Reduction of divisors and Kowalevski top
RReduction of divisors and Kowalevski top
A.V. Tsiganov
St. Petersburg State University, St. Petersburg, Russiaemail: [email protected]
Abstract
The Lax pair representation of Reyman and Semenov-Tian-Shansky is used to construct a finiteset of the equivalent divisors on a spectral curve associated with numerical normalization of theBaker-Akhiezer function. One of these divisors has degree two on the underlying elliptic curve andthe evolution of the corresponding poles is given by a pair of separable differential equations onthe elliptic curve.
In the realm of algebraic geometry usually associated with many Liouville integrable systems, theHamiltonian evolution equations are written as a Lax equation ddt L ( x ) = [ L ( x ) , A ( x )] , for two N × N matrix functions L ( x ) and A ( x ) on the phase space depending on the auxiliary spectralparameter x . The time-independent spectral equation L ( x ) ψ ( x, y ) = y ψ ( x, y ) (1.1)allows us to represent the vector Baker-Akhiezer function ψ in terms of the Riemann theta functionon a nonsingular compactification of the spectral curve defined by the equationΓ : f ( x, y ) = det( L ( x ) − y ) = 0 . (1.2)The second equation defines time evolution ddt ψ ( x, y ) = − A ( x ) ψ ( x, y ) , see [2, 3, 7, 16, 32] and references within.Divisor of poles of the vector Baker-Akhiezer function ψ (1.1) belongs to a class of equivalentdivisors on Γ. To get some representative D of this class we have to impose a linear constraint (cid:126)α · ψ = N (cid:88) i =1 α i ψ i = 1 , i.e. fix normalization (cid:126)α of vector function ψ [2, 17, 19, 24]. Substituting matrices L ( p ) = L (cid:16) tr L ( p − (cid:17) − ( p − L ( p − L , with L (1) ≡ L, into the N × N matrix B ( x ) = (cid:126)α · L (1) ( x ) L − ( x ) (cid:126)α · L (2) ( x ) L − ( x )12 (cid:126)α · L (3) ( x ) L − ( x ) · · · N − (cid:126)α · L ( N ) ( x ) L − ( x ) , (1.3)1 a r X i v : . [ n li n . S I] N ov o that (cid:126)α · ( L ( x ) − y ) ∧ ≡ (cid:0) ( − y ) N − , ( − y ) N − , . . . , (cid:1) · B ( x ) = 0 , we can determine first Mumford’s coordinate of divisor DU ( x ) = MakeMonic det B ( x ) , (1.4)and a finite set of second coordinates y m − V m ( x ) = 0 , y mi = V m ( x i ) , i = 1 , . . . , N , m = 1 , . . . , N − . As usual these second coordinates ( − y ) j − i = ( B ∧ ( x )) k,i ( B ∧ ( x )) k,j , (1.5)are equivalent up to mod U ( x ) and the MakeMonic means that we take the numerator of a rationalfunction and divide the corresponding polynomial by its leading coefficient.We have to underline that Jacobi [10] and Mumford [21] used only interpolation polynomials V m ( x ) as a second coordinate of divisor D . In [1] Abel used rational interpolation functions V m ( x ) = P ( x ) Q ( x ) + R ( x ) U ( x ) , where P ( x ) and Q ( x ) are polynomials and R ( x ) are rational function without poles coinciding withzeroes of U ( x ). Similar to modern cryptography on algebraic curves below we follow to original Abel’scalculations.Following to Abel [1], divisor of poles D is the part of a finite set of intersection divisors of thespectral curve Γ (1.2) with the family of plane curves defined by equations (1.5) D + D ( ijk ) + D ( ijk ) ∞ = 0 . Here D ( ijk ) are equivalent divisors, D ( ijk ) ∞ are suitable linear combination of points at infinity, and V m ( x ) is a rational interpolation function through points of both divisors D and D ( ijk ) ∞ . Eliminating y from equations of intersecting curves (1.2) and (1.5) we obtain Abel’s polynomialΨ = θU ( x ) U (cid:48) ( ijk ) ( x ) = 0 , where U (cid:48) ( ijk ) ( x ) are first coordinates of the divisors D ( ijk ) which are linearly equivalent to divisor D .According to the Riemann-Roch theorem dimension of the linear system | D | , which is the set ofall nonnegative divisors which are linearly equivalent to D | D | = { D (cid:48) ∈ Div( C ) | D (cid:48) ∼ D and D (cid:48) > } , is equal to dim | D | = deg D − g , at deg D = n > g , where g is a topological genus of spectral curve Γ, see definitions and other details in textbook [20].Thus, if deg D = n > g we can construct a finite chain of divisors D → D n − → · · · → D g , deg D i = i . For instance, such finite set of divisors on nonhyperelliptic spectral curve for SL (3) magnetic chainwas constructed in [31].The main aim of this note is the study time evolution of a chain of the equivalent divisors onelliptic curve associated with the 4 × Description of the model
Let two vectors (cid:96) and g are coordinates on the phase space M . As a Poisson manifold M is identifiedwith Euclidean algebra e (3) ∗ with the Lie-Poisson brackets (cid:8) (cid:96) i , (cid:96) j (cid:9) = ε ijk (cid:96) k , (cid:8) (cid:96) i , g j (cid:9) = ε ijk g k , (cid:8) g i , g j (cid:9) = 0 , (2.1)having two Casimir functions c = g + g + g , c = g (cid:96) + g (cid:96) + g (cid:96) . (2.2)Here ε ijk is the skew-symmetric tensor.The Euler-Poisson equations on e (3) ∗ are given by X : ˙ (cid:96) = (cid:96) × ∂H∂(cid:96) + g × ∂H∂g , ˙ g = g × ∂H∂(cid:96) , (2.3)where x × y means cross product of two vectors.The Kowalevski top is defined by the Hamiltonian H , H = (cid:96) + (cid:96) + 2 (cid:96) − bg , b ∈ R (2.4)and the second integral K , K = ( (cid:96) + (cid:96) ) + 4 b (cid:0) g ( (cid:96) − (cid:96) ) + 2 g (cid:96) (cid:96) (cid:1) + 4 b ( g + g ) , (2.5)which are in involution { H, K } = 0 with respect to the Poisson brackets (2.1). The Euler-Poisson equations (2.3) were integrated by S. Kowalevski by using change of variables whichreduced the problem to hyperelliptic quadratures [15]. Let us briefly discuss her calculations.At the first step, Kowalevski introduced two pairs of Lagrangian variables z , and ˙ z , such that H = − ˙ z ˙ z + R ( z , z )( z − z ) , K = (cid:0) ˙ z − R ( z , z ) (cid:1)(cid:0) ˙ z − R ( z , z ) (cid:1) ( z − z ) . (3.1)Here z = J + i J , z = J − i J (3.2)and R ( z , z ) = z z − H ( z + z ) − b c ( z + z ) − b c + K. To remove cross-terms ˙ z ˙ z in (3.1) Kowalevski proposed to use ”rotation” of divisor D = (cid:18) p p (cid:19) → D (cid:48) = (cid:18) p (cid:48) p (cid:48) (cid:19) = (cid:18) − (cid:19) D (3.3)on an elliptic curve defined by the equation E : Z = R ( z, z ) , R ( z, z ) = z − Hz − b c z − b c + K ≡ (cid:88) k =0 a k z k . (3.4)Here p = ( z , Z ) and p = ( z , Z ) are two points on E with abscissas z , and ordinates Z , whichsatisfy equation (3.4).Affine coordinates of points p (cid:48) = ( z (cid:48) , Z (cid:48) ) and p (cid:48) = ( z (cid:48) , Z (cid:48) ) on E are defined by two time-independent equations dz Z + dz Z = dz (cid:48) Z (cid:48) and dz Z − dz Z = dz (cid:48) Z (cid:48) , z (cid:48) , = − z − z − b b + b − a b b − a , and Z (cid:48) , = −P ( z (cid:48) , ) . Here a j are given by (3.4) and b j are defined by interpolation polynomial P ( z ) = b z + b z + b ≡ √ a ( z − z )( z − z ) + ( z − z ) Z z − z ± ( z − z ) Z z − z . Abscissas z (cid:48) and z (cid:48) commute to each other with respect to the Poisson brackets (2.1) { z (cid:48) , z (cid:48) } = 0and the corresponding velocities ˙ z (cid:48) and ˙ z (cid:48) satisfy to Abel’s differential equations.In [15] Kowalevski preferred to use unpublished Weierstrass lectures and popular in that timetheory of birational transformations, see historical remarks in [4, 5]. Therefore, at the second step,Kowalevski applied birational transformation which reduces equation (3.4) for elliptic curve to theshort Weierstrass form E : W = 4 w − (cid:18) K − b c + H (cid:19) w − (cid:18) H (36 b c + H − K )27 − b c (cid:19) ≡ w − g w − g . (3.5)Now any computer algebra system performs such reductions for a few seconds.Images w (cid:48) , of variables z (cid:48) , satisfy Abel’s differential equations but do not commute to each other { w (cid:48) , w (cid:48) } (cid:54) = 0 , and, therefore, at the third step, Kowalevski applied the second birational transformation w → s + H E (3.4) E : S = P ( s ) , P ( s ) = 4 s + 4 Hs + (4 b c + H − K ) s + 4 b c . (3.6)So, famous Kowalevski variables s , = R ( z , z ) ± (cid:112) R ( z , z ) (cid:112) R ( z , z )2( z − z ) , (3.7)are images of variables z (cid:48) , after a pair of birational transformations of elliptic curve E (3.4). Theyare solutions of time-independent equations on elliptic curve Edz Z + dz Z = dz (cid:48) Z (cid:48) = ds S and dz Z − dz Z = dz (cid:48) Z (cid:48) = ds S , associated with transformation of divisors D → D (cid:48) (3.3). Variables s , in involution with respect tothe Poisson brackets (2.1) { s , s } = 0 . The corresponding canonically conjugated momenta and separation relations are discussed in [18].Two points P (cid:48) = ( s , S ) and P (cid:48) = ( s , S ) determine a line S = s − s s − s S + s − s s − s S , which has an intersection with elliptic curve E (3.6) at finite third point P (cid:48) = ( s , S ) so that ds S + ds S + ds S = 0 . s = b (g − ig ) (cid:96) − (cid:96) (cid:96) ( z (cid:96) + b g ) (cid:0) z + b (g − ig ) (cid:1) − ( z − (cid:96) )( z (cid:96) + b g ) ( z + 2 b g − ) (cid:96) + ( z (cid:96) + b g ) At the fourth step, Kowalevski studied the time evolution of the divisor D (cid:48) and proved that˙ s (cid:112) P ( s ) + ˙ s (cid:112) P ( s ) = 0 , s ˙ s (cid:112) P ( s ) + s ˙ s (cid:112) P ( s ) = 1 . (3.8)where P ( s ) = (4 s + 4 Hs + H − K ) P ( s ) . Using inverse birational transformations one gets similar equations for w (cid:48) i and z (cid:48) i variables.In [13] K¨otter expressed solutions (cid:96) i ( t ) and g i ( t ) of the Euler-Poisson equations on e (3) ∗ (2.3) interms of solutions s , ( t ) of Abel’s hyperelliptic quadratures (3.8). In [6] using the Lax pair representa-tion and the machinery of finite-band integration theory, authors obtained concise explicit expressionsfor the solutions of the Kowalevski top which are much simpler than the original formulae of Kowalevskiand K¨otter. In [22] Reyman and Semenov-Tian-Shansky found Lax matrices for various generalizations of theKowalevski top. In [6] these Lax matrices were used to integrate the problem in terms of theta-functions, see also textbook [23].Let us take Lax matrix L (6.3) from [6] and multiply it’s first term on b that corresponds to scaling g i → bg i . As a result, we obtain the Lax matrix L ( λ ) = i bλ g − i g − g − g − i g g − g − g − i g g g − i g (4.1)+i (cid:96) − i (cid:96)
00 0 0 − (cid:96) − i (cid:96) (cid:96) + i (cid:96) − (cid:96) − λ − (cid:96) + i (cid:96) λ (cid:96) , i = √− . with the spectral curve defined by equationΓ : det ( L ( λ ) − µ ) = µ − (cid:18) λ − H + b c λ (cid:19) µ + K − b (cid:0) c H − c (cid:1) λ + b c λ = 0 , (4.2)where c , are the Casimir functions (2.2) and integrals of motion H and K are given by (2.4-2.5).Symmetries of Lax matrices give rise to two commuting involutions τ and τ on Γ, that allowsus to consider quotient elliptic curve E = Γ / ( τ , τ ) [6]. Indeed, substituting µ = v , and λ = u into (4.2) we obtain the following equation E : Φ( u, v ) = ( uv ) − (cid:0) u + Hu + b c (cid:1) uv + Ku − b (cid:0) c H − c (cid:1) u + b c = 0 , (4.3)which after birational transformation v → y − u − Hu − b c u and u = − x (4.4)5ooks like E : y = xP ( x ) , P ( x ) = 4 x + 4 Hx + (4 b c + H − K ) x + 4 b c . (4.5)Here P ( x ) (3.6) is the cubic polynomial from definition of the Kowalevski elliptic curve E (3.6).Using variables s , (3.7) we can construct divisor D s on the elliptic curve E (4.5) with coordinates U s ( x ) = ( x − s )( x − s ) , V s ( x ) = x − s s − s (cid:112) s P ( s ) + x − s s − s (cid:112) s P ( s ) mod U s ( x )and reconstruct the corresponding normalization (cid:126)α s from equations (1.4) and (1.5). This vector (cid:126)α s will be a function on the phase space M = e ∗ (3), i.e. we may obtain dynamical normalization of thevector Baker-Akhiezer function ψ (1.1) associated with the Kowalevski variables of separation. Let us consider the standard normalization vector (cid:126)α = (1 , , ,
0) (5.1)Matrix B (1.3) is equal to B ( λ ) = − i bg − λ − i (cid:96) − i bg λ − λ + (cid:96) + (cid:96) +4 (cid:96) − bg − b c λ (cid:96) (cid:96) − +2 bg (cid:96) − λ + 2 bc λ − i ( (cid:96) (cid:96) +2 bg (cid:96) + (cid:96) (cid:96) ) − i b g c λ b b i b g g − + i b(cid:96) + (cid:0) g (cid:96) − − g − (cid:96) (cid:1) λ − i b g c λ where b = 2i (cid:0) (cid:96) − + 2 bg − (cid:1) λ + b (cid:0) (cid:96) ( g (cid:96) − − g − (cid:96) ) + 2i b ( g + 2 g g − ) + i g + (cid:96) − (cid:1) λ + i b c g − λ ,b = − i (cid:96) + (cid:0) (cid:96) − + 2 bg − (cid:1) + i b (cid:0) g (cid:96) − − g − g (cid:96) − g − (cid:96) + (cid:1) λ , and g ± = g ± i g , (cid:96) ± = (cid:96) ± i (cid:96) , The corresponding polynomial U ( λ ) (1.4) has the following form U ( λ ) = λ + u λ + u λ + c u , (5.2)where coefficients u k may be recovered from the definition (1.4).At λ = u we have a cubic polynomial U ( u ) which is the first coordinate of divisor D = P + P + P on the elliptic curve E . According to the Riemann-Roch theoremdim | D | = deg D − g ( E ) = 3 − , and we have nontrivial space of equivalent divisors involving a chain of divisors D → D (cid:48) → D (cid:48)(cid:48) , deg D (cid:48) = 2 , deg D (cid:48)(cid:48) = 1 . Our aim is to construct these divisors and to study evolution of these divisors.6 .1 Semi-reduced divisor At c = 0 polynomial U ( λ ) (5.2) looks like U ( λ ) = λ ( λ − λ )( λ − λ ) . According to [19] poles λ , of the Baker-Akhiezer function ψ are variables of separation for the partialcase of the Kowalevski top at c = 0. Another variables of separation in this partial case were proposedin [25, 26].Below we study only case c = (cid:96) g + (cid:96) g + (cid:96) g (cid:54) = 0. Let us calculate six possible secondcoordinates (1.5) for µ µ − V ( λ ) , V ( λ ) = ( B ∧ ( λ )) k,j ( B ∧ ( λ )) k,j +2 , j = 1 , , k = 2 , , . Substituting µ = V ( λ ) into the equation for spectral curve Γ (4.2) we obtain rational function on λV ( λ ) − d ( λ ) V ( λ ) + d ( λ ) = 0 . Its numerator is Abel’s polynomial [1]Ψ( λ ) = θ U ( λ ) U (cid:48) ( λ ) = 0 , generating coordinates of divisors D and D (cid:48) so that D + D (cid:48) + D ∞ = 0 . It is easy to prove that • at k = 3 , j = 1 divisor D (cid:48) has degree more then degree of divisor D ; • at k = 2 , i = 1 and k = 3 , j = 2 or k = 4 , j = 2 divisor D (cid:48) is a constant divisor of degree twowith coordinate U (cid:48) ( λ ) = λ ; • at k = j = 2 divisor D (cid:48) has degree four and its first coordinate is equal to U (cid:48) ( λ ) = λ − (cid:18) (cid:96) + 2 b(cid:96) ( g (cid:96) + g (cid:96) ) + b g (cid:96) + (cid:96) (cid:19) λ + b c (cid:96) + (cid:96) . If we substitute any numerical vector (cid:126)α instead of (5.1) we also obtain the same constant divisor ofdegree two and the same equivalent divisors of degree four on a spectral curve Γ.Summing up, using numerical normalization vector (cid:126)α and Abel’s reduction of divisors we obtaindivisor of degree two D (cid:48) on the elliptic curve E (4.3) with coordinates U (cid:48) ( u ) = ( u − u )( u − u ) ≡ u − (cid:18) (cid:96) + 2 b(cid:96) ( g (cid:96) + g (cid:96) ) + b g (cid:96) + (cid:96) (cid:19) u + b c (cid:96) + (cid:96) . (5.3)and v − V (cid:48) ( u ) , V (cid:48) ( u ) = ( B ∧ ( λ )) , ( B ∧ ( λ )) , (cid:12)(cid:12)(cid:12)(cid:12) λ = u . (5.4)Because dim | D (cid:48) | = deg D (cid:48) − g ( E ) = 2 − , divisor D (cid:48) is semi-reduced divisor on elliptic curve, which can be reduced to one degree equivalentdivisor D (cid:48)(cid:48) . 7 .2 Reduced divisor Birational transformation ( v, u ) → ( x, y ) (4.4) transforms elliptic curve E to canonical form (4.5) thatallows as to directly apply Abel’s calculations [1]. Indeed, two points P (cid:48) = ( x , y ) and P (cid:48) = ( x , y )of divisor D (cid:48) on elliptic curve E : y = a x + a x + a x + a x + a determine parabolaΥ (cid:48) : y = √ a x + b x + b = √ a ( x − x )( x − x ) + ( x − x ) y x − x + ( x − x ) y x − x , which has a finite third point of intersection P (cid:48) = ( x , y ) with coordinates x = − x − x − (2 √ a b + b − a √ a b − a , y = √ a x + b x + b . These equations describe reduction of divisors D (cid:48) = P (cid:48) + P (cid:48) → D (cid:48)(cid:48) = P (cid:48) . In our case, a unique reduced divisor D (cid:48)(cid:48) in a class of equivalent divisors associated with normalization (cid:126)α = (1 , , ,
0) (5.1) consists of the point P (cid:48) ( x , y ) , dim | D (cid:48)(cid:48) | = deg D (cid:48)(cid:48) − g ( E ) = 1 − , with affine coordinates x = 4 b c b c − K , y = 4 b c (4 b c H − b c − HK )(4 b c − K ) . It is fixed point on E because { H, x } = { K, x } = 0 , { H, y } = { K, y } = 0 , Two variable points P (cid:48) , P (cid:48) and fixed point P (cid:48) lie on the parabola Υ (cid:48) and, therefore, there are geometricAbel’s integral [1] dx y + dx y + dx y = 0 , which determine evolution of the points P (cid:48) ( x , y ) and P (cid:48) ( x , y ) dx y + dx y ≡ dxy ( P (cid:48) ) + dxy ( P (cid:48) ) = 0 . (5.5)So, similar to Kepler problem and harmonic oscillator [27, 28, 29, 30], we have motion of parabola Υ (cid:48) around a fixed point on elliptic curve ' ' ' ' '' ' Figure 1: Rotation of the parabola with points P (cid:48) ( t ) and P (cid:48) ( t ) around fixed point P (cid:48) .8ive variable points of divisors D = P + P + P and D (cid:48) = P (cid:48) + P (cid:48) lie on the auxiliary curve Υ definedby equation (5.4) up to birational transformation (4.4)Υ : y = V (cid:48) ( x ) − x − Hx + b c x , and, therefore, geometric Abel’s integral dxy ( P ) + dxy ( P ) + dxy ( P ) + dxy ( P (cid:48) ) + dxy ( P (cid:48) ) = 0determines evolution of the points P , P and P dxy ( P ) + dxy ( P ) + dxy ( P ) = 0according to (5.5). This equation associated with holomorphic differentials on elliptic curve E coincideswith the first equation in the system of equations (7.69) obtained in [6]. Other two equations in [6]are associated with Prym differentials. In the next Section we try to find remaining two equations. Following Kowalevski [15] now we have to study equations of motion for variables u and u on thephase space M = e ∗ (3).Abscissas u , (5.3) and ordinates (5.4) v , = V (cid:48) ( u = u , ) , of the points in support of the reduced divisor D (cid:48) satisfy to equation for the elliptic curve E (4.3)Φ( u i , v i ) = 0 , It is easy to prove that variables u , do not commute to each other { u , u } (cid:54) = 0 , and, therefore, u , cannot be variables of separation in the Hamilton-Jacobi equation associated withKowalevski top. Proposition 1
Coordinates of poles u , and v , satisfy to the following differential equations Ω ( u , v ) ˙ u + Ω ( u , v ) ˙ u = 0 , Ω ( u , v ) ˙ u + Ω ( u , v ) ˙ u = 0 , (6.1) on the elliptic curve E (4.3). Here ˙ u k = { H, u k } and Ω ( u, v ) = 1 u ∂ H Φ( u, v ) ∂ u Φ( u, v ) = b c − uvu ( b c − Hu − uv + 2 u ) , Ω ( u, v ) = 1 u ∂ K Φ( u, v ) ∂ u Φ( u, v ) = − b c − Hu − uv + 2 u ) . (6.2)The proof is straightforward.Birational transformation ( v, u ) → ( x, y ) (4.4) transforms elliptic curve E to canonical form y = xP ( x ) , P ( x ) = 4 x + 4 Hx + (4 b c + H − K ) x + 4 b c dx y + dx y = 0 , (cid:18) dx x + 2 x dx y (cid:19) + (cid:18) dx x + 2 x dx y (cid:19) = 0 , (6.3)involving holomorphic 1-form ω and logarithmic 1-form ω ω = dxy , ω = dxx − xdxy . Differential of third kind ω = 1 /xy appears in the following equation˙ x x y + ˙ x x y = 12 ddt x x . For the Kowalevski top second equation in (6.3) (cid:18) x + 2 x y (cid:19) dx dt + (cid:18) x + 2 x y (cid:19) dx dt = 0does not define time in contrast with the third equation in the system (7.69) from [6]. The geometricinterpretation of this equation is also unknown. Our calculations can be directly generalized to the Kowalevski gyrostat with the Lax matrix˜ L = L + i γ diag( − , , − , , γ ∈ R , and integrals of motion˜ H = (cid:96) + (cid:96) + (cid:96) + ( (cid:96) + γ ) − bg , ˜ K = (4 g + 4 g ) b + (4 (cid:96) g + 8 (cid:96) (cid:96) g − γ(cid:96) g − (cid:96) g − γ g ) b + ( (cid:96) + (cid:96) − (cid:96) γ − γ ) . The corresponding spectral curve ˜Γ is also twofold coverings of elliptic curve ˜ E defined by the equation˜Φ( u, v ) = Φ( u, v ) − γ u = 0 , whereas first coordinate of reduced divisor ˜ D (cid:48) on ˜ E is equal to˜ U (cid:48) ( u ) = ( u − ˜ u )( u − ˜ u ) = U (cid:48) ( u ) − γu (cid:18) (cid:96) + γ + 2 b(cid:96) g (cid:96) + (cid:96) (cid:19) . As above coordinates of poles ˜ u , and ˜ v , satisfy equations of the form (6.1) and (6.2) on the ellipticcurve ˜ E . Similar equations appear also for other generalizations of the Kowalevski curve from [6] and[11] associated with elliptic curves.Because Kowalevski gyrostat is closely related to the Clebsch system, see [12] and referenceswithin, our next aim is to study reduced divisors for the Clebsch system which has a few Lax matrices.The corresponding spectral curves are also twofold coverings of elliptic curves [8], so we hope to getequations of the form (6.1) for the Clebsch system.The work was supported by the Russian Science Foundation (project 18-11-00032).10 eferences [1] Abel N. H., M´emoire sure une propri´et´e g´en´erale d’une classe tr`es ´entendue de fonctions tran-scendantes , Oeuvres compl´etes, Tome I, Grondahl Son, Christiania, (1881), pages 145-211.[2] Babelon O., Bernard D., Talon M.,
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