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This article is cited in 1 scientific paper (total in 1 paper) Asymptotic behaviour of the sphere and front of a flat sub-Riemannian structure on the Martinet distribution I. A. Bogaevskyabc a Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, Moscow, Russia b Scientific Research Institute for System Analysis of the Russian Academy of Sciences, Moscow, Russia c Ailamazyan Program Systems Institute of Russian Academy of Sciences, Ves'kovo, Pereslavl' district, Yaroslavl' oblast', Russia
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     IntroductionA sphere of a flat sub-Riemannian structure on the Martinet distribution is the singular surface shown in Figure 1, a. It is the boundary of the attainability set (from the origin, in unit time) of a control system in three-dimensional space whose admissible velocities at each point form a planar disc:
$$
\begin{equation*}
\dot{x}=u_1, \qquad \dot{y}= u_2, \qquad \dot{z}= \frac{u_1 y^2}2, \qquad u_1^2+u_2^2 \leqslant 1.
\end{equation*}
\notag
$$
The planes containing these discs form the Martinet distribution $dz=y^2\,dx/2$. The Martinet sub-Riemannian sphere (known results)The control system under consideration and, by implication, the Martinet sphere are symmetric relative to the plane $y=0$ and the straight line $x=z=0$. After the change of one coordinate $w=z-x y^2 /6$ the principal meridian of the Martinet sphere (the line of points at which the tangent plane to the sphere is vertical) flattens out and becomes a horizontal circle, as shown in Figure 1, b. Parametric formulae for the Martinet sub-Riemannian sphere, which involve Jacobi elliptic functions and elliptic integrals, were found in [1]; we present them in § 1. The Martinet sphere is subanalytic at all points apart from two, $x=\pm 1$, $y=z=0$; we call these points poles. It was shown in [1] that the Martinet sphere is not subanalytic at the poles. The intersection of the sphere and the plane $y=0$ is the union of the poles and two curves connecting them. The sphere has edges along these curves, where it is locally diffeomorphic to a dihedral angle. More precisely, the Martinet sphere is a subset of the image of the so-called exponential map, which is an analytic map from a cylinder (the product of a circle and a line) to three-dimensional space. The exponential map is expressed by the same formulae in [1] as the sphere itself, but no constraints are imposed on the parameters. The image of the exponential map is called the Martinet front. It contains the whole sphere, but the preimage of each pole of the sphere is a noncompact generator of the cylinder: this explains why the Martinet front is not analytic at the poles of the sphere. The Martinet front is not compact: its closure contains the interval of the $x$-axis connecting the poles of the sphere, as shown in Figure 2 and follows from the formulae in [1] mentioned above. In Figure 2, which was drawn with the use of the formulae in [1], we show (a) half of the Martinet front in the variables $(x,y,z)$ and (b) the cross-section of the front by the plane $x=0$. This figure substantiates the following quite plausible conjectures concerning the singularities of the Martinet front. Evidently, these singularities reduce to the poles of the sphere connected by lines of self-intersection and cuspidal edges. In a neighbourhood of each point of self-intersection the Martinet front is diffeomorphic to the pair of planes $x^2=z^2$, and in a neighbourhood of a point on a cuspidal edge, to the surface $x^3=z^2$. Any neighbourhood of the line segment with endpoints at the poles contains infinite countable sets of lines of self-intersection and cuspidal edges. The results of the paperIn this paper we calculate the asymptotics of the exponential map in neighbourhoods of the preimages of the poles of the Martinet sphere. (Due to symmetry relative to the line $x=z=0$, we only need to consider one pole.) We obtain an infinite countable number of distinct asymptotic formulae: in one of them (Theorem 1) the coordinate along the generator of the cylinder is fixed, while in the other formulae (Theorem 2) it tends to infinity at various rates with the decreasing angular coordinate on the cylinder. Using this result, we calculate (in Corollaries 1 and 2) the asymptotic behaviour of the cross-sections of the sphere and the Martinet front by a vertical plane as it approaches a pole. It turns out that the image of the cross-section of the Martinet front by the plane $x=1-\varepsilon$ under the quasihomogeneous dilation $(y,z) \mapsto(y/\sqrt{\varepsilon}, z/\varepsilon)$ converges as $\varepsilon \to 0+ $ to the curve shown in Figure 3, a; and its image under the dilation $(y,z) \mapsto (y/\varepsilon, z/\varepsilon^3)$ converges to the curve in Figure 3, b (we explain the precise meaning of ‘converges’ in Corollaries 1 and 2). The first limit curve (in Figure 3, a) consists of the origin and two mutually symmetric transcendental analytic curves; their parametric formulae are presented in Corollary 1 to Theorem 1. The limit curve is not analytic at the origin. The second limit curve (in Figure 3, b) consists of a finite number of symmetric algebraic curves which are tangent one to another; their parametric formulae are presented in Corollary 2 to Theorem 2. The noncompact lowest curve has the equation $z=y^4/16-y^2/8-1/48$. In Figure 4 we show the cross-sections of the Martinet front by the vertical planes $x=0.4$ (a) and $x=0.9$ (b); these correspond to $\varepsilon=0.6$ and $\varepsilon=0.1$. In Figure 5 we show their approximation by the compressions $(y,z) \mapsto(\sqrt{\varepsilon} y, \varepsilon z)$ of the limit curve from Figure 3, a. In Figure 6 we show the approximations of the section by the compressions $(y,z) \mapsto(\varepsilon y, \varepsilon^3 z)$ of the limit curve in Figure 3, b. The approximations in Figure 6 are not valid on the whole cross-section, but are quite accurate (with infinite-order error as $\varepsilon \to 0+$). Thus the angles between branches of the cross-section below the origin are infinitesimals of infinite order as $\varepsilon \to 0+$.  Figure 5.Approximations of cross-sections of the Martinet front by compressions of the curve from Figure 3, a.  Figure 6.Approximations of cross-sections of the Martinet front by compressions of the curve from Figure 3, b. To obtain approximations of the sphere from approximation of the front we simply need to remove the inner parts of the limit curves. For instance, in Figure 7 we show the approximations of the cross-section of the Martinet sphere by the vertical plane $x=0.9$ obtained in this way. The bottom part of the cross-section of the sphere is approximated by the graph of a fourth-order polynomial with an infinite-order error as $\varepsilon \to 0+$, despite its having a kink at $y=0$. This mean that the lower kink (by contrast to the higher one) straightens out quite rapidly as ${\varepsilon \to 0+}$. Motivations for the researchA distribution of planes in $\mathbb{R}^3$ can locally be defined as the field of null subspaces of a nonvanishing 1-form $\alpha$. If the 3-form ${\omega=\alpha \wedge d \alpha}$ is distinct from zero, then we have a contact distribution and by Darboux’s theorem it can be reduced to the form $y\,dx-dz=0$ by choosing appropriate smooth1 local variables. However, for a generic $\alpha$ the distribution is not contact on the smooth surface $\omega=0$. If a plane in the distribution is transversal to the surface $\omega=0$, then by Martinet’s theorem the distribution is reduced to the form $y^2 \, dx /2-dz=0$ by a choice of appropriate local coordinate variables, and it is called the Martinet distribution. In these variables the surface $\omega=0$ has the equation $y=0$. Integral curves of the distributions that lie on the Martinet surface are called abnormal geodesics: they are geodesics for any sub-Riemannian metric on the Martinet distribution. The sub-Riemannian distance has fairly complicated and mysterious singularities along abnormal geodesics, and the Martinet distribution is the simplest distribution that has abnormal geodesics. There are many sub-Riemannian structures on the Martinet distribution that are not pairwise equivalent under smooth changes of local coordinates: similarly to Riemannian structures they are distinguished by functional invariants. Nevertheless, all of them have isometric tangent spaces in the sense of Gromov-Hausdorff (see details in [2]). Such a tangent space is called a flat sub-Riemannian structure on the Martinet distribution. In this paper we calculate asymptotics for this structure in neighbourhoods of abnormal geodesics, because poles of the Martinet sphere are just points of its intersection with the abnormal geodesics from its centre. We stress that here we only investigate the sphere and front of a flat sub-Riemannian structure (given by $ds^2=dx^2+dy^2$) on the Martinet distribution ${dz=y^2 \, dx /2}$. In this case the equations of geodesics are integrated by Jacobi elliptic functions, and the sub-Riemannian sphere is described by explicit formulae found in [1] and underlying our calculations. However, the flat sub-Riemannian metric $ds^2$ on the Martinet distribution is not generic: the coefficients of a generic metric can depend on $x$, $y$ and $z$. In the generic case a sub-Riemannian sphere of small radius is slightly ‘more tame’ (for instance, see [3], Figure 4), and the asymptotic behaviour of its singularities at the poles is currently not known. Small spheres of a generic sub-Riemannian structure on the Martinet distribution were considered in [4]–[6], for instance. Using the methods developed in our paper the asymptotic behaviour of spheres can evidently be calculated for a nilpotent sub-Riemannian problem on the (four-dimensional) Engel group and the (five-dimensional) Cartan group. This is because the equations of geodesics of left-invariant sub-Riemannian metrics on these groups are also integrated by Jacobi elliptic functions, and the sub-Riemannian spheres are described by explicit formulae, which were found in [7] and [8] for the Engel and the Cartan group, respectively. As regards the simpler case of a contact distribution, similar questions were answered for it at the end of the last century: the sphere and front of the flat sub-Riemannian structure were investigated in [9], and for a generic structure they were considered in [10]. AcknowledgementsThe author is grateful to L. V. Lokutsievskiy, Yu. L. Sachkov and the referees, whose assistance—in one or another form—helped him considerably in his work on this paper. § 1. Exponential mapThe Martinet front is the image of the analytic map of a cylinder
$$
\begin{equation*}
x=X_\lambda (\varkappa), \quad y=Y_\lambda(\varkappa), \quad z=Z_\lambda (\varkappa), \qquad \lambda \in \mathbb{R}, \quad \varkappa \in \mathbb{R}/\pi \mathbb{Z},
\end{equation*}
\notag
$$
which is called the exponential map (for unit time) for the Hamiltonian of Pontryagin’s maximum principle. In what follows we use the standard notation of the theory of elliptic functions:
$$
\begin{equation*}
\begin{aligned} \, K &=\int_0^{\pi/2} \frac{d \theta}{\sqrt{1-k^2 \sin^2{\theta}}}, \qquad k=\sin{\varkappa}, \\ K' &=\int_0^{\pi/2} \frac{d \theta}{\sqrt{1-{k'}^2 \sin^2{\theta}}}, \qquad k'= \cos{\varkappa}, \end{aligned}
\end{equation*}
\notag
$$
where $K$ and $K'$ are complete elliptic integrals of the first kind, $k$ is the elliptic modulus, $k'$ is the complementary modulus, and $\varkappa$ is the modular angle. We denote the Jacobi elliptic functions with elliptic modulus $k$ (which we drop in formulae) by $\operatorname{sn}$, $\operatorname{cn}$ and $\operatorname{dn}$, and the ones with the complementary modulus $k'$ by ${\,}'\!\operatorname{sn}$, ${\,}'\!\operatorname{cn}$ and ${\,}'\!\operatorname{dn}$. By the results in [1] (see § 4 and Proposition 4.3 there) the exponential map has the following properties. $\bullet$ If $0<\lambda$ and $0 \leqslant \varkappa<\pi/2$, then
$$
\begin{equation}
{X}_\lambda (\varkappa)=-1+\frac{2}{\sqrt{\lambda}} \int_K^{u} \operatorname{dn}^2{v} \, dv, \qquad {Y}_\lambda (\varkappa)=- \frac{2 k}{\sqrt{\lambda}} \operatorname{cn}{u}
\end{equation}
\tag{1.1}
$$
and
$$
\begin{equation}
{Z}_\lambda (\varkappa)=\frac{2}{3 \lambda^{3/2}} \biggl\{(2 k^2-1) \int_K^{u} \operatorname{dn}^2{v} \, dv+2 k^2 \operatorname{sn}{u} \, \operatorname{cn}{u} \, \operatorname{dn}{u}+{k'}^2 \sqrt{\lambda} \biggr\},
\end{equation}
\tag{1.2}
$$
where $u=K+\sqrt{\lambda}$. $\bullet$ On the principal meridian $\lambda=0$
$$
\begin{equation*}
X_0 (\varkappa)=\cos{2 \varkappa}, \qquad Y_0 (\varkappa)=\sin{2 \varkappa}\quad\text{and} \quad Z_0 (\varkappa)=\frac16 \cos{2 \varkappa} \sin^2{2 \varkappa}.
\end{equation*}
\notag
$$
$\bullet$ The preimage of the pole $x=1$, $y=z=0$ is the generator $\varkappa=0$ of the cylinder; the preimage of the pole $x=-1$, $y=z=0$ is the generator $\varkappa=\pi/2$:
$$
\begin{equation*}
X_\lambda (0)=- X_\lambda \biggl(\frac\pi2\biggr)=1, \qquad Y_\lambda (0)=Y_\lambda\biggl(\frac\pi2\biggr)= Z_\lambda (0)= Z_\lambda\biggl(\frac\pi2\biggr)=0.
\end{equation*}
\notag
$$
$\bullet$ The image of the derivative of the map lies in the plane with equation
$$
\begin{equation}
{P}_\lambda(\varkappa) \, dx+{Q}_\lambda(\varkappa) \, dy+{R}_\lambda(\varkappa) \, dz= 0;
\end{equation}
\tag{1.3}
$$
its coefficients are the analytic functions on the cylinder such that
$$
\begin{equation}
{P}_\lambda(\varkappa)=\cos{2 \varkappa}, \quad {R}_\lambda(\varkappa)=\lambda \quad \text{for } \lambda \in \mathbb{R} \quad\text{and}\quad \varkappa \in \mathbb{R}/\pi\mathbb{Z},
\end{equation}
\tag{1.4}
$$
and
$$
\begin{equation}
{Q}_\lambda(\varkappa)=2 k \operatorname{sn}{u} \, \operatorname{dn}{u} \quad \text{for } 0<\lambda \quad \text{and}\quad 0\leqslant \varkappa<\frac\pi2.
\end{equation}
\tag{1.5}
$$
In other words, the substitution $x=X_\lambda(\varkappa)$, $y=Y_\lambda(\varkappa)$, $z= Z_\lambda(\varkappa)$ transforms (1.3) into an identity. $\bullet$ The image of the exponential map is symmetric relative to the straight line ${x=z=0}$ and the plane $y=0$. The map itself is invariant under the transformations
$$
\begin{equation}
(\lambda, \varkappa, X, Y, Z, P, Q, R) \mapsto \biggl(- \lambda, \frac\pi2-\varkappa, -X, Y, -Z, -P, Q, -R\biggr)
\end{equation}
\tag{1.6}
$$
and
$$
\begin{equation}
(\lambda, \varkappa, X, Y, Z, P, Q, R) \mapsto (\lambda,-\varkappa, X, -Y ,Z, P,-Q, R).
\end{equation}
\tag{1.7}
$$
$\bullet$ The Martinet sphere is the image of the part of the cylinder given by $- 4 {K'}^2 \leqslant \lambda \leqslant 4 K^2$. Remark 1. In [1] the quantity $\varphi=\pi/2-2 \varkappa$ was used in place of the modular angle $\varkappa$ as a coordinate on the cylinder. Formulae (1.1), (1.2), (1.4) and (1.5) were obtained in [1] (see § 4 and Proposition 4.3 there) by integrating explicitly the Hamiltonian equations of Pontryagin’s maximum principle with Hamiltonian
$$
\begin{equation*}
H=\frac12 \biggl(p_x+\frac12 y^2 p_z\biggr)^2+\frac12 p_y^2.
\end{equation*}
\notag
$$
The trajectories issuing from $x=y=z=0$ at time $t=0$ and lying on the level set $H=1/2$ of the Hamiltonian occur on the Martinet front at $t=1$. The coefficients of (1.3) are the values of the dual variables $p_x$, $p_y$ and $p_z$ for $t=1$; for $t=0$ they take values $\cos{2 \varkappa}$, $\sin{2 \varkappa}$ and $\lambda$, respectively. A priori, the right-hand sides of (1.1), (1.2) and (1.5) are not analytic at $\varkappa= \pi/2$ because they contain an elliptic integral of the first kind. The formulae in the proposition below, which we use in what follows, define the exponential map at all points on the cylinder away from the principal meridian $\lambda=0$ and do not have this drawback. Proposition. If $\lambda \neq 0$, $k=\sin{\varkappa}$ and $k'=\cos{\varkappa}$, then Remark 2. For fixed $\varkappa \in \mathbb{C}$ the functions in (1.8)–(1.10) are meromorphic in $\lambda \in \mathbb{C}$. They have removable singularities at $\lambda=0$ because of the following: Remark 3. It is not straightforward that formulae (1.8)–(1.10) are invariant under the substitution (1.6); we verify this in § 3. The Legendrian submanifoldConsider the singular Legendrian submanifold
$$
\begin{equation*}
\Lambda=\bigl\{(X_\lambda(\varkappa), Y_\lambda(\varkappa), Z_\lambda(\varkappa); \, P_\lambda(\varkappa):Q_\lambda(\varkappa):R_\lambda(\varkappa))\bigr\} \subset P T^\ast \mathbb{R}^3
\end{equation*}
\notag
$$
in the projectivization of the tangent bundle. The Martinet front is the image of it under the natural projection The closure of $\Lambda$ is compact and has singularities at points over the line segment joining the poles of the sphere and obtained by taking the limit as $\lambda \to \infty$. These singularities are not subanalytic. For a flat sub-Riemannian structure on the distribution $dz=y \, dx$ the submanifold $\Lambda$ (as well as its closure) is stable in the following sense: under a small perturbation of the coefficients of the metric this manifold stays contact diffeomorphic to the original manifold [11]. In our case of a flat structure on the Martinet distribution $dz=y^2 \, dx /2$ the Legendrian manifold $\Lambda$ is evidently no longer stable. Despite the stability of $\Lambda$ for a flat sub-Riemannian structure on $dz=y \, dx$, the front of this structure, although it is a projection of $\Lambda$, is no longer stable: it changes significantly under a generic perturbation of the metric and is not diffeomorphic to the original front, not even locally. The fronts of flat and generic sub-Riemannian structures on the distribution $dz=y \, dx$ were investigated in [9] and [10], respectively. § 2. Asymptotic behaviour of the front near the poles Theorem 1. For fixed $\lambda$ the following asymptotic expansions hold as $\varkappa \to 0$: Remark 4. In place of $\varkappa$ we can write $k$ on the right-hand sides of the formulae in Theorem 1, because $ \varkappa=\arcsin{k}=k+O(k^3)$ as $k \to 0$. Corollary 1. 1) The two mutually symmetric transcendental analytic curves with parametric equations 2) The distance of a fixed point on the curves (2.1) to the image of the cross-section of the Martinet front is an infinitesimal quantity $O(\varepsilon)$ as $\varepsilon \to 0+$. 3) The points on the curves (2.1) such that $\lambda \leqslant \pi^2$ and the origin are the limit points as $\varepsilon \to 0+$ of the image of the cross-section of the Martinet sphere by the plane $x=1-\varepsilon$ under the quasihomogeneous dilation $(y,z) \mapsto(y/\sqrt{\varepsilon}, z/\varepsilon)$. 4) The lower branches of the curves (2.1) (corresponding to $\lambda<0$) and the origin form the graph of a smooth nonanalytic function, which can be approximated by a polynomial up to a nontrivial flat remainder: $z=y^4/16+O(|y|^{N})$ for each $N \in \mathbb{N}$. Remark 5. From the limit curve (2.1) in Figure 3, a, the condition $\lambda \leqslant \pi^2$ removes its inner part, on which $\lambda > \pi^2$. Remark 6. A stronger result (which we do not prove here) must evidently be true: the union of the curve (2.1) with the origin is the Kuratowski limit as $\varepsilon \to 0+$ of the image of the cross-section of the Martinet front by the plane $x=1-\varepsilon$ under the quasihomogeneous dilation $(y,z) \mapsto (y/\sqrt{\varepsilon}, z/\varepsilon)$. For negative $\lambda$ of large absolute value the approximations in Theorem 1 produce poor results because the exponential map is bounded, whereas $\gamma(\lambda) \to+\infty$ as $\lambda \to-\infty$. In Theorem 2 we find a series of asymptotic formulae which approximate the exponential map uniformly for negative values of $\lambda$ which are bounded away from zero and lie on the intervals
$$
\begin{equation*}
\lambda \in \bigl[-4 n^2 {K'}^2,-4 (n-1)^2 {K'}^2 \bigr], \qquad n=1, 2, 3, \dots,
\end{equation*}
\notag
$$
whose endpoints (with the exception of the right-hand endpoint of the interval corresponding to $n=1$) tend to $- \infty$ as $\varkappa \to 0$ because $K' \to+\infty$. We obtain these formulae using well-known approximations of Jacobi elliptic functions with modulus close to $1$ by hyperbolic functions. Theorem 2. Let $ \mu_{b,\varkappa}=\sigma_n K'+b $, where $\sigma_n=2 n-1$, $n\in\mathbb{N}$. Then for fixed $b \in \mathbb{R}$ the following expansions hold as $\varkappa \to 0$: In Figure 8, a, we show the graph of the $Y$-component of the exponential map as a function of $\lambda$ for $\varkappa=0.05$, and in Figure 8, b, we show by a dashed line the graph of its approximation in Theorem 1. The solid lines in (b) are the approximations from Theorem 2 corresponding to $n=1$ and $n=2$. As $\varkappa \to 0$, the negative zeros of $Y$ tend to minus infinity, and the positive zeros have finite limits. For this reason in Theorem 2, in contrast to Theorem 1, we make a change of parameter. Corollary 2. The algebraic curves given parametrically by the equations The distance of any fixed point on these curves to the image of the cross-section of the Martinet front tends to zero more rapidly than any positive power of $\varepsilon$. The curve corresponding to $n=1$ ($\sigma_1=1$) has the explicit equation $z=y^4/16-y^2/8-1/48$. It consists of the limit points as $\varepsilon \to 0+$ of the image of the cross-section of the Martinet sphere by the plane $x=1-\varepsilon$ under the quasihomogeneous dilation $(y,z) \mapsto(y/\varepsilon, z/\varepsilon^3)$. Remark 7. A stronger result (which we do not prove here) is evidently true: the union of all curves (2.2) and the origin is the Kuratowski limit as $\varepsilon \to 0+$ of the image of the cross-section of the Martinet sphere by the plane $x=1-\varepsilon$ under the quasihomogeneous dilation $(y,z) \mapsto (y/\varepsilon, z/\varepsilon^3 )$. § 3. ProofsThe proofs of the proposition in § 1 and Theorems 1 and 2 are based on well-known properties of Jacobi elliptic functions and complete elliptic integrals. Apart from the books cited below, the reader can find these properties in the web handbook at https://functions.wolfram.com. Proof of the proposition. That (1.8)–(1.10) coincide with (1.1), (1.2) and (1.5) for $0<\lambda$ and $0 \leqslant \varkappa<\pi/2$ follows from reduction formulae for Jacobi elliptic functions (see [12], § 13.17, Table 7):
$$
\begin{equation*}
\operatorname{cn}{u}=- k' \frac{\operatorname{sn}{\sqrt{\lambda}}}{\operatorname{dn}{\sqrt{\lambda}}}, \qquad \operatorname{sn}{u}= \frac{\operatorname{cn}{\sqrt{\lambda}}}{\operatorname{dn}{\sqrt{\lambda}}}\quad\text{and} \quad \operatorname{dn}{u}=\frac{k'}{\operatorname{dn}{\sqrt{\lambda}}},
\end{equation*}
\notag
$$
where $u=K+\sqrt{\lambda}$.
That the functions (1.8)–(1.10) are invariant under the substitutions (1.6) and (1.7) follows from the symmetry of the Hamiltonian $H$ and the initial conditions. However, for more confidence we deduce this directly from the properties of Jacobi elliptic functions. The functions $X$ and $Z$ are even in $\varkappa$, while $Y$ and $Q$ are odd, because by changing the sign of $\varkappa$ we change the sign of $k$, but preserve $k'$, while the elliptic functions $\operatorname{sn}$, $\operatorname{cn}$ and $\operatorname{dn}$ are even in $k$. The pattern of sign changes of (1.8)–(1.10) when we change the sign of $\lambda$ and go from $\varkappa$ to $\pi/2-\varkappa$ is a consequence of the following properties of Jacobi elliptic functions (see [12], § 13.17, formulae (13), and [13], Ch. II, formulae (91)–(93)):
$$
\begin{equation*}
\operatorname{cn}{i \tau}=\frac{1}{{\,}'\!\operatorname{cn}{\tau}}, \qquad \operatorname{sn}{i \tau}=i \frac{{\,}'\!\operatorname{sn}{\tau}}{{\,}'\!\operatorname{cn}{\tau}}\quad\text{and} \quad \operatorname{dn}{i \tau}=\frac{{\,}'\!\operatorname{dn}{\tau}}{{\,}'\!\operatorname{cn}{\tau}},
\end{equation*}
\notag
$$
where dashes mean that the elliptic modulus is changed from $k$ to $k'$. In fact,
$$
\begin{equation*}
Y_{-\lambda} (\varkappa)=\frac{2kk'\operatorname{sn}\sqrt{-\lambda}}{\sqrt{-\lambda} \, \operatorname{dn}{\sqrt{-\lambda}}} =\frac{2 k k'\operatorname{sn}{i \sqrt{\lambda}}}{i \sqrt{\lambda} \, \operatorname{dn}{i \sqrt{\lambda}}} =\frac{2 k k'{\,}'\!\operatorname{sn}{\sqrt{\lambda}}}{\sqrt{\lambda} \, {\,}'\!\operatorname{dn}{\sqrt{\lambda}}} =Y_{\lambda} \biggl(\frac{\pi}2-\varkappa\biggr)
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
Q_{-\lambda} (\varkappa)=\frac{2 k k'\operatorname{cn}{\sqrt{-\lambda}}}{\operatorname{dn}^2{\sqrt{-\lambda}}} =\frac{2 k k'\operatorname{cn}{i \sqrt{\lambda}}}{\operatorname{dn}^2{i \sqrt{\lambda}}} =\frac{2 k k'{\,}'\!\operatorname{cn}{\sqrt{\lambda}}}{{\,}'\!\operatorname{dn}^2{\sqrt{\lambda}}} =Q_{\lambda} \biggl(\frac{\pi}2-\varkappa\biggr),
\end{equation*}
\notag
$$
because the transition from $\varkappa$ to $\pi/2-\varkappa$ interchanges $k$ and $k'$. Now, the chain of equalities
$$
\begin{equation*}
\begin{aligned} \, &\frac{{k'}^2}{\sqrt{-\lambda}} \int_0^{\sqrt{-\lambda}}{\frac{d\tau}{\operatorname{dn}^2{\tau}}} =\frac{{k'}^2}{i\sqrt{\lambda}} \int_0^{\sqrt{\lambda}}{\frac{i\,d\tau}{\operatorname{dn}^2{i\tau}}} = \frac{{k'}^2}{\sqrt{\lambda}} \int_0^{\sqrt{\lambda}}{\frac{{\,}'\!\operatorname{cn}^2{\tau}}{{\,}'\!\operatorname{dn}^2{\tau}}\,d\tau} \\ &\qquad =\frac{1}{\sqrt{\lambda}} \int_0^{\sqrt{\lambda}}{\biggl(1- \frac{k^2}{{\,}'\!\operatorname{dn}^2{\tau}}\biggr)\,d\tau} =1-\frac{k^2}{\sqrt{\lambda}} \int_0^{\sqrt{\lambda}}{\frac{d \tau}{{\,}'\!\operatorname{dn}^2{\tau}} } \end{aligned}
\end{equation*}
\notag
$$
implies that
$$
\begin{equation*}
\begin{aligned} \, X_{-\lambda} (\varkappa) &=-1+\frac{2 {k'}^2}{\sqrt{- \lambda}} \int_0^{\sqrt{- \lambda}}{\frac{d \tau}{\operatorname{dn}^2{\tau}}} =1-\frac{2 k^2}{\sqrt{\lambda}} \int_0^{\sqrt{\lambda}}{\frac{d \tau}{{\,}'\!\operatorname{dn}^2{\tau}} }=- X_{\lambda} \biggl(\frac\pi2-\varkappa\biggr), \\ \frac32 Z_{-\lambda} (\varkappa) &=\frac{{k'}^2 (k^2-{k'}^2) }{(-\lambda)^{3/2}} \int_0^{\sqrt{-\lambda}}{\frac{d \tau}{\operatorname{dn}^2{\tau}}}+\frac{{k'}^2}{(-\lambda)} -\frac{2 k^2 {k'}^2 \operatorname{sn}{\sqrt{-\lambda}} \, \operatorname{cn}{\sqrt{-\lambda}}}{(-\lambda)^{3/2} \operatorname{dn}^3{\sqrt{-\lambda}}} \\ &=\frac{ {k'}^2-k^2 }{\lambda} \biggl\{ \frac{{k'}^2}{\sqrt{-\lambda}} \int_0^{\sqrt{-\lambda}}\frac{d\tau}{\operatorname{dn}^2{\tau}}\biggr\} -\frac{{k'}^2}{\lambda}+ \frac{2 k^2 {k'}^2 \operatorname{sn}{\sqrt{-\lambda}} \, \operatorname{cn}{\sqrt{-\lambda}}}{i \lambda^{3/2} \operatorname{dn}^3{\sqrt{-\lambda}}} \\ &=\frac{ {k'}^2-k^2 }{\lambda} \biggl\{ 1-\frac{k^2}{\sqrt{\lambda}} \int_0^{\sqrt{\lambda}}{\frac{d \tau}{{\,}'\!\operatorname{dn}^2{\tau}}} \biggr\}-\frac{{k'}^2}{\lambda}+ \frac{2 k^2 {k'}^2 {\,}'\!\operatorname{sn}{\sqrt{\lambda}} \, \operatorname{cn}'{\sqrt{\lambda}}}{\lambda^{3/2} {\,}'\!\operatorname{dn}^3{\sqrt{\lambda}}} \\ &=-\frac{ k^2 ({k'}^2-k^2) }{\lambda^{3/2}} \int_0^{\sqrt{\lambda}}{\frac{d \tau}{{\,}'\!\operatorname{dn}^2{\tau}}}-\frac{k^2}{\lambda}+\frac{2 k^2 {k'}^2 {\,}'\!\operatorname{sn}{\sqrt{\lambda}} \, \operatorname{cn}'{\sqrt{\lambda}}}{\lambda^{3/2} {\,}'\!\operatorname{dn}^3{\sqrt{\lambda}}} \\ &=- \frac32 Z_{\lambda} \biggl(\frac\pi2- \varkappa\biggr). \end{aligned}
\end{equation*}
\notag
$$
Thus we have shown that the functions (1.8)–(1.10) are invariant under the substitutions (1.6) and (1.7). The proof is complete. Proof of Theorem 1. We use the well-known formulae expressing the degeneration of Jacobi elliptic functions as $k\to\infty$:
$$
\begin{equation*}
\operatorname{sn}{\tau}=\sin{\tau}+O(k^2) \quad\text{and} \quad \operatorname{cn}{\tau}=\cos{\tau}+O(k^2)
\end{equation*}
\notag
$$
(see [12], § 13.18, formula (6), and [13], Ch. IV, formulae (205)). They show that
$$
\begin{equation*}
\begin{aligned} \, \int_0^{\sqrt{\lambda}}{\frac{d \tau}{\operatorname{dn}^2{\tau}}} &=\int_0^{\sqrt{\lambda}} (1+ k^2 \sin^2{\tau})\,d \tau+O(k^4) \\ &=\sqrt{\lambda}+\frac{k^2 \sqrt{\lambda}}2-\frac{k^2 \sin(2\sqrt{\lambda})}4+ O(k^4), \end{aligned}
\end{equation*}
\notag
$$
because (see [12], § 13.17, formulae (2), and [13], Ch. II, formulae (77))
$$
\begin{equation*}
\operatorname{dn}^2{\tau}=1-k^2 \operatorname{sn}^2{\tau}=1-k^2 \sin^2{\tau} +O(k^4).
\end{equation*}
\notag
$$
Bearing in mind that
$$
\begin{equation*}
k=\sin{\varkappa} \to 0\quad\text{and} \quad k'=\cos{\varkappa}=\sqrt{1-k^2}=1+O(k^2)
\end{equation*}
\notag
$$
as $\varkappa \to 0$, from this and formulae (1.4) and (1.8)–(1.10) we obtain
$$
\begin{equation*}
\begin{gathered} \, \begin{aligned} \, X_\lambda (\varkappa) &=-1+\frac{2 (1-k^2)}{\sqrt{\lambda}} \int_0^{\sqrt{\lambda}}{\frac{d \tau}{\operatorname{dn}^2{\tau}}} \\ &=- 1+2 (1-k^2) \biggl(1+\frac{k^2}{2}- \frac{k^2 \sin(2\sqrt{\lambda})}{4 \sqrt{\lambda}}\biggr)+O(k^4) \\ &=1-k^2- \frac{k^2 \sin{\sqrt{4 \lambda}} } {\sqrt{4 \lambda}}+O(k^4), \end{aligned} \\ Y_\lambda (\varkappa) =\frac{2 k k'\operatorname{sn}{\sqrt{\lambda}}}{\sqrt{\lambda} \, \operatorname{dn}{\sqrt{\lambda}}}= \frac{2 k \sin{\sqrt{\lambda}}}{\sqrt{\lambda}}+O(k^3), \\ \begin{aligned} \, Z_\lambda (\varkappa) &=\frac{2 (1-{k'}^2)}{3 \lambda^{3/2}} \biggl\{ (2 k^2-1) \int_0^{\sqrt{\lambda}}{\frac{d \tau}{\operatorname{dn}^2{\tau}}}-\frac{2 k^2 \operatorname{sn}{\sqrt{\lambda}} \, \operatorname{cn}{\sqrt{\lambda}}}{\operatorname{dn}^3{\sqrt{\lambda}}}+\sqrt{\lambda} \biggr\} \\ &=\frac{2}{3 \lambda} \biggl\{ (2 k^2-1) \biggl(1+\frac{k^2}{2}- \frac{k^2\sin(2\sqrt{\lambda})}{4 \sqrt{\lambda}}\biggr) -\frac{k^2\sin(2\sqrt{\lambda})}{\sqrt{\lambda}}+1 \biggr\}+O(k^4) \\ &=\frac{k^2}{\lambda} \biggl\{ 1-\frac{\sin{\sqrt{4 \lambda}}}{\sqrt{4 \lambda}} \biggr\}+ O(k^4), \end{aligned} \\ P_\lambda(\varkappa)=1-2 k^2=1+O(k^2), \qquad Q_\lambda (\varkappa)=\frac{2 k k'\operatorname{cn}{\sqrt{\lambda}}}{\operatorname{dn}^2{\sqrt{\lambda}}}=2 k \cos{\sqrt{\lambda}}+O(k^3). \end{gathered}
\end{equation*}
\notag
$$
Substituting in the expansion $k= \varkappa+O(\varkappa^3)$ (which holds because $k=\sin{\varkappa}$), we obtain the asymptotic formulae for $X_\lambda$, $Y_\lambda$, $Z_\lambda$, $P_\lambda$ and $Q_\lambda$ from the theorem. The asymptotic formula for $R_\lambda$ coincides with the explicit formula (1.4) for this coefficient.
Theorem 1 is proved. Proof of Corollary 1. Fix $\lambda \in \mathbb{R}$ and set $A=1+\gamma(4 \lambda)$. By Theorem 1 we have $1- X_\lambda(\varkappa)=A \varkappa^2+O(\varkappa^4)$ as $\varkappa \to 0$. Since $A > 0$ for all $\lambda$, the equation ${X_\lambda (\varkappa)=1-\varepsilon}$ has just two roots $\varkappa=\varkappa_\pm(\varepsilon)$ as $\varepsilon \to 0+$, and we have $\varkappa_\pm^2(\varepsilon)=\varepsilon/A+O(\varepsilon^2)$. Hence $\varkappa_\pm (\varepsilon)=\pm \sqrt{\varepsilon}/\sqrt{A}+O(\varepsilon^{3/2})$; in particular, $\varkappa_\pm (\varepsilon)=O(\sqrt{\varepsilon})$. Now from Theorem 1 we obtain
$$
\begin{equation*}
\frac{Y_\lambda(\varkappa_\pm(\varepsilon))}{\sqrt{\varepsilon}} =\frac{2 \varkappa_\pm(\varepsilon) \gamma(\lambda)+O(\varkappa_\pm^3(\varepsilon))}{\sqrt{\varepsilon}} =\pm \frac{2 \gamma(\lambda)}{\sqrt{A}}+O(\varepsilon)
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\frac{Z_\lambda(\varkappa_\pm(\varepsilon))}{\varepsilon} = \frac{\varkappa_\pm^2(\varepsilon)}{\varepsilon} \,\frac{1-\gamma(4 \lambda)}{\lambda}+ \frac{O(\varkappa_\pm^4(\varepsilon))}{\varepsilon} =\frac{1-\gamma(4 \lambda)}{\lambda A}+ O(\varepsilon).
\end{equation*}
\notag
$$
These formulae prove parts 1)–3) of Corollary 1.
Now we use the following expansions as $\lambda \to+\infty$, which hold to within infinitesimals of infinite order, which we conventionally denote by $O(\lambda^{-\infty})$:
$$
\begin{equation*}
\gamma(-\lambda)=\frac{\sinh{\sqrt{\lambda}}}{\sqrt{\lambda}}=\frac{e^{\sqrt{\lambda}}}{2 \sqrt{\lambda}}+O(\lambda^{-\infty})\quad\text{and} \quad \gamma(- 4 \lambda)=\frac{e^{2 \sqrt{\lambda}}}{4 \sqrt{\lambda}}+O(\lambda^{-\infty}).
\end{equation*}
\notag
$$
Then we obtain
$$
\begin{equation*}
y=\pm \frac{2 \gamma(-\lambda)}{\sqrt{1+\gamma(-4 \lambda)}}=\pm \frac{2}{\lambda^{1/4}}+ O(\lambda^{-\infty})
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
z=\frac{1-\gamma(- 4 \lambda)}{- \lambda(1+\gamma(- 4 \lambda))} = \frac{1}{\lambda}+ O(\lambda^{-\infty}),
\end{equation*}
\notag
$$
so that $z=(y/2)^4+O (|y|^{+\infty})$, that is, $z=y/16+O(|y|^{N})$ for each $N \in \mathbb{N}$. The proof of part 4) of Corollary 1 is complete. Proof of Theorem 2. Let $k=\sin{\varkappa}$ and $k'=\cos{\varkappa}$, and let $\varkappa \to 0+$. Then $k \to 0+$, $k' \to 1-$ and $K' \to+\infty$, and the well-known asymptotic formula (see [12], § 13.8, formula (10), or [13], Ch. I, formula (144)) yields
$$
\begin{equation*}
k^2=O(e^{-2 K'}), \qquad-\log{k^2}=O(K')\quad\text{and} \quad k'=\sqrt{1-k^2}=1+ O(e^{-2 K'}).
\end{equation*}
\notag
$$
Hence from the well-known asymptotic formula (see [13], Ch. I, formula (149)) for the complete elliptic integral of the second kind we obtain
$$
\begin{equation*}
E'=\int_0^{\pi/2}\sqrt{1-{k'}^2 \sin^2{\theta}} \, d \theta =1+O(-k^2 \log{k^2})=1+O(K'e^{-2 K'}).
\end{equation*}
\notag
$$
On the other hand, well-known formulae (see [12], § 13.18, formula (4), and [13], Ch. IV, formulae (217)) for the degeneration of Jacobi elliptic functions of $u' = K' + \mu_{b,\varkappa} = 2 n K' + b$ show that
$$
\begin{equation*}
\begin{gathered} \, {\,}'\!\operatorname{sn}{u'}=(-1)^n {\,}'\!\operatorname{sn}{b}=(-1)^n \tanh{b}+O(k^2)=(-1)^n \tanh{b}+O (e^{-2 K'}), \\ {\,}'\!\operatorname{cn}{u'}=(-1)^n {\,}'\!\operatorname{cn}{b}=\frac{(-1)^n}{\cosh{b}}+O(k^2)=\frac{(-1)^n}{\cosh{b}}+O(e^{-2 K'}) \end{gathered}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
{\,}'\!\operatorname{dn}{u'}={\,}'\!\operatorname{dn}{b}=\frac{1}{\cosh{b}}+O(k^2) =\frac{1}{\cosh{b}}+O(e^{-2K'}),
\end{equation*}
\notag
$$
because
$$
\begin{equation*}
{\,}'\!\operatorname{sn}{(2 K'+b)}=- {\,}'\!\operatorname{sn}{b}, \qquad {\,}'\!\operatorname{cn}{(2 K'+b)}=- {\,}'\!\operatorname{cn}{b}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
{\,}'\!\operatorname{dn}{(2 K'+b)}={\,}'\!\operatorname{dn}{b}
\end{equation*}
\notag
$$
(see [12], § 13.17, formulae (8) and (9), and [13], Ch. II, formulae (6), (10), (21), (32) and (37)). The function ${\,}'\!\operatorname{dn}$ is even and $2 K'$-periodic, so
$$
\begin{equation*}
\begin{aligned} \, \int_{K'}^{u'} {\,}'\!\operatorname{dn}^2{v} \, dv &=\int_{K'}^{2 n K'+b} {\,}'\!\operatorname{dn}^2{v} \, dv=\sigma_n \int_{0}^{K'} {\,}'\!\operatorname{dn}^2{v} \, dv+\int_{0}^{b} {\,}'\!\operatorname{dn}^2{v} \, dv \\ &=\sigma_n E'+\int_{0}^{b} {\,}'\!\operatorname{dn}^2{v} \, dv=\sigma_n+\int_{0}^{b} \frac{dv}{\cosh^2{b}}+O(K'e^{-2K'}) \\ &=\sigma_n+\tanh{b}+O(K'e^{-2K'}), \end{aligned}
\end{equation*}
\notag
$$
because (see [13], Ch. II, formulae (148), (1) and (2))
Using Theorem 1 and (1.1), from the above asymptotic expressions we obtain
$$
\begin{equation*}
\begin{aligned} \, 1-X_{- \mu_{b,\varkappa}^2} (\varkappa) &=1+X_{\mu_{b,\varkappa}^2} \biggl(\frac\pi2-\varkappa\biggr) =\frac{2}{\mu_{b,\varkappa}} \int_{K'}^{u'} {\,}'\!\operatorname{dn}^2{v} \, dv \\ &=\frac{2}{\mu_{b,\varkappa}} \bigl(\sigma_n+\tanh{b}+O(K'e^{-2K'})\bigr) =\frac{2 (\sigma_n+\tanh{b})}{\mu_{b,\varkappa}}+O(e^{-2K'}), \\ Y_{- \mu_{b,\varkappa}^2} (\varkappa) &=Y_{\mu_{b,\varkappa}^2} \biggl(\frac\pi2-\varkappa\biggr) =- \frac{2 k'}{\mu_{b,\varkappa}} {\,}'\!\operatorname{cn} {u'} \\ &=-\frac{2}{\mu_{b,\varkappa}} \biggl(\frac{ (-1)^{n}}{\cosh{b}}+O(e^{-2K'})\biggr) =\frac{ 2 (-1)^{n+1}}{\mu_{b,\varkappa} \cosh{b}}+O\biggl(\frac{e^{-2K'}}{K'}\biggr), \end{aligned}
\end{equation*}
\notag
$$
because $\mu_{b,\varkappa}=\sigma_n K'+b$. In a similar way, using Theorem 1 and (1.2), from the above asymptotic expressions we obtain
$$
\begin{equation*}
\begin{aligned} \, &Z_{- \mu_{b,\varkappa}^2} (\varkappa) =- Z_{\mu_{b,\varkappa}^2} \biggl(\frac\pi2-\varkappa\biggr) \\ &\qquad=- \frac{2}{3 \mu_{b,\varkappa}^3} \biggl\{ (2 {k'}^2-1) \int_{K'}^{u'} {\,}'\!\operatorname{dn}^2{v} \, dv+ 2 {k'}^2 {\,}'\!\operatorname{sn}{u'} \, {\,}'\!\operatorname{cn}{u'} \, {\,}'\!\operatorname{dn}{u'}+k^2 \mu_{b,\varkappa} \biggr\} \\ &\qquad=- \frac{2}{3 \mu_{b,\varkappa}^3} \biggl\{ \sigma_n+\tanh{b}+\frac{2 \tanh{b}}{\cosh^2{b}}+ O(K'e^{-2K'}) \biggr\} \\ &\qquad=- \frac{2 (\sigma_n+3 \tanh{b}-2 \tanh^3{b})}{3 \mu_{b,\varkappa}^3} +O\biggl(\frac{e^{-2 K'}}{{K'}^2}\biggr). \end{aligned}
\end{equation*}
\notag
$$
Finally, using Theorem 1 and formulae (1.4) and (1.5), from the above asymptotic expressions we obtain
$$
\begin{equation*}
\begin{gathered} \, Q_{- \mu_{b,\varkappa}^2} (\varkappa) =Q_{\mu_{b,\varkappa}^2} \biggl(\frac\pi2-\varkappa\biggr) =2 k' {\,}'\!\operatorname{sn}{u'} \, {\,}'\!\operatorname{dn}{u'} =\frac{2 (-1)^{n} \tanh{b}}{\cosh{b}} +O(e^{-2K'}), \\ P_{-\mu_{b,\varkappa}^2} (\varkappa)=1-2 k^2=1+O(e^{-2K'})\quad\text{and} \quad R_{-\mu_{b,\varkappa}^2} (\varkappa)=- \mu_{b,\varkappa}^2. \end{gathered}
\end{equation*}
\notag
$$
Theorem 2 is proved for $\varkappa \to 0+$. In the case when $\varkappa \to 0-$ we use Theorem 1; if the sign of $\varkappa$ is changed, then $Y$ and $Q$ change sign, while $X$, $P$, $Z$ and $R$ do not.
Theorem 2 is proved. Proof of Corollary 2. Since $\mu_{b,\varkappa}=\sigma_n K'+b$, we can write the expansions in Theorem 2 up to infinitesimals of infinite order as $\mu_{b,\varkappa} \to+\infty$, which we conventionally denote by $O(\lambda^{-\infty})$:
$$
\begin{equation*}
\begin{gathered} \, X_{- \mu_{b,\varkappa}^2} (\varkappa) =1-\frac{2(\sigma_n+\tanh{b})}{\mu_{b,\varkappa}}+O (\mu_{b,\varkappa}^{-\infty}), \\ Y_{- \mu_{b,\varkappa}^2} (\varkappa)= \frac{2(-1)^{n+1} \operatorname{sgn}{\varkappa}}{\mu_{b,\varkappa} \cosh{b}}+O(\mu_{b,\varkappa}^{-\infty}), \\ Z_{- \mu_{b,\varkappa}^2} (\varkappa)=- \frac{2(\sigma_n+3 \tanh{b}-2 \tanh^3{b})}{3 \mu_{b,\varkappa}^3}+O(\mu_{b,\varkappa}^{-\infty}). \end{gathered}
\end{equation*}
\notag
$$
From the equation $X_{- \mu_{b,\varkappa}^2} (\varkappa)=1-\varepsilon$ we obtain
$$
\begin{equation*}
\mu_{b,\varkappa} =\frac{2(\sigma_n+\tanh{b})}{\varepsilon}+O(\varepsilon^{+\infty}),
\end{equation*}
\notag
$$
because $\sigma_n+\tanh{b} > 0$ for all $b$. Hence
$$
\begin{equation*}
\frac{Y_{- \mu_{b,\varkappa}^2}^2 (\varkappa)}{\varepsilon^2} = \frac{4}{\varepsilon^2 \mu_{b,\varkappa}^2 \cosh^2{b}}+O(\mu_{b,\varkappa}^{-\infty}) = \frac{1- \tanh^2{b}}{(\sigma_n+\tanh{b})^2}+O(\varepsilon^{+\infty})
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\frac{Z_{- \mu_{b,\varkappa}^2} (\varkappa)}{\varepsilon^3}=- \frac{2(\sigma_n+3 \tanh{b}-2 \tanh^3{b})}{12(\sigma_n+\tanh{b})^3}+O(\varepsilon^{+\infty}).
\end{equation*}
\notag
$$
After the substitution $\tanh{b}=\beta$ we obtain the curves in Corollary 2. The fact that for $n=1$ ($\sigma_1=1$) we have $z=y^4/16-y^2/8-1/48$ is verified by a substitution.
Corollary 2 is proved. 
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\Bibitem{Bog22} |
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