Generalized Trigonometric functions and elementary applications Giuseppe Dattoli, Emanuele Di Palma, Federico Nguyen and Elio Sabia Abstract. The Trigonometric functions of second and higher degree order are introduced in this paper by taking advantage from the notion of generalized complex number. It is shown that the Trigonometric functions are associated with the exponentiation of the roots of any algebraic equation, whose degree fixes that of the relevant trigonometry. A few elementary applications (friction problems, Bloch evolution equations and Volterra integral equations. . . ) are briefly discussed. Mathematics Subject Classification (2010). 14A20, 44Axx, 45J05. Keywords. Generalized Euler identities, higher order trigonometries, evolution operator, Bessel functions.
1. Introduction In a previous paper [1], it has been noticed that the notion of generalized complex number is an ideally suited tool to introduce further families of trigonometric functions (Trig-functions, hereafter). The pivoting argument of such a generalization is the use of an appropriate extension of the Euler exponential formula. The argument developed in [1] goes as follows. a) The algebraic number h such that h2 = bh + a
(1.1)
defines a generalized imaginary unit, reducing to the ordinary hyperbolic or circular imaginary units for b = 0, a = ±1. b) The identity ehθ = C(θ) + hS(θ)
(1.2)
2 Giuseppe Dattoli, Emanuele Di Palma, Federico Nguyen and Elio Sabia allows to introduce the cosine- and sine-like functions plicitly defined as 2 C(θ)
=
S(θ)
=
1
C(θ), S(θ) ex-
h− eh+ θ − h+ eh− θ h− − h+ h+ θ e − eh− θ h+ − h−
(1.3)
√ where h± = b ± b2 + 4a /2 are the self-conjugated units (SCU) corresponding to the roots of eq.(1.1). c) The derivatives and other relevant identities of the Trig-functions are obtained from eq.(1.2). Indeed, by keeping the derivative with respect to θ of both sides, we find (the apex denotes derivative with respect to θ) d hθ e = C 0 (θ) + hS 0 (θ) dθ = hC(θ) + h2 S(θ)
=
(1.4)
= aS(θ) + h (C(θ) + bS(θ))
and after equating the h-like coefficients we eventually get C 0 (θ) 0
S (θ)
=
aS(θ)
=
C(θ) + bS(θ)
(1.5)
d) The addition formulae can finally be derived from eh(θ1 +θ2 ) = C(θ1 + θ2 ) + hS(θ1 + θ2 ) eh(θ1 +θ2 ) = [C(θ1 ) + hS(θ1 )] [C(θ2 ) + hS(θ2 )] =
(1.6)
= C(θ1 )C(θ2 ) + aS(θ1 )S(θ2 ) + h [C(θ1 )S(θ2 ) + C(θ2 )S(θ1 ) + bS(θ1 )S(θ2 )] and by comparing real and imaginary parts, eq.(1.6) provides the following result [2] C(θ1 + θ2 )
=
C(θ1 )C(θ2 ) + aS(θ1 )S(θ2 )
(1.7)
S(θ1 + θ2 )
=
C(θ1 )S(θ2 ) + C(θ2 )S(θ1 ) + bS(θ1 )S(θ2 )
which can also be written in the more transparent form C(θ1 + θ2 ) ˆ 1 ) C(θ2 ) , = R(θ S(θ1 + θ2 ) S(θ2 ) ˆ 1) = R(θ
1 Whenever
C(θ1 ) aS(θ1 ) S(θ1 ) C(θ1 ) + bS(θ1 )
(1.8)
necessary to emphasize the dependence on the constants a, b we will use the notation C(θ|a, b), S(θ|a, b). 2 The Trig-functions in eq.(1.2) are neither even nor odd under parity transformation and it is understood that C(−θ|a, b) = C(θ|a, −b) and S(−θ|a, b) = −S(θ|a, −b).
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e) Finally, by taking into account that e(h+ +h− )θ = ebθ we end up with the identity 3 C 2 (θ) − aS 2 (θ) + bC(θ)S(θ) = ebθ
(1.9)
coinciding with the determinant of the matrix on the r.h.s. of eq.(1.8). The last equation reduces to the fundamental trigonometric and hyperbolic identities for b = 0, a = ∓1. The family of functions discussed so far will be called second degree Trigfunction, to distinguish them from the higher order degree families we will introduce in the following. From the geometrical point of view, the matrix in eq.(1.8) can be interpreted as a rotation not preserving the Euclidean norm. The following example may provide further insight on the above statement. We consider indeed the dynamical system d C(θ) 0 a C(θ) = (1.10) S(θ) 1 b S(θ) dθ whose solution reads C(θ) C(θ0 ) ˆ = eM θ , S(θ) S(θ0 )
ˆ = M
ˆ is easily seen to satisfy the identity The matrix M 1 0 ˆ 2 = bM ˆ + aˆ ˆ M 1, 1= 0 1
0 1
a b
(1.11)
(1.12)
ˆ ˆ which allows to conclude that eM θ = R(θ). ˆ It is now evident that the non-unitary matrix R(θ) can be viewed as the evolution operator associated with the problem in eq.(1.10). After an appropriate redefinition of the associated parameters, we will see in the following that the system can be adopted for the description of physical problems involving e.g. damped harmonic oscillators. The arguments developed so far suggest that we can safely conclude that the trigonometric functions of order two are associated with the exponentiation of the roots of any second degree equation. Within such a context the two roots provide a conjugated pair according to the identity
h+ h− = a It is evident that for b = 0 the functions C(θ), S(θ) reduce to the ordinary circular or hyperbolic functions. is worth noticing that enhθ = C(nθ) + hS(nθ) = [C(θ) + hS(θ)]n , which can be exploited to derive identities mimicking those of the circular (and hyperbolic) trigonometry. Regarding the duplication formulae we find e2hθ = C 2 (θ)+aS 2 (θ)+2hC(θ)S(θ), eventually providing the second degree trigonometric duplication formula C(2θ) = C 2 (θ) + aS 2 (θ), S(2θ) = 2S(θ)C(θ) + bS 2 (θ). 3 It
4 Giuseppe Dattoli, Emanuele Di Palma, Federico Nguyen and Elio Sabia Before analyzing more specific properties of these families of generalized Trig-functions, we discuss a further general property involving identities of the type f (hx) = A(x) + hB(x) (1.13) where the A, B functions are defined as h− f (h+ x) − h+ f (h− x) A(x) = (1.14) h− − h+ f (h+ x) − f (h− x) B(x) = h+ − h− In the b = 0, a = +1 case they are just the even and odd components of the f (x) function itself. If we define the x → h± x generalization of the parity transformation we obtain for instance h− f (h2+ x) − h+ f (h− h+ x) h− f ((a + bh+ )x) − h+ f (ax) A(h+ x) = = h− − h+ h− − h+ f (h2+ x) − f (h− h+ x) f ((a + bh+ )x) − f (ax) B(h+ x) = = (1.15) h+ − h− h+ − h− The concepts associated with a more general definition of the parity transformation are beyond the scope of the present article, but the theory of generalized rotations is however an appropriate context to place in the relevant concepts and will be explored in a forthcoming work.
2. Third degree Trig-Functions and their applications In the previous section we have stated that the ordinary trigonometric (and hyperbolic functions as well) belong to the family of second degree trigonometry, associated with a generalization of the Euler exponential formulae. It is also evident that the same functions can be defined by the exponentiation of any 2 × 2 matrix and, although straightforward, we consider instructive to discuss this aspect of the problem with some detail, also to simplify the forthcoming discussion. We accordingly set α β ˆ ˆ ˆ= eQθ = C(θ)ˆ 1 + S(θ)Q, Q (2.1) γ δ ˆ matrix The functions C(θ), S(θ) are defined through the eigenvalues of the Q supposed not singular as in eq. (1.3) with h± replaced by λ± . The relevant addition theorems are derived from eq. (2.1) by following the same procedure outlined in eq. (1.6), namely ˆ ˆ + S(θ1 )S(θ2 )Q ˆ2 eQ(θ1 +θ2 ) = C(θ1 )C(θ2 )ˆ 1 + (C(θ1 )S(θ2 ) + C(θ2 )S(θ1 )) Q (2.2) the use of the identity [3] n−1 n−1 λn − λn+ ˆ ˆ n = λ− λ+ (λ− − λ+ ) ˆ Q 1+ − Q λ− − λ+ λ− − λ+
(2.3)
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which for n = 2 reduces to ˆ2 Q
=
ˆ ∆Q ˆ 1 + TrQ Q
∆Q
=
αδ − βγ
TrQ
=
α+δ
(2.4)
which finally yields C(θ1 + θ2 )
= C(θ1 )C(θ2 ) + ∆Q S(θ1 )S(θ2 )
S(θ1 + θ2 )
= C(θ1 )S(θ2 ) + C(θ2 )S(θ1 ) + TrQ S(θ1 )S(θ2 )
(2.5)
Furthermore, the relevant differential equations can be expressed as ˆ ˆ Qθ ˆ = C 0 (θ)ˆ1 + S 0 (θ)Q ˆ Qe = S(θ)∆Q ˆ 1 + [C(θ) + TrQ S(θ)] Q
(2.6)
which allows to specify the following coupled pair of first order differential equations C 0 (θ)
=
∆Q S(θ)
S 0 (θ)
=
C(θ) + TrQ S(θ)
(2.7)
We have already mentioned that the rotation matrix, associated with the second order Trig-functions, does not preserve the Euclidean norm. This is not surprising at all, since in the case of b = 0, a = 1, they reduce to the hyperbolic functions which are the essential tool describing the Lorentz transformation in the (non-Euclidean) Minkowski space-time geometry [4]. In general, if TrQ 6= 0, the determinant of the transformation matrix is not an invariant but depends on the rotation angle itself, we cannot define any invariant under generalized rotation, unless we invoke for other invariant forms, whose discussion is beyond the scope of the present paper. It is now almost natural to introduce higher degree trigonometric functions as associated to the exponentiation of the roots of an algebraic equation with degree larger than 2. Accordingly if we define the algebraic number roots of the third degree equation λ3 = cλ2 + bλ + a,
(2.8)
we can define the third degree Trig-functions [5] eλθ = C0 (θ) + C1 (θ)λ + C2 (θ)λ2 ,
(2.9)
which are easily recognized to satisfy the differential equations [1] C00 (θ) =
aC2 (θ)
C10 (θ) C20 (θ)
=
C0 (θ) + bC2 (θ)
=
cC2 (θ) + C1 (θ)
and the addition formulae C0 C0 aC2 C1 = C1 C0 + bC2 C2 θ +θ 0 (1 + b)C1 1
2
(2.10)
aC1 C0 (b + cb + a)C1 C1 (b + c2 )C2 C2 θ θ 2
(2.11) 1
6 Giuseppe Dattoli, Emanuele Di Palma, Federico Nguyen and Elio Sabia It is furthermore evident that the third degree Trig-functions read λ θ e 1 C0 1 λ1 λ21 C1 = 1 λ2 λ22 eλ2 θ C2 θ 1 λ3 λ23 eλ3 θ
(2.12)
Where λα , α = 1, 2, 3 are the roots of eq.(2.8). If b = c = 0, a = 1 in eq. (2.8), the third order Trig-functions are those associated with the cubic roots of unity and are just provided by the pseudo-hyperbolic functions introduced in ref. [6] ∞ X θ3n+k ck (θ) = , k = 0, 1, 2 (2.13) (3n + k)! n=0 The possibility of exploiting ck (θ) as a basis for the Ck (θ) will be discussed in the forthcoming concluding section. It is accordingly evident that any third order ordinary differential equations with constant coefficients (the apex denotes derivation with respect to x) y 000 + my 00 + py 0 + qy = 0 00
(2.14)
0
y (0) = 2 k0 , y (0) = 1 k0 , y(0) = 0 k0 can be solved as y A y 0 = Aλ1 y 00 x Aλ21 A B C
B C Bλ2 Cλ3 Bλ22 Cλ23 1 = 1 1
λ1 λ21 C0 λ2 λ22 C1 λ3 λ23 C2 x λ21 0 k0 λ22 1 k0 λ23 2 k0
1 1 1 λ1 λ2 λ3
(2.15)
with λ being the roots of the characteristic polynomial associated with eq.(2.14). Before considering other aspects of the problem we would like to consider a further aspect of third order differential equations, which can always be cast in the form of a Volterra integro-differential equation (see Appendix A) (2 k0 = 0, 1 k0 = 0) Z x y0 = (Aτ + B)ekτ y(x − τ )dτ, (2.16) 0
m 3 mp m 2 m A= − + − q, k = − , B = −p 2 2 2 2 We have dwelt on this aspect of the problem because the interest for this type of equation is manifold and it is often exploited in electromagnetism in the study of free electron laser devices [7], albeit the relevant link to the ordinary differential equations is not well known as it should. Let us now consider the 3 × 3 matrix 0 α β ˆ = −α 0 γ Ω (2.17) −β −γ 0
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whose exponentiation yields ˆ
ˆ + C2 (θ)Ω ˆ2 eΩθ = C0 (θ)ˆ 1 + C1 (θ)Ω Being the eigenvalues of the matrix of eq.(2.17) given by p λ1 = 0, λ2,3 = ±i|Ω|, |Ω| = α2 + β 2 + γ 2
(2.18)
(2.19)
the explicit form of the Trig-functions reads C0 (θ) = 1, C1 (θ) = 2
sin(|Ω|θ) 1 − cos(|Ω|θ) , C2 (θ) = − |Ω| |Ω|2
We have dwelt on this particular case because the matrix generated exponentiation of eq.(2.17) provides the so called Rodriguez rotation sociated with the solution of vector equations of the type d~ ~ × L, ~ ~ = (−γ, β, −α) L=Ω Ω dt The “evolution operator” h i h i h i2 ~ t Ω ~ = Ω×, ~ ~ =Ω ~ × Ω× ~ U (t) = e , Ω Ω ,...
(2.20) by the [8], as(2.21)
(2.22)
can be easily handled using the cyclic properties of the vector product and written in the form [9] h i h i h i2 ~ t Ω ~ Ω ~ + C2 (t|Ω|) ~ Ω ~ e =ˆ 1 + C1 (t|Ω|) (2.23) which is the vector counterpart of eq.(2.18). The solution of non-homogeneous vector equations of the type [9] d~ ~ ×L ~ + F~ L=Ω (2.24) dt can take benefit from the outlined procedure and indeed we easily end up with (see ref. [9] for a more complete treatment of the problem) Z t h i h i ~ ~ t Ω −t0 Ω 0 ~ ~ ~ L = e L0 + e F dt (2.25) 0 h i h i2 ~ Ω ~ + C2 (t|Ω|) ~ Ω ~ ~0 + ˆ1 + C1 (t|Ω|) = L Z t h i h i2 ~ Ω ~ + C2 ((t − t0 )|Ω|) ~ Ω ~ ˆ 1 + C1 ((t − t0 )|Ω|) F~ dt0 0
If the entries of the matrix in eq.(2.17) are not all real (e.g. β → βeiδ ), the relevant eigenvalues are provided by the third degree equation ~ 2 λ − 2iγαβ sin δ = 0 λ3 + |Ω| (2.26) Being the three roots all different from zero, the explicit derivation of the third degree Trig-functions requires more analytical work. It is however worth noticing that they are solution of the differential equation ~ 2 z 0 = (2iγαβ sin δ)z z 000 + |Ω| (2.27)
8 Giuseppe Dattoli, Emanuele Di Palma, Federico Nguyen and Elio Sabia which can be cast in the Volterra-like form Z x 0 ~ 2 z(x − τ )dτ z = (2iγαβ sin δ)τ − |Ω|
(2.28)
0
suitable for a perturbative treatment. ˆ matrix with a complex entry has been It is worth noticing that the Ω exploited to study the CP violations in weak decay processes [10]. The associated third order Trig-functions provides the elements of the weak rotation matrix and simplify most of the algebraic steps involved in the process, specific comments on this aspect of the problem can be found in refs. [11].
3. Final Remarks The most important element of this paper has been the use of the exponential function and the generalization of the Euler formula. The formalism can be pushed even further by simply taking advantage from the analogy with the properties and the geometrical interpretation of ordinary complex numbers. To clarify this point we consider the following analog of the Fresnel functions 2 e−hx = C(−x2 ) + hS(−x2 ) (3.1) If we are interested in evaluating e.g. the integral of the Fresnel functions on the rhs of eq.(3.1) we proceed by first evaluating the Gaussian integral r Z +∞ 2 π e−hx dx = h −∞ assuming that it exists (namely that 0), then we write r Z +∞ Z +∞ π 2 = IC + hIS , IC = C(−x )dx, IS = S(−x2 )dx h −∞ −∞
(3.2)
∗
Furthermore, by setting 4 h = ehφ = C(φ∗ ) + hS(φ∗ ), and since hα = C(αφ∗ ) + hS(αφ∗ ), α ∈ C we can write r π √ φ∗ φ∗ = π C(− ) + hS(− ) (3.3) h 2 2 thus finally obtaining √ φ∗ φ∗ ), IS = πS(− ) (3.4) 2 2 It is now worth stressing that also nh is an imaginary unit defined by (nh)2 = nb(nh) + n2 a, therefore we find IC =
√
πC(−
C(nθ|a, b) = C(θ|n2 a, nb),
S(nθ|a, b) = nS(θ|n2 a, nb)
For the sake of completeness we note: C(iθ|a, b) = C(θ| −a, ib), 4 In
S(iθ|a, b) = iS(θ| −a, ib)
the case of the ordinary imaginary unit φ∗ = iπ/2.
(3.5)
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Analogous tricks can be exploited to derive integrals involving C, S functions. To this aim we note that Z θ Z θ Z θ ehθ − 1 hθ 0 0 0 0 e dθ = = C(θ )dθ + h S(θ0 )dθ0 (3.6) h 0 0 0 which can easily be manipulated to obtain (see eq.(1.5)) Z θ C(θ) − 1 S(θ0 )dθ0 = a 0 Z θ [C(θ0 ) + bS(θ0 )] dθ0 = S(θ)
(3.7)
0
The use of the same procedure and of the duplication formulae allows the derivation of the further identity Z θ C(2θ) − 1 S(θ0 )C(θ0 )dθ0 = (3.8) 4a 0 Z θ 1 2bS(θ0 )C(θ0 ) + C 2 (θ0 ) + aS 2 (θ0 ) dθ0 = S(2θ) 2 0 It is instructive to use the formalism of the previous sections to treat a largely well- known topic, as the damped harmonic oscillator, for which the second degree Trig-functions seem to be tailor-suited. By keeping indeed the successive derivatives of eq.(1.5), we obtain the following differential equation C 00 − bC 0 − aC = 0
(3.9)
which after setting θ = ωt, κ = −bω, a = −1, can be cast in the form (the dot denotes derivative with respect to time) Z¨ + κZ˙ + ω 2 Z = 0 ˙ Z(0) = Z0 , Z(0) = Z˙ 0
(3.10)
It is evident that the solution of the previous problem can be written in terms of the second degree Trig-functions as specified below κ 1 ˙ κ Z(t) = Z0 C ωt −1, − + Z0 S ωt −1, − (3.11) ω ω ω If we express the initial conditions in terms of a pseudo amplitude and phase κ Z0 = AC Φ −1, − (3.12) ω 1 ˙ κ Z0 = −AS Φ −1, − ω ω we can finally cast the solution of eq.(3.10) in the more compact form (see eq.(1.6)) κ Z(t) = AC ωt + Φ −1, − (3.13) ω
10Giuseppe Dattoli, Emanuele Di Palma, Federico Nguyen and Elio Sabia where A and Φ are pseudo-amplitude and pseudo-phase. Regarding the specific definition of these quantities we introduce the second degree tangent κ S (Φ, −1, −κ/ω) Z˙ 0 T Φ −1, − = =− (3.14) ω C (Φ, −1, −κ/ω) ωZ0 and note that from eq.(1.9) we obtain √κ 1 − h− ω ∆ 1 − h+ T A= v !2 u u κZ˙ 0 t1 + Z˙ 0 + 2 ωZ0 ω Z0
(3.15)
It is evident that we have just restyled elementary notions, by the use of a slightly more elaborated formalism. It is however fairly interesting that, instead of exploiting combinations of ordinary trigonometric functions and exponential, we can concisely describe damped oscillations by the use of more general functions, which include the ordinary cases and fulfill definite addition theorems and differential equations. Regarding the third degree case we have emphasized the existence of trigonometric functions associated with the roots of unit, it is accordingly evident that ck (θ) = Ck (θ|1, 0, 0) (3.16) It is furthermore evident that, since any third degree algebraic equation can be reduced to a Cardanic form (c = 0), all the third degree Trig-functions can be written in terms of third degree Cardan Trig-functions (namely those associated with the roots of eq.(2.8) with c = 0. It is, on the other side, well known that the roots of a Cardanic equation can be expressed as [12] √ √ x1 = 3 x+ + 3 x− √ √ x2 = ω 3 x+ + ω 2 3 x− √ √ x3 = ω 2 3 x+ + ω 4 3 x− p α ± α2 + 4γ x± = (3.17) 2 b3 α = 2a, γ = 27 ω 3 = 1, ω 2 + ω = −1 √ where ω = −1 + i 3 /2 is the principal root of unity, and therefore the third degree Trig-function can be expressed in terms of the ck (θ). We note indeed that ex1 θ = e− −
√ 3
√ 3
e
√ 3
x+ −
√ 3
x− )θ
=
[c0 (φ) + ωc1 (φ) + ω 2 c2 (φ)] √ √ φ = 3 x+ − 3 x− θ =e
x− θ
x− θ ω ˆ(
(3.18)
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This conclusion can be extended to any higher order (> 3) expressed in terms of the roots of higher order Cardan equations [12]. Before concluding this paper we will dwell on some points which we have just touched on in the previous sections as the meaning of identities of the type of eq.(1.13). To this aim we consider continuous and infinitely differentiable functions which can be represented through the expansion ∞ X cn n x n! n=0
f (x) = ecˆx =
(3.19)
Where cˆ is an umbral notation for cˆn = cn
(3.20)
f (hx) = ecˆhx = C(ˆ cx) + hS(ˆ cx)
(3.21)
Accordingly we can write
from which we can extend the associated second degree Trig-functions to the case of a generic function f (x). The relevant addition theorems cannot be straightforwardly inferred since, in general, the f (x) does not possess the semigroup property of the exponential, namely f (x + y) 6= f (x)f (y) (3.22) Furthermore the relevant differential equation can be derived only after that having identified an operator playing the role of derivative, namely such that ˆ (µx) = µf (µx) Of
(3.23)
To find a way to overcome these problems, we consider a specific example cn =
1 n!
(3.24)
for which the function f (x) is the Bessel-like function ecˆx = l e(x) =
∞ X xr (r!)2 r=0
5
(3.25)
ˆ can be identified with the It is easily verified that the associated operator O Laguerre derivative [13, 14] l ∂x
= ∂x x∂x
(3.26)
And accordingly we find the associated differential equations in the form
5 It
l ∂xl C(x)
=
al S(x)
l ∂xl S(x)
=
l C(x)
(3.27)
+ bl S(x)
can be expressed in terms of the 0 − th order modified Bessel through the identity √ = I0 (2 η).
l e(η)
12Giuseppe Dattoli, Emanuele Di Palma, Federico Nguyen and Elio Sabia In the case of b = 0, a = 1 the corresponding Trig-functions are defined by the series expansion: l ch(x)
=
∞ X r=0
l sh(x)
=
∞ X r=0
x2r [(2r)!]2 x2r+1 [(2r + 1)!]2
Finally, regarding the addition formula, we note that the semi-group property of the exponential function can be replaced by l e(y)l e(x)
= l e(x ⊕l y)
(3.28)
where the following composition rule has been defined (see Appendix B, for further comments) n 2 X n n (x ⊕l y) = xn−r y r (3.29) r r=0 we find therefore l C(x
⊕l y)
=
l C(x)l C(y)
+ al S(x)l S(y)
(3.30)
l S(x
⊕l y)
=
l C(x)l S(y)
+ l C(y)l S(x) + bl S(x)l S(y)
(3.31)
It is now evident that the concepts of Trig-function, addition theoremcan be generalized ad libitum, the theory of algebraic equations offer a powerful complementary tool of investigation and, albeit academic, the exercise we have presented may offer seeds of speculations relevant to the possibility of getting new and perhaps unsuspected links between apparently uncorrelated topics. Acknowledgment We would like to thank Danilo Babusci and Robert Yamaleev for their kind interest and encouragement in this work.
Appendix A. The link between Volterra integro-differential equations and higher order ODE is fairly natural and can be easily stated. Setting x − τ = ξ in eq.(2.16) we obtain Z x
e−kx y 0 =
(A(x − ξ) + B)e−kξ y(ξ)dξ
(A.1)
0
The use of the formalism of negative order derivative, according to which the differintegral of order ν of the function f (x) can be written as 6 Z x ˆ −ν (f (x)) = 1 D (x − ξ)ν−1 f (ξ)dξ (A.2) x Γ(ν) 0 6 For
integer ν, eq.(A.2) is nothing but a Cauchy-repeated integral.
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allows to cast eq.(A.1) in the form ˆ x−2 (e−kx y(x)) + B D ˆ x−1 (e−kx y(x)) e−kx y 0 = AD
(A.3)
By keeping the second order derivative of both sides of eq.(A.3) we end up with y 000 − 2ky 00 + (k 2 − B)y 0 − (A − Bk)y = 0 (A.4) and eventually eq.(2.16) after a comparison with eq.(2.14).
Appendix B. In this appendix we will clarify the role and the genesis of the composition rule of eq.(3.28). The Laguerre derivative satisfies the identity n l ∂ξ
= ∂ξn ξ n ∂ξn
(B.1)
and, accordingly, the pseudo-shift operator ecˆyl ∂x = l e(y l ∂x ) acting on a monomial xn yields "∞ # n X yr X yr (n!)2 n r r r n n−r ∂ x ∂ x = = (x ⊕l y)n l e(y l ∂x )x = x 2x 2 x 2 (r!) (r!) [(n − r)!] r=0 r=0 (B.2) It can now be argued that the composition of eq.(3.28) can be considered as the Newton binomial associated with the algebraic structure due to the specific form of the operator cˆn . According to the previous identities we can also state that l e(y l ∂x )l e(x)
=
l e(y)l e(x)
l e(y l ∂x )l e(x)
=
l e(x
⊕l y)
(B.3)
allowing the derivation of the following semi-group property of the l-exponential l e(y)l e(x)
= l e(x ⊕l y)
(B.4)
In full analogy with the ordinary Euler formula we introduce the l-trigonometric (l-t)-functions through the identity l e(ix)
= l c(x) + i l s(x)
(B.5)
where the l-t cosine and sine functions are specified by the series l c(x)
l s(x)
= =
∞ X (−1)r x2r r=0 ∞ X r=0
[(2r)!]2
(B.6)
(−1)r x2r+1 [(2r + 1)!]2
It is easily checked that they satisfy the identities l ∂x [l c(αx)]
=
−α l s(αx)
l ∂x [l s(αx)]
=
α l c(αx)
(B.7)
14Giuseppe Dattoli, Emanuele Di Palma, Federico Nguyen and Elio Sabia and therefore the harmonic equation (l ∂x )2 [l c(αx)] 2
(l ∂x ) [l s(αx)]
= −α2 l c(αx)
(B.8)
2
= −α l s(αx)
The use of the properties of the l-exponential function allows the derivation of the following addition theorems for the functions in eq.(B.6) l c(x
⊕l y)
=
l c(x)l c(y)
− l s(x)l s(y)
l s(x
⊕l y)
=
l c(x)l s(y)
+ l s(x)l c(y)
(B.9)
The proof is given below, by observing that l e(ix)l e(iy)
= [l c(x) + i l s(x)] [l c(y) + i l s(y)] =
(B.10)
[l c(x)l c(y) − l s(x)l s(y)] + i [l c(x)l s(y) + l s(x)l c(y)] and since l e(ix)l e(iy)
= l e (i(x ⊕l y)) = l c(x ⊕l y) + i l s(x ⊕l y)
(B.11)
we can equate real and imaginary parts to infer the identities of eq.(B.9).
References [1] D. Babusci, G. Dattoli, E. Di Palma and E. Sabia, Adv. Appl. Clifford Algebras 22 (2) (2012) 271. [2] P. Fjelstad and S.G. Gal, Adv. Appl. Clifford Algebras 11 (1) (2001) 81. [3] G. Dattoli, C. Mari and A. Torre, Nuovo Cim. B 108 (1993) 61. [4] R.M. Yamaleev, Journal of Mathematics 2013 (2013) Article ID 920528. G. Dattoli and M. Del Franco, “Hyperbolic and Circular Trigonometry and Application to Special Relativity”, arXiv:1002.4728 [math-ph] (2010). [5] R.M. Yamaleev, Adv. Appl. Clifford Algebras 10 (1) (2005) 15. [6] G. Dattoli, M. Migliorati and P.E. Ricci, “The Eisentein group and the pseudo hyperbolic function”, arXiv:1010.1676 [math-ph] (2010). [7] F. Ciocci, G. Dattoli, A. Renieri and A. Torre, “Free electron lasers as insertion devices”, 237-286, from Insertion Devices for Synchrotron Radiation and Free Electron Laser, World Scientific Singapore, 2000. [8] F. De Zela, Symmetry 6 (2) (2014) 329. [9] D. Babusci, G. Dattoli and E. Sabia, J. Phys. Math. 3 (2011) 1. [10] N. Cabibbo, Phys. Rev. Lett. 10 (1963) 531; M. Kobayashi and T. Maskawa, Prog. Theor. Phys. 49 (1973) 652. [11] G. Dattoli and K. Zhukovsky, Eur. Phys. J. C 50 (2007) 817. G. Dattoli and E. Di Palma, “The exponential parameterization of the quark mixing matrix”, arXiv:1301.5111 [hep-ph]. [12] G. Dattoli, E. Di Palma and E. Sabia, Adv. Appl. Clifford Algebras 25 (1) (2015) 81. [13] K.A. Penson, P. Blasiak, A. Horzela, G.H.E. Duchamp and A.I. Solomon, J. Math. Phys. 50 (2009) 083512. [14] G. Dattoli, A.M. Mancho, M. Quattromini and A. Torre, Radiat. Phys. Chem. 61 (2001) 99.
GenTrig2015 Giuseppe Dattoli, Emanuele Di Palma, Federico Nguyen ENEA – Fusion Department – Via Enrico Fermi 45, I-00044 Frascati (Roma), Italy e-mail:
[email protected] e-mail:
[email protected] e-mail:
[email protected] Elio Sabia ENEA – Fusion Department Piazzale Enrico Fermi 1, I-80055 Portici (Napoli), Italy e-mail:
[email protected]
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