Supplementary material of the paper: Linking the rotation of a rigid body to the Schrödinger equation: The quantum tennis racket effect and beyond

L. Van Damme, D. Leiner, P. Mardešić, S. J. Glaser, and D. Sugny1,4∗ 1 Laboratoire Interdisciplinaire Carnot de Bourgogne (ICB), UMR 6303 CNRS-Université Bourgogne Franche-Comté, 9 Av. A. Savary, BP 47 870, F-21078 DIJON Cedex, FRANCE 2 Department of Chemistry, Technical University of Munich, Lichtenbergstrasse 4, D-85747 Garching, Germany 3 Institut de Mathématiques de Bourgogne, UMR 5584 CNRS-Université de Bourgogne Franche-Comté, 9 Av. A. Savary, BP 4787

This supplementary material gives a theoretical description of the tennis racket effect in a three-dimensional rigid body and describes the analogy that can be established between the free rotation of a rigid body and the control of the Bloch equations by magnetic fields.
This work is organized as follows. Section I summarizes the main results about the two dynamical systems. Two different identifications, corresponding to the cases (a) and (b), are proposed. Sections II and III focus on the analytical derivation of the solutions of the differential systems. We study the robustness of a state to state transfer in the two situations. We also show how the Euler angles can be used to parameterize the global phase of a quantum state. This analysis completes the formal mathematical link between the free rotation of a rigid body and the dynamics of a two-level quantum system. The classical and the quantum tennis racket effects are discussed in Sec. IV. We show in Sec. V how the dynamics of the rigid body can be used to implement one qubit quantum gates. The example of the Hadamard gate is investigated in details. The robustness issue against experimental imperfections of the gates is also discussed. The Montgomery phase, a geometric feature of the free rotation of a rigid body, is derived to realize a non-adiabatic geometric phase gate.
A Matlab code computing the trajectories of a rigid body and of the corresponding Bloch vector is also provided and a short description given in Sec. VI. Some standard properties of Jacobi's elliptic functions are detailed in Sec. VII.

A. A classical rigid body
The free rotation of a rigid body in classical mechanics is based on the motion of its angular momentum L, which has a constant norm | L| = ℓ [1,2]. This norm can be set to ℓ = 1 without loss of generality [2]. We introduce the frame ( e 1 , e 2 , e 3 ) attached to the rigid body. These three vectors define the principal axes of inertia of the body. An example is given below with the tennis racket. The time evolution of L in the frame ( e 1 , e 2 , e 3 ) is ruled by Euler's equations: 2 where Ω is the angular velocity vector. In matrix form, Eq. (1) reads: The components of Ω = (Ω 1 , Ω 2 , Ω 3 ) in ( e 1 , e 2 , e 3 ) can be written in terms of the ones of L = (L 1 , L 2 , L 3 ) as follows: where I 1 , I 2 and I 3 are the principal moments of inertia.
For a homogeneous rigid body, the principal moments of inertia are related to the shape of the solid. More precisely, they correspond to the repartition of the mass along the three principal axes of inertia e 1 , e 2 and e 3 [2]. For a tennis racket, the principal axes of inertia are such that e 1 is along the handle, e 2 is perpendicular to the handle and in the plane defined by the head of the racket, and e 3 is perpendicular to the head of the racket [8]. In this configuration, we have I 1 < I 2 < I 3 . The frame attached to the racket is represented in  Substituting Eq. (3) into Eq. (1), we can integrate the dynamical system and derive the solutions for L 1 (t), L 2 (t) and L 3 (t). The system has two constants of motion, the total mechanical energy and the angular momentum: 3 The conservations of the energy and of the angular momentum correspond geometrically to the equation of an ellipsoid of radii I 1 √ 2E, I 2 √ 2E and I 3 √ 2E, and to a sphere of radius 1, respectively. The classical trajectory L(t) lies on the intersection of the two surfaces, and it can be expressed in terms of Jacobi's elliptic functions [2].

B. The Bloch equation
We consider a general two-level quantum system defined by the state |λ(t) whose dynamics is governed by the Schrödinger equation i ∂ t |λ =Ĥ|λ , where the HamiltonianĤ is given by: Ω(t) and η(t) are respectively the real amplitude and phase of the control field, and ∆(t) is the offset with respect to the Larmor frequency of the system [3,4]. We can first consider the case for which η(t) = 0. The corresponding HamiltonianĤ A reads: A second option consists in working at resonance, setting ∆(t) = 0. The HamiltonianĤ B is then given by:Ĥ with ω 1 = Ω cos η and ω 2 = Ω sin η the two real control fields. The Bloch vector M (t) can be defined in terms of the components of the density matrixρ = |λ λ| as M 1 =ρ 21 +ρ 12 , It is then straightforward to identify these differential equations with the free rotation of a rigid body given by Eq. (1), with M = L, defining thus some particular families of control fields. Sections II and III will be dedicated to the integration of the Euler equations and to the derivation of the corresponding control fields.

II. INTEGRATION OF THE BLOCH EQUATION: CASE (A)
A. Analytical derivation of the solutions In case (a), the identification between the two dynamics leads to ∆(t) = Ω 3 (t) = M 3 (t)/I 3 , Ω(t) = Ω 1 (t) = M 1 (t)/I 1 and Ω 2 (t) = 0. Note that the condition Ω 2 (t) = 0 induces a constraint on the classical rigid body, I 2 → ∞. The two other principal moments of inertia can be set to I 1 = 1 and I 3 = 1/k 2 , k ∈ [0, 1], without loss of generality. We obtain the following Bloch equation: The solutions of this system lie in the intersection of the angular momentum sphere and of the energy ellipsoid. Here, the angular momentum sphere is simply the Bloch sphere, and the energy ellipsoid becomes an elliptic cylinder since I 2 → ∞. More precisely, the equations of the two surfaces are given by:  : that is a standard constant pulse along e 1 . The separatrix corresponds to the case 2E = k 2 as shown in Tab. I. This particular pulse given in terms of hyperbolic functions is a general Allen-Eberly solution, which is discussed in Sec. II D. The second stable point occurs in the

Solutions of the Euler equations
Oscillating Rotating Separatrix k sn(t + ρ, m)

Control fields of the Bloch equation
Oscillating Rotating Separatrix  oscillating mode, when 2E = 0. This value leads to a zero field and is not really interesting.
The oscillating solutions give a smooth transition between a zero field and an Allen-Eberly type pulse sequence [6]. and k → 1 (right). The separatrix is associated with some Allen-Eberly type control fields given in Sec. II D.
of Tab. I, we get: which is a constant pulse about e 1 of amplitude 1. The trajectory along the separatrix is here of the form: with a field area A given by: A = +∞ −∞ |Ω|dt = π. For k = 0, we deduce that the transfer from the north pole to the south pole on the Bloch sphere (called here the inversion of the state) is realized by a π pulse with a hyperbolic secant shape.
For k = 1, the rotating solutions do not exist and the different trajectories are parallel to the plane ( e 1 , e 3 ). If we substitute k = 1 in the oscillating fields of Tab. I, we obtain: The components of the Bloch vector can be expressed as: For a trajectory in the neighborhood of the separatrix, 2E is close to 1 and the area of the field Ω goes to infinity. This transfer corresponds to an adiabatic inversion [9].
To summarize, the choice of the value of the parameter k allows us to make a compromise between a constant pulse (k → 0) and an adiabatic pulse (k → 1). Along the separatrix, the solution is an Allen-Eberly solution going from a hyperbolic π pulse for k = 0 to an adiabatic inversion when k → 1.

D. The Allen-Eberly solution
The control fields of Tab. I along the separatrix can be viewed as Allen-Eberly type solutions [6,[11][12][13] of the form: where 2δ is the magnitude of the frequency sweep and τ an arbitrary pulse length. In order to simplify the different expressions, we introduce the normalized quantities: We then obtain: Using δτ = k/ √ 1 − k 2 , we get the solution on the separatrix (see Tab. I). Starting at t = 0 on the north pole of the Bloch sphere, i.e. ρ → −∞, the pulse brings the system to the south pole at t → +∞. The area A of the pulse is given by: If k → 0, we obtain a standard π pulse which naturally inverts the population of the twolevel quantum system. In the case k > 0, the area is larger than π, but the transfer is still an inversion. The larger the area is, the more robust the control fields are [9]. Note that Fig. 2 gives an instructive geometrical interpretation of the Allen-Eberly solution. The separatrix connects the north and the south poles of the sphere in an infinite time.

E. Efficiency and robustness of the inversion transfer
In this paragraph, we analyze the properties of three trajectories realizing an inversion process on the Bloch sphere. We define a small positive parameter ǫ corresponding to the initial polar angle of the chosen trajectory. We compare the fields attached to a rotating curve, an oscillating curve and the separatrix as illustrated in Fig. 4. The expression of the fields is given in Tab. I. The phase ρ is computed so that M 1 (0) = 0 for the oscillating field and M 2 (0) = 0 for the rotating one, which leads to ρ = K(m), where K denotes an  Tab. II. Table III gives TABLE II. Expressions of the control fields corresponding to the three curves represented in Fig. 4.
The function K(·) denotes the elliptic integral of the first kind.
the final time t f and the area A = t f 0 |Ω|dt for the three sets of fields. Note that a series expansion around ε = 0 is used to derive simple expressions for t f and A. Table III shows that the control time and the area can be changed to some extent by a judicious choice of the parameters ε and k. We observe that, for k 2 < 1/2, the rotating field

Oscillating
Rotating Separatrix allows us to make an inversion in a shorter time t f , but needs more energy. In contrast, the oscillating mode offers the lowest energy and the shortest time if k 2 > 3/4.
Another important feature of the control fields is their robustness property. Here, we focus on the rotating solutions and the parameter ǫ is set to ǫ = 10 −2 . Figure with respect to a scaling factor α on the amplitude of the field and an arbitrary offset term, δ, defined as follows: We  In summary, the parameter k can be mainly used to adjust the robustness of the fields.
The parameter ǫ allows to change the area of the field and the control time.
(we recall that in case (a), I 2 = +∞). We set I 1 = 1 and I 2 = 1/k 2 and we obtain the differential system:Ṁ The energy surface is an elliptic cylinder along the vector e 3 : The corresponding trajectories are plotted on the Bloch sphere in Fig. 7. We observe that the separatrix makes a transfer on the equator from the state M 0 = (0, 1, 0) to M f = (0, −1, 0). This process has some robustness properties, as for the inversion process in case (a). The Oscillating Rotating Separatrix We follow here the same approach as the one used in Sec. II E. We consider the transfer from M 0 = (0, 1, 0) to M f = (0, −1, 0), through a rotating curve so that: 13 The fields can be expressed as follows: with m given in Tab. IV. We set ǫ = 10 −2 and we investigate the robustness properties. In this case, the Bloch equation reads: (c) and (d). In each case, the parameter ǫ is set to 10 −2 .

C. Global phase
The Bloch vector M (t) does not take into account the global phase of the quantum state.
We show for the case (b) how to derive this phase by introducing the Euler angles, which completes the analogy between the two-level quantum system and the classical rigid body.
14 We introduce three Euler angles θ, φ and ψ as shown in Fig. 9. The set of angles θ, φ and ψ can be used to parameterize the wave function |λ(t) as: The dynamics of the three angles can be obtained by substituting Eq. (19) into the Schrödinger equation (6). We arrive at: In the Bloch representation, the two angles θ and φ define the vector M (t) such that: The two control fields can be expressed in terms of the Euler angles as follows: 15 The final dynamical system to solve can be written as:

D. Rotation matrix
We introduce a 3 × 3 matrix R defined by R(t) = ( Q(t), P (t), M(t)), and we denote by The propagator R f is given by: The matrix R can be written as a function of the Euler angles, R = R φ R θ R ψ with: We introduce the angles (θ 0 , φ 0 , ψ 0 ) and (θ f , φ f , ψ f ), which are respectively the initial and final values of the Euler angles. The initial global phase ψ 0 is irrelevant, and can be set to 0 without loss of generality.
We will see in the next section that for ψ f = 3π (n = 1), the dynamics of this process is associated with the tennis racket effect [7]. This effect is described in Sec. IV.

IV. THE CLASSICAL AND THE QUANTUM TENNIS RACKET EFFECTS
The classical tennis racket effect (TRE) is a particular motion which occurs for some trajectories in the neighbourhood of the separatrix [7]. We recall that, for a classical rigid body, e 1 , e 2 and e 3 are the principal axes of inertia of the solid. If the principal moments of inertia are such that I 1 < I 2 < I 3 (it is the used convention in this work) then a rotation about the e 1 -and e 3 -axes is stable, but a rotation about e 2 is unstable. This point can be checked in Fig. 7.
The TRE is easier to figure out by considering the classical case, where ( X, Y , L) is the laboratory frame and ( e 1 , e 2 , e 3 ) is the frame attached to the racket [7,8]. In this representation, e 1 is along the handle, e 2 is perpendicular to the handle and belongs to the head of the racket, and e 3 is perpendicular to the head. The corresponding process is displayed in Fig. 10. For a standard tennis racket, this process occurs for trajectories starting in the neighbourhood of the unstable equilibrium point, i.e. L ≃ e 2 at time t = 0. Note that if we apply two times this motion then the racket goes back to its initial position. In other words, if we consider the frame S = ( e 1 , e 2 , e 3 ), we have: We have now all the tools in hand to describe the quantum TRE. In the (θ,φ,ψ)-represen- tation, the TRE satisfies: If we substitute these relations into the wave function of Eq. (19), we obtain that the wave function goes back to its initial state after the application of four TRE, i.e.
Among all the possible trajectories, only one exactly satisfies the quantum TRE process of Eq. (25). Since θ 0 = π/2, we can deduce from Fig. 7 that this curve is a rotating one, starting The corresponding fields are given by: We now apply this set of control fields to the quantum system, in order to implement the quantum tennis racket effect. The control process is a unitary transformation which does not depend on the initial conditions. We consider the following initial point: The TRE-pulse makes a global rotation about the axis e 1 . The three states are given at time t f by: We consider the case k = 0.7. The control fields are computed from Eq. (26). The dynamics of the vectors Q, P and M are plotted in Fig. 11. We also represent in Fig. 11 the motion of the vectors e 1 , e 2 and e 3 in the ( Q, P , M)-frame, which is more suitable to describe the classical TRE (in the classical problem, ( e 1 , e 2 , e 3 ) is the frame attached to the racket).

V. IMPLEMENTATION OF QUANTUM GATES
The goal of this section is to show how the dynamics of a rigid body can be used to implement one-qubit quantum gates. In Sec. V A, we consider the example of the Hadamard gate before generalizing this result to any one-qubit quantum gate in Sec. V B. Section V C is dedicated to the case of geometric phase gates. The robustness issue of the gates with respect to experimental imperfections is investigated in Sec. V D. Generalizing the BIR approach used in NMR [27,28], we show that the quantum gates can be made robust.
As an illustrative example, we consider the case of a NOT gate and its robustness against control field inhomogeneities.

A. The Hadamard gate
The Hadamard gate U H is a unitary transformation which can be decomposed into a π phase gate and a π 2 rotation gate. It reads: We can show that this gate corresponds to a transformation G H in SO (3): In a classical rigid body, the Hadamard gate realizes the following transfer: where R is the rotation matrix given in terms of Euler's angles. Since the classical angular momentum L must also satisfy the transfer (31), we deduce that: We choose a trajectory such that L 1 (0) = L 3 (0) and L 1 (t f ) = L 3 (t f ) at time t f , which ensures that the Hadamard transformation is satisfied for L, due to the symmetries of the Jacobi functions. These conditions can be verified for a rotating extremal if k < 1/ √ 2, an oscillating one if k > 1/ √ 2 and along the separatrix if k = 1/ √ 2. As an illustrative example, we set k = 0.5. The solution is a rotating extremal of the form: 20 It can be shown that the initial condition L 1 (0) = L 3 (0) implies that the parameter ρ is given by: The condition at time t f is satisfied if we choose t f such that: Equations (34) we obtain that θ(t f ) = θ(0) and φ(t f ) = −φ(0). Moreover, since the rotation matrix R can be written as R = R φ R θ R ψ , where R φ , R θ and R ψ are defined in Eq. (23), straightforward computations lead to the following global phase ψ(t f ): The Hadamard gate is associated with a trajectory of the rigid body satisfying Eqs. (34), (35) and (37). However, the global phase ψ can only be expressed in terms of elliptic integrals of the third kind. A numerical solution for k = 0.5 is given by the following initial angular momentum: The corresponding trajectories of the frames ( X, Y , L) and ( Q, P , M) of the classical and quantum systems are displayed in Fig. 12.

B. Generalization to any one-qubit gate
The goal of this paragraph is to show that the trajectories of the free rotation of a rigid body allow us to realize any one-qubit gate. We first present a proof showing that the reachable set of the control protocol is SO(3) and then a numerical method that can be applied for any gate. The rotation matrix R(t) associated with the classical system can be viewed as the optimal solution of the following dynamical system: with the constraint u 2 1 + u 2 2 = 1. Note that this system is fully controllable, which means that any rotation matrix R can be realized.
We denote by a ij the element of the ith row and jth column of the matrix R. We introduce r the vector defined as r = t (a 11 , a 21 , · · · , a 33 ), and we define the energy minimum pseudo-Hamiltonian H p of the Pontryagin Maximum Principle (See Ref. [14] for mathematical details): where p = t (b 11 , b 21 , · · · , b 33 ) is the adjoint state vector. We define the following angular momentum: The pseudo-Hamiltonian can be written in terms of the component of this angular momen-22 tum as: The Pontryagin Maximum Principle states that the dynamics and the associated control fields are optimal if this pseudo-Hamiltonian is maximum. The Pontryagin Hamiltonian H is defined so that H = max u 2 1 +u 2 2 =1 H p . This optimality condition is satisfied for the following control fields u 1 = . In this case, we get that 2E = L 2 1 + k 2 L 2 2 is constant and we can show that the dynamics of the angular momentum L satisfies the Euler's equation (15).
In order to illustrate the fact that any gate can be realized, we propose a numerical method to obtain the following gate: We define the function ∆ψ (see Fig. 9), which is the Montgomery phase. This phase can be expressed as the sum of a dynamical and a geometric parts as follows: where M is the norm of the angular momentum, T the period of the motion of M and S is the solid angle swept out by the angular momentum vector in the frame ( e 1 , e 2 , e 3 ). Starting from the conservation of the energy 2E = M 2 1 + k 2 M 2 2 , and using Eq. (22) we have: From cos 2 θψ = − cos θφ, we get: Thus, the variation ∆ψ of the phase for one period is given by: π/2 sin θdθdφ is the solid angle swept out by M . This formula is of the same form as the well-known Berry phase [17,18] in quantum mechanics which can be used for adiabatic [18] and periodic [19] trajectories on the Bloch sphere. Such geometric phases have been recently the subject of a large interest in quantum computing as a way to implement geometric and thus robust quantum gates [20,21]. One of the main difficulties to implement phase gates using geometric phases is to find a way to cancel the dynamical contribution of the phase. Different techniques have been proposed up to date [20,21]. For instance, Jones et al. [22] showed that the dynamical part of Berry's phase can be removed by using two cyclic trajectories of M(t) on the Bloch sphere, the second cycle being surrounded by a pair of π pulses. In [22], the geometric phase gate was implemented in the adiabatic regime, but it is possible to generalize this process to consider non-adiabatic cyclic evolution [23][24][25][26].
We propose here a method based on the free rotation of a rigid body and the TRE to implement a geometric phase gate in the non-adiabatic regime. We first recall that each trajectory M(t) of Fig. 7 can evolve in the backward direction by changing the sign of the control field ω 1 (t) (see Sec. I). If we consider two identical cycles, the second being followed in the backward direction along the same trajectory, then all the phases vanish, the dynamical ones as well as the geometric ones. However, if for the second cycle, the value of k is different from the first cycle, the total phase at the end of the process is given by ∆ψ cycle 1 − ∆ψ cycle 2 . This process does not cancel automatically the dynamical phases. To do so, a particular trajectory for the second cycle has to be chosen.
The method can be described as follows. To simplify the discussion and the analytical computations, we assume here that the system follows trajectories along the separatrices. A similar process can be designed by considering trajectories associated with the TRE, which are close to the separatrices. More precisely, we first make a transfer along the separatrix from the point (0, 1, 0) to the point (0, −1, 0) with a finite (long enough) time, and a fixed value k a . The second step consists in bringing the system from (0, −1, 0) to (0, 1, 0) also along the separatrix but with a different value of the parameter k, i.e. k b = k a . Since the time T b of the second process can be chosen arbitrarily long, we choose T b such that the two dynamical phases cancel each other. The global process is shown in Fig. 14 and described in Tab. VI.
In order to derive the geometric phase, we consider the case of a separatrix such that M 1 (t) > 0 and M 3 (t) > 0 for t > 0. We start from the following expression of the geometric 25 phase: For this separatrix, we have θ ∈ [0, π/2], which gives cos θ = 1 − sin 2 θ. Along the separatrix, 2E = k 2 and we can express sin 2 θ in terms of the angle φ using Eq. (45). We deduce that: Finally, the integration gives: We recall that the system evolves in the forward and backward directions for the cycles (a) and (b), respectively. Moreover, on the separatrix we have ω = k √ 1 − k 2 and 2E = k 2 .
Thus, for each cycle, the variation ∆ψ is given by: If we choose T b such that: then the two dynamical phases cancel each other and we get a purely geometric phase given by: Going back to the expression of the quantum state of Eq. (19), this corresponds to a phase gate of the form  The global process with the applied control fields is described in Tab. VI. The final geometric phase is equal to the area displayed in Fig. 14. In Fig. 14, we consider the following parameters T a = 100, k a = ( 2 − √ 2)/2 and k b = ( 2 + √ 2)/2 in order to implement a geometric phase equal to ∆ψ tot = π/2. changes in the cycle (b) in order to follow the second trajectory in the backward direction.
FIG. 14. (Color online) Trajectories on the Bloch sphere for implementing a π/2 phase gate. The brown (gray) surface is the total geometric phase at the end of the process i.e. S tot = π/2.

D. Robustness of the NOT gate
We show in this paragraph how the robustness of the control fields used to implement one-qubit gates can be improved. The approach is inspired from the BIR-pulses method in NMR [27,28]. We design a pulse which allows us to realize a NOT gate in a robust manner with respect to the control fields-inhomogeneities, which is equivalent to a BIR-1 pulse in the adiabatic limit [27,28]. We use the fact that the global phase of the state monotonically increases or decreases when the system follows a specific trajectory of the phase portrait.
This property allows us to cancel the global phase of the dynamics (both dynamical and geometrical terms) and then to realize a robust NOT gate. As an example, we consider here the separatrix of the configuration (a) (see Fig. 2).

27
The separatrix is first followed from the north pole to the equatorial plane of the sphere in the direction given in Fig. 2. We then transfer the state to another separatrix followed in the opposite direction in order to reach the south pole of the Bloch sphere. The two separatrices are connected with a constant pulse ∆(t) about e 3 . The global phase is cancelled leading to an inversion with a phase equal to zero, i.e. a NOT-gate along the e 2 -axis. The trajectory is represented in Fig. 15. We can show that the azimuthal angle between the two separatrices is equal to 2 arccos k. We thus need to apply a 2 arccos k-pulse along e 3 when the Bloch vector belongs to the equator. Note that in the adiabatic limit, which corresponds here to k → 1, this angle tends to zero. In the latter case, the method is equivalent to a 180 • BIR-1 pulse [27], with an Allen-Eberly hyperbolic shape. The details of the pulse are given in Tab. VII. The robustness of the control process is shown in Fig. 16 for a control field of the TABLE VII. Details of the pulse used to implement robust-NOT gates. T is the total duration of the process and T ∆ is the duration of the constant pulse along e 3 . form: 28 where α is a scaling factor corresponding to the fields inhomogeneities. The fidelity J is defined such that J = Trace( t R(t f ) · R f ), where R f is the target rotation matrix given by: The fidelity has been also computed for a standard π-pulse and for the TRE-pulse adapted to configuration (a) in order to compare the three methods.

VI. NUMERICAL IMPLEMENTATION OF THE TRAJECTORIES OF A RIGID BODY AND OF A CONTROL PULSE SEQUENCE
We provide in this supplementary material a Matlab code to compute the trajectories of a rigid body as a function of the parameters k and E. We also deduce the associated control fields and the dynamics of the Bloch vector for a state to state transfer. This code can be used by the interested reader to test some of the control protocols proposed in this work.

VII. JACOBI'S ELLIPTIC FUNCTIONS
We recall in this paragraph some standard properties of Jacobi's elliptic functions used in this paper [10]. Jacobi's elliptic functions generalize the standard trigonometric functions. This is a family of functions which includes cosine, sine and hyperbolic functions. They are written as follows: sn(u, m), cn(u, m) and dn(u, m). Their periods are related to the complete elliptic integral of the first kind K(m) as 4K(m), 4K(m) and 2K(m), respectively, with m a parameter belonging to the interval [0, 1]. The elliptic integral is defined as: In the case m = 0, we have K(m) = π/2 and the functions sn(u, m), cn(u, m) and dn(u, m) become sin(u), cos(u) and 1, respectively. In the case m = 1, we obtain K(m) = +∞ and the elliptic functions can be identified with tanh(u), sech(u) and sech(u). For a value of m such that m = 1 − ε with ε a small positive parameter, the function K(m) is, to the first order, given by: (57) Figure 17 shows Jacobi's elliptic functions for m = 1/2.