

(24)

is called Phragmén–Lindelöf indicator or just indicator of f of order $\varrho$; it is continuous and always assumed to be extended to the real axis as a $2\pi$-periodic function. Having $h(\vartheta )$ at hand, the Nevanlinna functions may be computed explicitly (for definitions and results we refer to Hayman [3] and Nevanlinna [9]):

where as² usual $x^+=\max \{x,0\}$ and $x^-=\max \{-x,0\}.$ In connection with our distinguished fundamental system (22) only local indicators

$$\begin{aligned} \textstyle h_j(\vartheta )=-\dfrac{1}{\varrho }\cos \!\left( \varrho \vartheta +\dfrac{2j\pi }{n-q}\right) , \quad 0\le j<n-q, \quad \mathrm{and}\quad h_{n-q}(\vartheta )=0 \end{aligned}$$

(25)

have to be considered.

5 Results on Normal and Sub-Normal Solutions

The Airy integral has an asymptotic representation

$$\begin{aligned} \mathrm{Ai}(z)\sim \frac{1}{2\sqrt{\pi }}e^{-(2/3)z^{3/2}}z^{-1/4}\sum _{j=0}^\infty c_jz^{-j/2}\quad \mathrm{even~on~}|\arg z|<\pi , \end{aligned}$$

hence

• 1. satisfies $\displaystyle \frac{\log \mathrm{Ai}(z)}{z^{3/2}}\sim -\frac{2}{3}$ as $z\rightarrow \infty$ on $|\arg z|\le \pi -\epsilon$ for every $\epsilon >0$;

• 2. has Phragmén–Lindelöf indicator $\displaystyle h(\vartheta )=-\frac{2}{3}\cos \Big (\frac{3}{2}\vartheta \Big )$ on $|\vartheta |<\pi$;

• 3. has infinitely many zeros, all on the negative real axis.

• 4. $\displaystyle T(r,\mathrm{Ai})\sim \frac{8}{9\pi }r^{3/2}$ and $\displaystyle N(r,1/\mathrm{Ai})\sim \frac{4}{9\pi }r^{3/2}.$

In the general case one cannot expect results of this high precision, but the following theorem seems to be a good approximation.

5.1 Theorem

Any Laplace contour integral $\Lambda _L$of order $\varrho =1+1/(n-q)\le 3/2$ (hence $n-q\ge 2$)

• 1. satisfies $\displaystyle \frac{\log \Lambda _L(z)}{z^\varrho }\sim -\frac{1}{\varrho }$ on $|\arg z|\le \pi -\epsilon$ for every $\epsilon >0;$

• 2. has Phragmén-Lindelöf indicator $\displaystyle h(\vartheta )=-\frac{1}{\varrho }\cos \Big (\varrho \vartheta \Big )$ on $|\vartheta |<\pi$;

• 3. has only finitely many zeros in $|\arg z-\pi |<\epsilon$ for every $\epsilon >0$;

• 4. $\displaystyle T(r,\Lambda _L)\sim \frac{1}{\pi \varrho ^2}(1+|\sin (\pi \varrho )|)r^\varrho$ and $\displaystyle N(r,1/\Lambda _L)\sim \frac{1}{\pi \varrho ^2}r^\varrho .$

$\Lambda _L$ has ‘many’ zeros by the last property; they are distributed over arbitrarily small sectors about the negative real axis. The analogue of Theorem 5.1 for $q=n-1$ reads as follows.

5.1a Theorem. Suppose $q=n-1$.  Then either $\Lambda _L(z)=e^{-z^2/2}P(z)$, Psome non-trivial polynomial, or else the following is true:

• 1. $\displaystyle \frac{\log \Lambda _L(z)}{z^2}\sim -\frac{1}{2}$ on $|\arg z|<\dfrac{3}{4}\pi -\epsilon$;

• 2. $\displaystyle h(\vartheta )=-\frac{1}{2}\cos (2\vartheta )$ on $|\vartheta |<\displaystyle \frac{3}{4}\pi$ and $h(\vartheta )=0$ on $\displaystyle \frac{3}{4}\pi \le |\vartheta |\le \pi$;

• 3. $\Lambda _L$ has only finitely many zeros in $|\arg z|\le \displaystyle \frac{3}{4}\pi -\epsilon$, and at most $o(r^2)$ zeros in $|\arg z-\pi |<\displaystyle \frac{\pi }{4}-\epsilon$, $|z|<r$, for every $\epsilon >0;$

• 4. $\displaystyle T(r,\Lambda _L)\sim \frac{1}{2\pi }r^2$ and $\displaystyle N(r,1/\Lambda _L)\sim \frac{1}{4\pi }r^2$.

In the second case, $\Lambda _L$ again has ‘many’ zeros; they are distributed over arbitrarily small sectors about $\arg z=\pm 3\pi /4$.

The proof of Theorem 5.1 and 5.1 a will be given in Sect. 9.

5.2 Subnormal Solutions

Solutions of maximal order $\varrho =1+1/(n-q)$ always exist, while the question of the existence of so-called sub-normal solutions having order $\varrho =1$, $\varrho =1-1/p$, and $\varrho =0$ (polynomials), respectively, remains open; necessary but by far not sufficient conditions are $q>p$, $p>1$, and $p=1$, respectively. Sufficient conditions 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 $p\ge 1$ 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 $p\ge 1$with integer residue $\lambda _0$.   Then the solution

$$\begin{aligned} w(z)=\mathop {{\text {res}}}_{0}\big [\phi _L(t)e^{-zt}\big ] \end{aligned}$$

has order of growth $\varrho =1-1/p$  if $t=0$ is an essential singularity of $\phi _L$, that is, if $p>1$,   and otherwise $(p=1)$ is either a non-trivial polynomial of degree $\lambda _0\ge 0$ or else vanishes identically $(\lambda _0<0)$.

5.2b Theorem. Let $t=t_0\ne 0$ be a pole of $Q_0/Q_1$ of order m with integer residue $\lambda _0$.   Then the solution

$$\begin{aligned} w(z)=\mathop {{\text {res}}}_{t_0}\big [\phi _L(t)e^{-zt}\big ] \end{aligned}$$

has the form

$$\begin{aligned} w(z)=e^{-t_0z}W(z), \end{aligned}$$

where Wis a transcendental entire function of order of growth at most $1-1/m$  if $m>1$,   and otherwise is a polynomial of degree $\lambda _0\ge 0$  or vanishes identically $(\lambda _0<0)$.

Proof of 5.2a. First assume that $t=0$ is a simple pole of $Q_0/Q_1$. Then $p=1$ (since $Q_0(0)=a_0\ne 0$) and

$$\begin{aligned} \phi _L(t)e^{-zt}=\frac{H(t)e^{-zt}}{(t-t_0)^{1+\lambda _0}} \end{aligned}$$

holds, where H is holomorphic at $t=0$ and $H(0)\ne 0$. Then (assuming $\lambda _0\ge 0$)

$$\begin{aligned} w(z)=\sum _{k=0}^{\lambda _0}\frac{H^{(\lambda _0-k)}(0)}{k!(\lambda _0-k)!}(-z)^k \end{aligned}$$

is a polynomial of degree $\lambda _0$. If, however $t=0$ is an essential singularity for $\phi _L$, then $p>1$ and

$$\begin{aligned} |\phi _L(t)|\le \frac{Ae^{B|t|^{-p+1}}}{|t|^{1+\lambda _0}} \end{aligned}$$

holds on $|t|<\delta ,$ hence

$$\begin{aligned} |w(z)|\le \frac{1}{2\pi }\int _{|t|=|z|^{-1/p}}\frac{Ae^{B|t|^{-p+1}+|z||t|}}{|t|^{1+\lambda _0}}\,|dt|\le A|z|^{\lambda _0/p}e^{(B+1)|z|^{1-1/p}} \end{aligned}$$

holds as $z\rightarrow \infty$. This yields $\varrho (w)\le 1-1/p$. Since w is not a polynomial, the only possibility is $\varrho =1-1/p$. This proves 5.2a.

To deal with 5.2b write

$$\begin{aligned} w(z)=\mathop {{\text {res}}}_{t_0}\big [\phi _L(t)e^{-zt}\big ]=e^{-t_0z}\mathop {{\text {res}}}_{t_0}\big [\phi _L(t)e^{-z(t-t_0)}\big ]=e^{-t_0z}W(z). \end{aligned}$$

Then by part a of the proof, either W is a polynomial of degree $\lambda _0\ge 0$ or vanishes identically $(\lambda _0<0$) 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. $\square$

Remark. By $w=e^{-t_0 z}W$ ($t_0\ne 0$), Eq. (5) is transformed into

$$\begin{aligned} W^{(n)}+\sum _{j=0}^{n-1}(A_j+B_jz)W^{(j)}=0 \end{aligned}$$

(26)

with $A_\nu +B_\nu z=\sum _{j=\nu }^n{j\atopwithdelims ()\nu }(-t_0)^{j-\nu }(a_j+b_jz);$ in particular, $B_q=b_q$ and $A_0+B_0z=Q_0(t_0)+Q_1(t_0)z$ 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 $B_m\ne 0$, but $B_j=0$ for $j<m$, hence W has the order $1-1/m$ by part a.

6 Rotational Symmetries

The functions $\mathrm{Ai}(ze^{-2\nu \pi i/3})$ also solve Airy’s equation and coincide up to some non-zero factor with the Laplace contour integrals

$$\begin{aligned} \frac{1}{2\pi i}\int _{\mathfrak {C}_{(2\nu +1)\pi /3,(2\nu -1)\pi /2}}\!\!\!\!\!\!\!\!\!\!\!\! e^{t^3/3}e^{-zt}\,dt; \end{aligned}$$

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.   $\Lambda _L(z e^{-2\nu \pi i/(n-q+1)})$ is not necessarily a solution to $L[w]=0$.

Nevertheless one may consider the contour integral solutions

$$\begin{aligned} \Lambda _{\nu }(z)=\frac{1}{2\pi i}\int _{\mathfrak {C}_{R,\theta _{2\nu +1},\theta _{2\nu -1}}}\!\!\!\!\!\!\!\!\!\!\!\!\phi _L(t)e^{-zt}\,dt,\quad \textstyle \theta _k=\dfrac{k\pi }{n-q+1}, \end{aligned}$$

(27)

and their Phragmén-Lindelöf indicators $h_\nu$ for $0\le \nu \le n-q$, but one has to take care since $\phi _L$ 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 $\Lambda _0=\Lambda _L$ is extended to the real line as a $2\pi$-periodic function. Although $\Lambda _\nu (z)$ is distinct from $\Lambda _L(ze^{-2\nu \pi i/(n-q)})$ in general, both functions share their main properties as are stated in Theorem 5.1 and 5.1a for $\Lambda _L$.

6.1 Theorem

Each contour integral (27) of order $\varrho =1+1/(n-q)$   has the same properties³as the corresponding function $\Lambda _L(e^{-2\nu \pi i/(n-q+1)}z)$   in Theorem 5.1and 5.1a.

If $\phi _L$ has no finite singularities, the sum

$$\begin{aligned} \Lambda (z)=\Lambda _L(z)+\Lambda _{1}(z)+\cdots + \Lambda _{n-q}(z)=-\frac{1}{2\pi i}\int _{|t|=R}\phi _L(t)e^{-zt}\,dt \end{aligned}$$

(28)

vanishes identically. On the other hand, poles $t_k$ of $Q_0/Q_1$ are poles or essential singularities of $\phi _L$; if these singularities are non-critical, that is, if the residues $\lambda _k=\mathop {{\text {res}}}_{t_k}[Q_0/Q_1]$ are integers, the following holds.

6.2 Theorem

Suppose the residues $\lambda _k$ are integers. Then the sum $\Lambda$ either vanishes identically or may be written as a linear combination of sub-normal solutions

$$\begin{aligned} \mathop {{\text {res}}}_{t_k}[\phi _L(t)e^{-zt}]=e^{-t_k z}W_k; \end{aligned}$$

$W_k$ has order of growth less than one.

      2.   $\phi _L$ may be many-valued on $\mathbb {C}{\setminus }\{\text {poles of } Q_0/Q_1\}$.

This is the case if some of the residues $\lambda _k$ are not integers. Nevertheless $\phi _L$ may be single-valued on $|t|>R_0$: if t goes around once the positively oriented circle $|t|=R>R_0$, $(t-t_k)^{\lambda _k}$ takes the value $(t-t_k)^{\lambda _k}e^{2\pi i\lambda _k}$, hence analytic continuation of $\phi _L(t)$ along $|t|=R$ yields the value $\phi _L(t)e^{-2\pi i\sum _{k}\lambda _k}$ (see the general form (15) of $\phi _L$), and $\phi _L$ is single-valued on $|t|>R_0$ if $\sum _{k}\lambda _k$ is an integer. Assuming this, $\Lambda$ is a sub-normal solution of order at most one. This follows from (28) and $|\phi _L(t)e^{-zt}|\le Ae^{R|z|}$ on $|t|=R,$ hence

$$\begin{aligned}|\Lambda (z)|\le ARe^{R|z|}.\end{aligned}$$

We have thus proved

6.3 Theorem

If the sum of residues $\sum _k\lambda _k$ is an integer, the sum $\Lambda$  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 $\Lambda _\nu$, $0\le \nu \le n-q$.

6.4 Theorem

Any set of $n-q$  functions $\Lambda _\nu$ is linearly independent.

Proof. Set

$$\begin{aligned} h(\vartheta )=\max _{0\le \nu \le n-q} h_\nu (\vartheta )=\max _{0\le \nu \le n-q} h_0(\vartheta -\theta _{2\nu }) \end{aligned}$$

and $I_\nu =(-\pi +\theta _{2\nu },-\pi +\theta _{2\nu +2})$, $0\le \nu \le n-p$. Then

$$\begin{aligned} h_k(\vartheta )=h_{k+1}(\vartheta )=h(\vartheta )>h_j(\vartheta )\quad \mathrm{holds~on~}I_{n-p-k} \end{aligned}$$

for every $j\ne k,k+1$ mod $(n-p+1)$. Now let

$$\begin{aligned} \sum _{\nu =0}^{n-p}c_\nu \Lambda _\nu \end{aligned}$$

be any non-trivial linear combination of the trivial solution, if any, and assume $c_k\ne 0$ for some k. Then

$$\begin{aligned} h_k(\vartheta )\le \max \{h_\nu (\vartheta ): c_\nu \ne 0,~\nu \ne k\} \end{aligned}$$

holds on one hand, and

$$\begin{aligned} h_k(\vartheta )=h(\vartheta ) \end{aligned}$$

on $I_{n-p-k}\cup I_{n-p-k+1}$ on the other. This implies $c_{k\pm 1}\ne 0$, hence $c_\nu \ne 0$ for every $\nu$, and proves that any collection of $n-q$ functions $\Lambda _\nu$ is linearly independent(Fig. 1). $\square$

Fig. 1

The indicators $h_k$ on $[-\pi ,\pi ]$ ($0\le k\le n-q$, from bottom to top), and the intervals $I_\nu$ ($0\le \nu \le n-q$, from left to right) for $n-q=2, 3, 4$

Remark. Note that $\phi _L$ has no singularities at all only in the case of

$$\begin{aligned} L[w]=w^{(n)}+\sum _{j=0}^{n-1}a_jw^{(j)}+(-1)^{n+1}zw=0 \end{aligned}$$

(with our normalisation). The most simple three-term example is Eq. (2). The Laplace solutions to

$$\begin{aligned} w^{(n)}+(-1)^{n+1}zw=0 \end{aligned}$$

seem to be qualified candidates to be named special functions. We do not know what happens if $\sum _k\lambda _k$ is not an integer. Can one guarantee the existence of sub-normal solutions? Is the sum $\Lambda$ itself sub-normal?

7 More Examples

• 1. The differential equation $w^{(5)}-zw'''+7w''+zw'-2w=0$ with $n=5,~q=3,~p=1$, $Q_0(t)=-t^5+7t^2-2$, $Q_1(t)=t^3-t$ has the fundamental system

  $$\begin{aligned} \qquad w_1= & {} \Lambda _L(z)=\frac{1}{2\pi i}\int _{\mathfrak {C}_{\frac{\pi }{3}}}\frac{e^{t^3/3+t-zt}}{t^3(t-1)^3(t+1)^4}\,dt\\ w_2= & {} \Lambda _1 \mathrm{~or~} \Lambda _2\\ w_3= & {} (2z^3-39z^2+264z-635)e^z\\ w_4= & {} (z^2+6z+15)e^{-z}\\ w_5= & {} z^2+7.\end{aligned}$$

  The functions $\Lambda _0$, $\Lambda _1$, and $\Lambda _2$ are linearly dependent.

• 2. The differential equation $w^{(6)}-zw^{(4)}-zw''+w=0$ with $n=6$, $q=4$, $p=2$, $Q_0(t)=t^6+1$ and $Q_1(t)=-t^4-t^2$, hence

  $$\begin{aligned} \phi _L(t)=\frac{1}{t^4+t^2}\exp \Bigg [\frac{t^3}{3}-t-\frac{1}{t}\Bigg ], \end{aligned}$$

  has solutions $w_{1,2}=e^{\pm iz}$ (associated with the poles at $t=\mp i$) of order $\varrho =1$ and three solutions of order $\varrho =3/2$, namely $\Lambda _0$, $\Lambda _1$, and $\Lambda _1$. The residue theorem yields one more solution $w_3(z)=\mathop {{\text {res}}}_0\big [\phi _L(t)e^{-zt}\big ]$ that has order of growth 2/3. It is not clear how complete the five solutions just discussed for obtaining a basis since

  $$\begin{aligned} \Lambda _0(z)+\Lambda _1(z)+\Lambda _2(z)=\frac{i}{2}e^{i(z+1/3)}-\frac{i}{2}e^{-i(z+1/3)}-w_3(z) \end{aligned}$$

  is a linear combination of $w_1,$ $w_2$, and $w_3$.

• 3. [1, Sect. 6] is devoted to the differential equation

  $$\begin{aligned} v^{(4)}-zv'''-v=0 \end{aligned}$$

  (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

  $$\begin{aligned} w^{(4)}+zw'''-w=0 \end{aligned}$$

  with $Q_0(t)=t^4-1$, $Q_1(t)=-t^3$, and $\displaystyle \phi _L(t)=\frac{1}{t^3}\exp \Big [\frac{t^2}{2}+\frac{1}{2t^2}\Big ]$. The solutions

  $$\begin{aligned} \Lambda _j(z)=\frac{1}{2\pi i}\int _{\mathfrak {C}_{1,(2j-1)\pi /2,(2j+1)\pi /2}}\!\!\!\!\!\!\!\!\!\!\!\!\phi _L(t)e^{-zt}\,dt,\quad j=0,1, \end{aligned}$$

  have order of growth 2, and the sum

  $$\begin{aligned} -\Lambda _0(z)-\Lambda _1(z)=\frac{1}{2\pi i}\int _{|t|=1}\phi _L(t)e^{-zt}\,dt=\mathop {{\text {res}}}_0[\phi _L(t)e^{-zt}] \end{aligned}$$

  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

  $$\begin{aligned} \sum _{m=0}^\infty \frac{C_m}{2^{m-1}m!}z^{2m}\quad \mathrm{with}\quad C_m=\sum _k\frac{1}{4^k k!(k+m-1)!}; \end{aligned}$$

  k runs over every non-negative integer such that $k+m-1\ge 0$.

• 4. To every $n\ge 2$ there exist a unique family of differential equations

  $$\begin{aligned} w^{(n)}+zw^{(n-1)}+\displaystyle \sum _{j=0}^{n-2}(a_j+b_jz)w^{(j)}=0 \end{aligned}$$

  depending on $n-2$ parameters with solution $w=e^{-z^2/2}$.

• 5. The differential equation $w^{(4)}+(z-1)w'''-8w''-zw'+2w=0$ with $n=4, q=3, Q_0(t)=t^4+t^3-8t^2+2, Q_1(t)=-t^3+t$ has the fundamental system

  $$\begin{aligned} \begin{aligned} w_1(z)=&~\Lambda _L(z)=\frac{1}{2\pi i}\int _{\mathfrak {C}_{2,\frac{\pi }{2}}}\frac{e^{t^2/2+t-zt}}{t^3(t-1)^3(t+1)^4}\,dt\\ w_2(z)=&~(2z^3-27z^2+150z-324)e^z\\ w_3(z)=&~(z^2+6z+14)e^{-z}\\ w_4(z)=&~z^2+8. \end{aligned} \end{aligned}$$

  $\Lambda _0+\Lambda _1$ is a non-trivial linear combination of $w_2, w_3,$ and $w_4$.

• 6. The differential equation $w^{(5)}-(z+1)w'''+w''+(z+1-\lambda -2\mu )w'+\lambda w=0$ with $Q_0(t)=-t^5+t^3+t^2+(\lambda +2\mu -1)t+\lambda$, $Q_1(t)=t^3-t$ ($\lambda$ and $\mu$ arbitrary) has the distinguished solutions $\Lambda _0=\Lambda _L$, $\Lambda _1$ and $\Lambda _2$. Although

  $$\begin{aligned} \phi _L(t)=\frac{e^{t^3/3}}{t^{1-\lambda }(t-1)^{1+\lambda +\mu }(t+1)^{2-\mu }} \end{aligned}$$

  may have transcendental singularities at $t=0,1,-1$, $\phi _L$ is single-valued on $|t|>1$, and $\Lambda _0(z)+\Lambda _1(z)+\Lambda _2(z)$ has order of growth at most 1.

8 Preparing the Proof of Theorem 5.1

Our proof of Theorem 5.1 will be based on the following.

8.1 Proposition

Suppose

$$\begin{aligned} \phi (t)=\exp \Bigg [\frac{t^{m+1}}{m+1}+\psi (t)\Bigg ] \end{aligned}$$

(29)

is holomorphic on $\{t:|t|>R_0,~|\arg t|<\pi \}$, where $\psi$ satisfies $|\psi (t)|\le C|t|^{m}$ and $m\ge 1$ is an integer. Then the Phragmén-Lindelöf indicator of

$$\begin{aligned} f(z)=\frac{1}{2\pi i}\int _{\mathfrak {C}_{R,\pi /(m+1)}}\phi (t)e^{-zt}\,dt \end{aligned}$$

of order $\varrho =1+1/m$is negative on

$$\begin{aligned} |\theta |<\frac{m\pi }{2(m+1)}. \end{aligned}$$

(30)

Remark. Actually Proposition 8.1 was stated and proved in [1, Thm. 3] for

$$\begin{aligned} \phi (t)=\exp \Bigg [\frac{t^{m+1}}{m+1}+\frac{(-1)^{k+1}b t^{k+1}}{k+1}\Bigg ],\quad 0<k<m, \end{aligned}$$

without reference to the Phragmén-Lindelöf indicator. Examining the proof shows that it works for any $\phi$ given by (29) and $m\ge 2$, but not for $m=1$, which was out of sight in [1]. For the proof of Theorem 5.1 we only need $m\ge 2$ ($m=n-q$ 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

$$\begin{aligned} \frac{1}{2\pi i}\int _{\mathfrak {C}_{R,\frac{\pi }{2}}}\phi (t)e^{-zt}\,dt \end{aligned}$$

(31)

with

$$\begin{aligned} \phi (t)=\exp \Bigg [\frac{t^2}{2}+\psi (t)\Bigg ]\quad \mathrm{and}\quad |\psi (t)|\le C|t|. \end{aligned}$$

First of all we notice that $R>R_0$ may take any value since for $R_1\ge R_2>R_0$, the simple closed curve $\mathfrak {C}_{R_2,\pi /{2}}\circleddash \mathfrak {C}_{R_1,\pi /{2}}$ is contained in the domain of $\phi$. We choose $R=\epsilon |z|$, where $\epsilon >0$ depends on $\theta =\arg z$ and will be determined during the proof. Secondly we choose $\lambda >1$ such that $\lambda \theta <\pi /4$ and show that for $0<\theta <\pi /4$, say, the contour $\mathfrak {C}_{\epsilon |z|,\pi /2}$ may be replaced with $\mathfrak {C}_{\epsilon |z|,\pi /2-\lambda \theta ,-\pi /2}$ (of course, nothing has to be done if $\theta =0$). To this end we have to show that the integral over the arc $\Gamma _r: t=ire^{-i\vartheta }$, $0\le \vartheta \le \lambda \theta$ vanishes in the limit $r\rightarrow \infty$. This follows from

$$\begin{aligned} \begin{aligned} \left| \frac{1}{2\pi i}\int _{\Gamma _r}\phi (t)e^{-zt}\,dt\right| \le&~\frac{r}{2\pi }\int _0^{\lambda \theta }\exp \!\left[ -\frac{r^2}{2}\cos (2\vartheta )+|z|r+Cr\right] \,d\vartheta \\ \le&~\frac{r}{8}\exp \!\left[ -\frac{r^2}{2}\cos (2\lambda \theta )+r(|z|+C)\right] \rightarrow 0~\mathrm{as~} r\rightarrow \infty \end{aligned} \end{aligned}$$

since $\cos (2\lambda \theta )>0$. To finish the proof we have to estimate the integrals

$$\begin{aligned} \int _{\mathfrak {C}}|\phi (t)e^{-zt}|\,|dt| \end{aligned}$$

over

• 1. $\mathfrak {C}: t=-i\epsilon |z|\tau$, $1\le \tau <\infty$;

• 2. $\mathfrak {C}: t=i\epsilon |z|\tau e^{-i \lambda \theta }$, $1\le \tau <\infty$;

• 3. $\mathfrak {C}: t=\epsilon |z|e^{i\vartheta }$, $-\pi /2\le \vartheta \le \pi /2-\lambda \theta$.

• ad 1. From Re $t^2=-\epsilon ^2|z|^2\tau ^2\le -\epsilon ^2|z|^2\tau$, $\mathrm{Re}(-zt)=-\epsilon |z|^2\tau \sin \theta \le 0$ (since $0\le \theta <\pi /4$), and $|dt|=\epsilon |z|\,d\tau$ we obtain the upper bound

  $$\begin{aligned} \qquad \epsilon |z|\int _1^\infty \exp \Big [-\frac{1}{2}\epsilon |z|(\epsilon |z|-2C)\tau \Big ]\,d\tau \le \frac{1}{C}e^{-\epsilon ^2|z|^2/4},\quad |z|\ge 4C/\epsilon . \end{aligned}$$

  Note that every $\epsilon >0$ works and $\lambda$ doesn’t appear.

• ad 2. Here it follows from $\mathrm{Re} t^2=-\epsilon ^2|z|^2\tau ^2\cos (2\lambda \theta )\le -\epsilon ^2|z|^2\tau \cos (2\lambda \theta )$ and $\mathrm{Re}(-zt)=-|z|^2\tau ^2\sin ((\lambda -1)\theta )\le 0$ that

  $$\begin{aligned} \epsilon |z|\int _1^\infty \exp \Big [-\frac{1}{2}\epsilon |z|(\epsilon |z|\cos (2\lambda \theta )-2C)\tau \Big ]\,d\tau \le \frac{1}{C}e^{-\epsilon ^2|z|^2/4} \end{aligned}$$

  is an upper bound provided $\displaystyle |z|\ge 4C/(\epsilon \cos (2\lambda \theta ))$. Again we note that every $\epsilon >0$ works and $\lambda$ does not play an essential role.

• ad 3. We first assume $0<\theta<\lambda \theta <\pi /4$. From Re $t^2\le \epsilon ^2|z|^2$, $\mathrm{Re} (-zt)=-\epsilon |z|^2\cos (\vartheta +\theta )$, and $|dt|=\epsilon |z|\,d\vartheta$ we get the bound

  $$\begin{aligned} \epsilon |z|\int _{-\pi /2}^{\pi /2-\lambda \theta }\exp \Big [\frac{1}{2}\epsilon |z|^2(\epsilon -2\cos (\vartheta +\theta ))+\epsilon C|z|\Big ]\,d\vartheta \end{aligned}$$

  To control the term $\cos (\vartheta +\theta )$ note that

  $$\begin{aligned} -\frac{\pi }{2}<-\frac{\pi }{2}+\theta \le \vartheta +\theta \le \frac{\pi }{2}-\lambda \theta +\theta =\frac{\pi }{2}-(\lambda -1)\theta <\frac{\pi }{2}, \end{aligned}$$

  hence $\cos (\vartheta +\theta )\ge \min \{\sin \theta , \sin ((\lambda -1)\theta )\}=\kappa (\theta )>0;$ here we need $\lambda$! This yields the upper bound

  $$\begin{aligned} \pi \epsilon |z|\exp \Bigg [\frac{1}{2}\epsilon |z|^2(\epsilon -2\kappa (\theta ))+\epsilon C|z|\Bigg ] \end{aligned}$$

  for our integral, and all we have to do is to choose $\epsilon =\kappa (\theta )$ to obtain the bound $\pi \epsilon |z|e^{-\epsilon ^2|z|^2/4}$ for $|z|\ge 4C/\epsilon$. It remains to discuss the case $\theta =0$, hence $z=x>0$, where we can work with $\epsilon =1$. From

  $$\begin{aligned} \mathrm{Re}\Big [\frac{t^2}{2}-xt\Big ]=\frac{1}{2}\mathrm{Re}(t-x)^2-\frac{1}{2}x^2=2x^2(\cos \vartheta -1)\cos \vartheta -\frac{1}{2}x^2\le -\frac{1}{2}x^2 \end{aligned}$$

  we obtain the very last upper bound

  $$\begin{aligned} \pi \exp \Big [-\frac{1}{2}x^2+Cx\Big ]\le \pi \exp \Big [-\frac{1}{4}x^2\Big ],\quad x\ge 4C. \end{aligned}$$

This proves Proposition 8.1 for $m=1$. To verify the case $m\ge 2$ the reader is referred to [1]. $\square$

9 Proof of Theorem 5.1 and 5.1a

Let h denote the Phragmén–Lindelöf indicator of our special solution $\Lambda _L$. To prove

$$\begin{aligned} h(\vartheta )=-\frac{1}{\varrho }\cos (\varrho \vartheta ) \end{aligned}$$

(32)

on $-\pi /(2\varrho )\le \vartheta \le \pi /(2\varrho )$ we use the following facts taken from [8, p. 53 ff.] which hold for arbitrary indicators of order $\varrho$.

• 1. h has one-sided derivatives everywhere, and $h'(\vartheta -)\le h'(\vartheta +)$ holds;

• 2. h is $\varrho$-trigonometrically convex;

• 3. $h(\varphi )+h(\varphi +\pi /\varrho )\ge 0$ holds for every $\vartheta$.

Since, however, $h(\pm \pi /(2\varrho ))\le 0$, the third property implies $h(\pm \pi /(2\varrho ))=0$, and the second leads to

$$\begin{aligned} h(\vartheta )=h(0)\cos (\varrho \vartheta )\quad \mathrm{on }\textstyle |\vartheta |\le \dfrac{\pi }{2\varrho } \end{aligned}$$

(again [8, p. 53 ff.]). Then $h(0)=-1/\varrho$ follows from the fact that the only local indicators available are given by (25). By the third property (note $h'(\pi /(2\varrho ))=1$), (32) remains true on $(\pi /(2\varrho ),\theta ]$, where $\theta >\pi /(2\varrho )$ is the smallest number such that

$$\begin{aligned} h_0(\theta )=h_k(\theta ),~ h'_0(\theta )<0, \mathrm{~and~}h'_k(\theta )>0 \end{aligned}$$

holds for some $k\ne 0$. This happens soonest at $\theta =\pi$. Finally, the claim about the distribution of zeros follows from the fact that

$$\begin{aligned} h(\vartheta )=\lim _{r\rightarrow \infty }\frac{\log |\Lambda _L(re^{i\vartheta })|}{r^\varrho } \end{aligned}$$

holds uniformly on $|\vartheta |\le \pi -\epsilon$, and

$$\begin{aligned} \frac{\log \Lambda _L(z)}{z^\varrho }\rightarrow -\frac{1}{\varrho }\quad \mathrm{as}~z\rightarrow \infty ~\mathrm{on~}|\arg z|\le \pi -\epsilon . \end{aligned}$$

Thus all but finitely many zeros are contained in $|\arg z-\pi |<\epsilon$ for every $\epsilon >0$. This proves Theorem 5.1, and the arguments may be repeated step-by-step to prove Theorem 5.1a until $\vartheta =\pm 3\pi /4$ is reached. Then either $h(\vartheta )=-(1/2)\cos (2\vartheta )$ holds on $[-\pi ,\pi ]$ or only on $[-3\pi /4,3\pi /4]$, while $h(\vartheta )=0$ on $3\pi /4\le |\vartheta |\le \pi$. Then

$$\begin{aligned} \displaystyle \frac{\log \Lambda _L(z)}{z^2}\rightarrow -\frac{1}{2}\quad \mathrm{as~}z\rightarrow \infty \end{aligned}$$

either holds on the whole plane or on $|\arg z|\le 3\pi /4-\epsilon$. In the first case $\Lambda _L$ has only finitely many zeros and $\Lambda _L(z)=e^{-z^2/2}P(z)$ holds. In the second case, $\Lambda _L$ has only finitely many zeros on $|\arg z|\le 3\pi /4-\epsilon$, and from

$$\begin{aligned} \log |\Lambda _L(re^{i\vartheta })|=o(r^2) \end{aligned}$$

as $r\rightarrow \infty$, uniformly on $|\vartheta |\le \pi /4-\epsilon$, it follows that $\Lambda _L$ has $o(r^2)$ (probably only O(r)) zeros on $|\arg z-\pi |\le \pi /4-\epsilon$, $|z|\le r$ (cf. [6, 7, p. 150]). The asymptotic formulae for $T(r,\Lambda _L)$ and $N(r,1/\Lambda _L)$ are obvious. $\square$