n -Dimensional dual complex numbers

n-DIMENSIONAL DUAL COMPLEX NUMBERS Paul Fjelstad 32991 Dresden Ave., Northfield, MN, USA e-mail: [email protected]

and Sorin G. Gal Department of Mathematics, University of Oradea, Str. Armatei Romˆ ane 5, 3700 Oradea, Romania e-mail: [email protected] (Received: April 30, 1998;

Accepted: June 10, 1998)

Abstract and Introduction. It is well-known that the quadratic algebras Qa,b = {z| z = x + qy, q 2 = a + qb, a, b, x, y ∈ IR, q 6∈ IR}, also expressible as IR[x]/(x2 − bx − a), are, up to isomorphism, equivalent to just three algebras, corresponding to elliptic, parabolic and hyperbolic. These three types are usually represented by Q−1,0 , Q0,0 , Q1,0 and called complex numbers, dual complex numbers and hyperbolic complex numbers, respectively. Each in turn describes a Euclidian, Galilean and Minkowskian plane. The hyperbolic complex numbers thus provide a 2-dimensional spacetime for special relativity physics (see e.g. [6]) and the dual complex numbers a 2-dimensional spacetime for Newtonian physics (see e.g. [17]). The present authors considered extensions of the hyperbolic complex numbers to n dimensions in [8], and here, in somewhat parallel fashion, some elements of algebra (in Section 1) and analysis (in Section 2) will be presented for n-dimensional dual complex numbers. Advances in Applied Clifford Algebras 8 No. 2, 309-322 (1998)

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1. Algebra For the hyperbolic case [8], some unconventional notation was used to exploit symmetries found there. Although the dual case does not benefit in the same way, similar notation will be used because of the parallel development, and just as “h” was used for the hyperbolic case, so “d” is used for the dual case. Definition 1.1. The n-dimensional dual complex number system is an abelian ring D(+, ·), where ( D=

x| x = x0 +

n−1 X

) x0 , xi ∈ IR .

xi i d,

1

Addition and multiplication are defined by x + x0 = (x0 + x00 ) +

n−1 X

(xi + x0i ) i d,

1

xx0 = x0 x00 +

n−1 X

(x0 x0i + x00 xi ) i d,

1

from which it follows that id j d

= 0,

1 ≤ i, j ≤ n − 1.

The dual conjugate of x is x defined by x = x0 −

n−1 X

xi i d,

1

so xx = x20 , and

1 1x x = = 2, x xx x0

for x0 6= 0.

The elements with x0 = 0, namely the elements of the form

n−1 X

xi i d, are

1

the zero divisors, and they form an ideal of the ring D(+, ·). They partition

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the space of non zero divisors into 2 parts: those with x0 > 0 and those with x0 < 0. The zero divisor set is 0D

= {x ∈ D | x0 = 0}.

The norm of x is defined by |x| = (xx)1/2 = (x20 )1/2 = |x0 |,

so

|xx0 | = |x||x0 |.

Polar coordinates are developed in much the same way as in [8], with the following results. Definition 1.2. Henceforth gamma, γ (also γ 0 ), will be the variable γ ∈ {1, −1}. The radial coordinate ρ(x) is just the norm of x and it is used to define the unit hypersurface U and its subsets γ U , ρ(x) = |x0 |,

U = {x| ρ(x) = 1},

γU

= {γx| x0 = 1}.

The zero divisor hyperplane 0 D partitions U into 2 discrete pieces 1 U and Clearly U (·) is a group and 1 U (·) is a subgroup.

−1 U .

Definition 1.3. An angle set Φ and its subsets γ Φ are defined by Φ = {1, −1} × IRn−1 = {γ θ| γ ∈ {1, −1}, θ = (θ1 , . . . , θn−1 ) ∈ IRn−1 }, n−1 , γ Φ = {γ} × IR where the ordered pair (γ, θ) is written γ θ. With IRn−1 (+) the direct product group (relative to addition) and {1, −1}(·) also a group, define Φ(+) the direct product group of these two groups by considering addition of angles in the following way, 0 0 γ θ + γ 0 θ = γγ 0 (θ + θ ). The identity is 1 0 = 1 (01 , . . . , 0n−1 ) = 1 (0, . . . , 0). The angle function α : D \ 0 D → Φ is defined by   xn−1 x1 α(x) = ,..., , x0 sgn x0 x0 where sgn is the sign function. From

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 x0 x0n−1 + x00 xn−1 x0 x01 + x00 x1 , . . . , = x0 x00 x0 x00 sgn(x0 x00 )    0  x0n−1 x1 xn−1 x1 = ,..., + ,..., 0 = α(x) + α(x0 ), x0 x0 sgn x0 x0 sgn x00 x00 

α(xx0 ) =

it is clear that α is a homomorphism from (D \ 0 D)(·) to Φ(+). Further, since n−1 X x ∈ 1 U if and only if x0 = 1, and since α(1 + xii d) = 1 (x1 , . . . , xn−1 ) for 1

all xi ∈ IR, α is an isomorphism, from 1 U (·) to 1 Φ(+), and similarly from U (·) to Φ(+). The restriction of α to U thus has an inverse µ : Φ → U , called the unit measure for angles, and notationally the argument γ θ will be applied as an exponent, namely µγ θ = γ +

n−1 X

n−1 X

γθi i d = γ(1 +

1

θi i d) = γ µ1 θ ,

1

which, besides being a useful relation, provides the following equivalent expressions, n−1 P n−1 n−1 n−1 θi i d Y Y X 1θ µ =1+ θii d = e 1 = eθi i d = (1 + θii d). 1

1

1

The final result is x = x0 +

n−1 X

xii d = x0

1+

1

n−1 X 1

xi id x0

( xx10 ,...,

sgnx0

= |x0 | µ

!

xn−1 x0

= |x0 | sgnx0 µ1 )

( xx10 ,...,

xn−1 x0

)

=

= ρ(x) µα(x) .

One can now come up with a generalization of the Euler formula by defining the trigonometric functions cd0 and cdi , 1 ≤ i ≤ n − 1, from Φ to IR, where, for x ∈ U , cd0 α(x) = x0 , cdi α(x) = xi , so that µα(x) = x = x0 +

n−1 X 1

xii d = cd0 α(x) +

n−1 X 1

cdi α(x) i d.

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The simplicity of trigonometry for the n = 2 case extends to the general case, namely cd0 γ θ = γ, cdi γ θ = γ θi . Similarly, geometric concepts, as developed for n = 2 in [17], extend to the general case. For example, the way to a special equation for some lines is the following. For 2 distinct points x0 , x00 , let (x0 ; x00 ) designate the directed line segment joining them. When x00 − x0 ∈ / 0 D, an angle is associated with this line segment by translating it by −x0 to the line segment (0; x00 − x0 ) whose angle is α(x00 − x0 ). For 2 such line segments (x0 ; x00 ), (x0 ; x) meeting at x0 , the directed angle between them, in moving from (x0 ; x00 ) to (x0 ; x), is   x − x0 0 00 0 α(x − x ) − α(x − x ) = α , x00 − x0 where the last term follows from α being a homomorphism. This allows one to come up with an equation for the line through x0 and x00 , because if x is on that line, α[(x − x0 )/(x00 − x0 )] is either 1 0 or −1 0, which means (x − x0 )/(x00 − x0 ) equals its dual conjugate, i.e. x − x0 x − x0 . = 00 00 0 x −x x − x0 Upon rearranging, this results in (x00 − x0 )x − (x00 − x0 )x + (x00 x0 − x00 x0 ) = 0,

x00 − x0 ∈ / 0 D,

which generalizes to ax − ax + b = 0,

a∈ / 0 D, b ∈ 0 D.

In somewhat similar fashion equations for higher dimensional hyperplanes, and also hyperspheres (of the dual kind), can be developed. With |x − x0 | interpreted as distance between x and x0 , the mappings of D to itself which are distance-preserving (also called motions) are the dual conjugate x, translations x + b, and rotations (of the dual kind) µγ θ x. These latter are of particular interest when D is interpreted as spacetime. With x an n−1 X event and x(t) = t + xi (t)i d a world line, one has for the world velocity 1 n−1

X dxi dx 1 =1+ id = µ dt dt 1

dxn−1 dx1 ,..., dt dt




= µ1 (v1 , . . . , vn−1 ) = µ1 v ,

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where v is the space velocity. Note that dx/d(−t) = µ−1 v . In general, γ = 1 means events are considered going forward in time, while γ = −1 considers them going backward in time. Upon applying µ1 v as a transformation to x, i.e. ! ! n−1 n−1 n−1 X X X 0 1v x =µ x= 1+ v d t+ x d =t+ (x + tv ) d, ii

ii

1

1

i

i i

1

one sees from t0 = t and x0i = xi + vi t that for n = 4 this is just the conventional Galilean transformation for Newtonian physics in 3-dimensional space, where the reference frame for x moves relative to that for x0 with velocity v. For 2 successive transformations, the exponential notation gives directly the Newtonian rule for addition of velocities 0 0 0 µ1 v µ1 v = µ1 v +1 v = µ1 (v + v) .

1) 2)

Remarks. The Galilean transformations for the case n = 4 are considered in [10]. It would be interesting to extend other results for n = 2, such as those in [2-5], to the general case.

2. Analysis Some results in [7,9,12] for the 2-dimensional case will be extended here to the n-dimensional case. The definitions of limit, continuity, and derivative are extended in the obvious way. For example, the derivative of f at x b is the limit (in D endowed with the Euclidean convergence from IRn ) lim x→b x |x−b x|6=0

f (x) − f (b x) = f 0 (b x). x−x b

Because of the subscript notation used in this paper, x b replaces the customary x0 . Theorem 2.1 (Cauchy-Riemann conditions). Let f : A → D, A ⊂ D, f (x) = V0 (x0 , . . . , xn−1 ) +

n−1 X

Vk (x0 , . . . , xn−1 )k d,

1

x = x0 +

n−1 X 1

xkk d ∈ A,

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where for x b=x b0 +

n−1 X

315

x bkk d, vb = (b x0 , . . . , x bn−1 ) and v = (x0 , . . . , xn−1 ), there

1

exists a neighborhood N1 of (b x1 , . . . , x bn−1 ) (in IRn−1 ) such that for each fixed (x1 , . . . , xn−1 ) ∈ N1 , Fk (x0 ) = Vk (x0 , x1 , . . . , xn−1 ) are continuous at x b0 , for all 0 ≤ k ≤ n − 1. If the derivative f 0 (b x) exists at x b, then the partial derivatives of the first order for all arguments of the Vk must exist at x b and satisfy the following conditions there, ∂V0 ∂Vk = , 1 ≤ k ≤ n − 1, (1) ∂x0 ∂xk ∂Vk = 0, ∂xj

0 ≤ k ≤ n − 1, 1 ≤ j ≤ n − 1, k 6= j.

(2)

Proof. Assume that f 0 (b x) = a0 +

n−1 X

akk d.

1

Then by hypothesis we get n−1 X V (v) − V (b v ) + [Vk (v) − Vk (b v )]k d 0  0  1 lim  −  n−1 X x→b x  x0 − x b0 + (xk − x bk )k d x0 6=b x0



 !  n−1 X  a0 + akk d   = 0,  1

1

i.e. V0 (v) − V0 (b v) + x0 − x b0 +

n−1 X 1 n−1 X

[Vk (v) − Vk (b v )]k d − a0 +

(xk − x bk )k d

n−1 X

! akk d

n−1 X

= h0 (v)+

1

hk (v)k d,

1

1

where lim hk (v) = 0, 0 ≤ k ≤ n − 1. v→b v x0 6=b x0 By simple calculation we obtain V0 (v) − V0 (b v ) = a0 (x0 − x b0 ) + h0 (v)(x0 − x b0 ),

(3)

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Vk (v)−Vk (b v ) = ak (x0 −b x0 )+a0 (xk −b xk )+hk (v)(x0 −b x0 )+h0 (v)(xk −b xk ). (4) Taking in (3) xk = x bk , 1 ≤ k ≤ n − 1, dividing by x0 − x b0 and passing ∂V0 to limit with x0 → x b0 (x0 6= x b0 ), we get (b v ) = a0 , since lim h0 (v) = ∂x0 v→b v x0 6=b x0 lim h0 (x0 , x b1 , . . . , x bn−1 ) = 0. x0 →x b0 x0 6=b x0 Then, passing to limit with x0 → x b0 , x0 6= x b0 in (3), for all (x1 , . . . , xn−1 ) ∈ N1 , the neighborhood of (b x1 , . . . , x bn−1 ), we get by hypothesis, V0 (b x0 , x1 , . . . , xn−1 ) − V0 (b x0 , x b1 , . . . , x bn−1 ) = lim [h0 (v)(x0 − x b0 )]. x→b x0 x0 6=b x0 But by the above reasonings, for any ε > 0, there is a δ > 0 such that |h0 (v)| < ε, for |xk − x bk | < δ, x0 6= x b0 , 0 ≤ k ≤ n − 1. For δ sufficiently small, |xk −b xk | < δ, 1 ≤ k ≤ n−1 implies (x1 , . . . , xn−1 ) ∈ N1 and therefore we get that V0 (b x0 , x1 , . . . , xn−1 ) − V0 (b x0 , x b1 , . . . , x bn−1 ) = 0, for |xk − x bk | < δ, 1 ≤ k ≤ n − 1. ∂V0 (b v ) = 0, 1 ≤ j ≤ n − 1. ∂xj Now, let k ∈ {1, . . . , n − 1} be fixed. Taking in (4) xk = x bk , passing to limit with x0 → x b0 , x0 6= x b0 and reasoning for hk (v)(x0 − x b0 ) as above for h0 (v)(x0 − x b0 ), we get that there exists a δ > 0 such that This obviously implies that

Vk (b x0 , x1 , . . . , xk−1 , x bk , xk+1 , . . . , xn−1 ) = Vk (b x0 , x b1 , . . . , x bn ), for |xj −b xj | < δ, j, k ∈ {1, . . . , n−1}, j 6= k. This obviously implies

∂Vk (b v ) = 0, ∂xj

for j, k ∈ {1, . . . , n − 1}, j 6= k. On the other hand, taking in (4) x1 = x b1 , xk−1 = x bk−1 , xk+1 = x bk+1 , . . . ,xn−1 = x bn−1 (where k ∈ {1, . . . , n − 1} is fixed) and then passing to limit with x0 → x b0 , x0 6= x b0 , we get Vk (b x0 , . . . , x bk−1 , xk , x bk+1 , . . . , x bn−1 ) − Vk (b x0 , . . . , x bn−1 ) = a0 (xk − x bk )+ + lim [h0 (x0 , x b1 , . . . , x bk−1 , xk , x bk+1 , . . . , x bn−1 )(xk − x bk )]. x0 →b x0 x0 6=b x0

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Dividing by xk − x bk , xk 6= x bk and then passing to limit with xk → x bk , xk 6= x bk (since |hk (v)| < ε, for all |xk − x bk | < δ, 0 ≤ k ≤ n − 1, x0 6= x b0 , we have     lim  lim h0 (x0 , x b1 , . . . , x bk−1 , xk , x bk+1 , . . . , x bn−1 )(xk − x bk )   = 0) xk →b xk x0 →b x0 x0 6=b x0 xk 6=b xk the result is

∂Vk (b v ) = a0 , for 1 ≤ k ≤ n − 1, which completes the proof. [] ∂xk

Corollary 2.2. Let f : A → D, A ⊂ D be of the form in Theorem 2.1 and assume the Vk , 0 ≤ k ≤ n − 1, have partial derivatives of order one continuous on A. Then f is holomorphic on A (i.e. the derivative exists at each x ∈ A) if and only if it is of the form V0 (x0 , . . . , xn−1 ) = P (x0 ), Vk (x0 , . . . , xn−1 ) = xk · P 0 (x0 ) + Rk (x0 ),

1 ≤ k ≤ n − 1,

where P 0 and Rk , 1 ≤ k ≤ n − 1 are differentiable on their domains of definition. Proof. Assume that f is holomorphic on A. Then by Theorem 2.1 we get that (1) and (2) are satisfied for each x ∈ A. By (2) we easily obtain that each Vk , 1 ≤ k ≤ n − 1, depends only on x0 and xk , i.e. we have Vk (v) = Pk (x0 , xk ),

1 ≤ k ≤ n − 1, v = (x0 , . . . , xn−1 ) ∈ A.

(5)

But by (2) we have that V0 (v) = P (x0 ), for all v ∈ A, which combined with (1) immediately implies, 0=

∂ 2 V0 ∂ 2 Vk (v) = (v). ∂x0 ∂xk ∂x2k

This and (5) imply ∂Vk ∂Pk (v) = (x0 , xk ) = Qk (x0 ), ∂xk ∂xk i.e. Pk (v) = xk Qk (x0 ) + Rk (x0 ).

(6)

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On the other hand by (1) we have ∂Vk ∂Pk ∂V0 (v) = (v) = = P 0 (x0 ) = Qk (x0 ), ∂xk ∂xk ∂x0 i.e. Vk (v) = xk P 0 (x0 ) + Rk (x0 ). Now, since for x bk 6= x b0k fixed, we have the hypothesis that Vk ∈ C 1 (A), subtracting one value of Vk from the other, we get that (b xk − x b0k )P 0 (x0 ) ∈ C 1 (A), 0 i.e. P is differentiable on its domain of definition (as function of the variable x0 ). This immediately implies the Rk also are differentiable on their domains of definition. Conversely, by simple calculations we obtain f (x) − f (b x) P (x0 ) − P (b x0 ) = + x−x b x0 − x b0 ( n−1 X P 0 (x0 )(x0 − x b0 ) − [P (x0 ) − P (b x0 )] (xk − x bk ) + + 2 (x0 − x b0 ) 1 x0 ) P 0 (x0 ) − P 0 (b x0 ) Rk (x0 ) − Rk (b + +b xk x0 − x b0 x0 − x b0

) k d,

x0 6= x b0 .

Passing to limit with x → x b, |x − x b| = 6 0 (i.e. x0 → x b0 , x0 6= x b0 ), since by the l’Hopital’s rule lim x0 →b x0 x0 6=b x0

P 00 (b x0 ) P 0 (x0 )(x0 − x b0 ) − [P (x0 ) − P (b x0 )] = , 2 (x0 − x b0 ) 2

finally we obtain that there exists the limit n−1

X f (x) − f (b x) lim = P 0 (b x0 ) + [b xk · P 00 (b x0 ) + Rk0 (b x0 )]k d, x − x b x0 →b x0 1 x0 6=b x0 which proves the corollary. []

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Remarks. For n = 2, Theorem 2.1 and Corollary 2.2 were proved in [9] and [7,12], respectively. 2) Reasoning exactly as in [12, §2.9] it is easy to show that the natural domains of definition for holomorphic functions are the streaks of the form ( ) n−1 X S[a, b] = x = x0 + xkk d ∈ D| a ≤ x0 ≤ b, xk ∈ IR . 1)

1

In what follows some simple applications of the derivative will be given. Theorem 2.3. Let f : S[a, b] → D, f (x) = V0 (x0 , . . . , xn−1 ) +

n−1 X

Vk (x0 , . . . , xn−1 )k d,

1

with the Vk having all partial derivatives of order one continuous (on their corresponding domains of definition). If f is holomorphic on S[a, b] and |f 0 (x)| = 6 0, x ∈ S[a, b], then f is one-toone on S[a, b] (i.e. univalent on S[a, b]). Proof. By Corollary 2.2 we have V1 (x0 , . . . , xn−1 ) = P (x0 ), Vk (x) = xk P 0 (x0 )+Rk (x0 ) with P 0 , Rk differentiable on [a, b]. Associate f with the vectorial function F = (F0 , . . . , Fn−1 ), where the Fk : [a, b] × IRn−1 → IR are defined by F1 (v) = P (x0 ),

Fk (v) = xk P 0 (x0 ) + Rk (x0 ),

1 ≤ k ≤ n − 1,

v = (x0 , x1 , . . . , xn−1 ),

such that the Jacobian J(F )(x) = [P 0 (x0 )]n 6= 0, by hypothesis. As a consequence of a well-known theorem of vector analysis (see e.g. [1, p.97]) it follows that F is one-to-one, which means that f is one-to-one. [] Since x = ρ(x)µα(x) when x0 6= 0, the idea of a conformal mapping can be n−1 X applied. An arc γ : [a, b] → D, with γ(t) = γ0 (t) + γk (t)k d, t ∈ [a, b], is 1 Pn−1 smooth if γk ∈ C 1 [a, b], for all 0 ≤ k ≤ n−1, and |γ 0 (t)| = γ00 (t) + 1 γk0 (t)k d = |γ00 (t)| = 6 0, for all t ∈ [a, b]. This means that if γ is a smooth arc then α[γ 0 (t)] is defined (see Section 1) and has a tangent whose direction is determined by the angle α[γ 0 (t)].

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A function f : A → D, A ⊂ D, x b ∈ A, is dual conformal at x b if for each pair of smooth arcs γ, β : [a, b] → D with γ(b t ) = β(b t) = x b, the following holds, α[γ 0 (b t )] − α[β 0 (b t )] = α[(f ◦ γ)0 (b t )] − α[(f ◦ β)0 (b t )]. Theorem 2.4. If f : A → D has a derivative at x b ∈ A and |f 0 (b x)| 6= 0 then f is dual conformal at x b. Proof. Let γ, β be a pair of smooth arcs as above. Since |f 0 (b x)| = 6 0, α[f 0 (b x)] is defined. By hypothesis it follows that there exists a neighborhood of b t, V (b t ), such that for all t ∈ V (b t ) \ {b t } we have |γ(t) − γ(b t )| = 6 0, so we can write f (γ(t)) − f (γ(b t )) f (γ(t)) − f (γ(b t )) γ(t) − γ(b t) = · , b b b t−t γ(t) − γ(t ) t−t and similarly for β. Passing to the limit and following standard arguments we get the theorem. Remark. A simple example of dual conformal mapping is f (x) = ax+b cx+d , a, b, c, d ∈ D, |ad − bc| = 6 0, x ∈ D \ {x| |cx + d| = 0}. At the end of this section some considerations on the integral will be given. Definition 2.5. Let f : A → D, A ⊂ D and let γ : [a, b] → D be an arc in D. Considering a division d : γ(a) = z0 , z1 , . . . , zm = γ(b) and the points ξk “between” zk and zk+1 on γ, we can define the integral in the standard way as Z m−1 X f (x)dx = lim f (ξk )(zk+1 − zk ) ν(d)→0

γ

k=0

where ν(d) is the norm of d, and where the limit exists independent of the divisions d and the ξk -points. By simple standard calculations (as for the case n = 2 in [12]) we get the following Theorem 2.6. If f : A → D, Pn−1 f (x) = V0 (x0 , . . . , xn−1 ) + 1 Vk (x0 , . . . , xn−1 )k d, x = x0 +

Pn−1 1

xkk d ∈ A,

is continuous on A and if γ : [a, b] → A is a rectifiable path in A then f is integrable on γ and we have   Z Z n−1 X Z  Vk (v)dx0 + V0 (v)dxk  k d, f (x)dx = V0 (v)dx0 + γ

γ∗

1

γ∗

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where v = (x0 , x1 , . . . , xn−1 ) and γ ∗ is the attached curve in IRn to γ, defined by the parametric equations (γ ∗ )

{xk = γk (t) | 0 ≤ k ≤ n − 1, t ∈ [a, b]}, γ(t) = γ0 (t) +

n−1 X

γk (t)k d.

1

Theorem 2.7 (Cauchy’s theorem). Let f : S[a, b] → D be holomorphic on the streak S[a, b]. Then for each closed rectifiable arc γ : [a, b] → S[a, b] we have Z f (x)dx = 0. γ

Proof. By Corollary 2.2 and Theorem 2.6 we have   Z Z Z n−1 X Z  (xk P 0 (x0 ) + Rk (x0 ))dx0 + P (x0 )dxk  k d, f (x)dx = P (x0 )dx0 + 1

γ∗

γ

γ∗

γ∗

where P 0 and Rk are differentiable on [a, b]. But since γ(a) = γ(b), it follows γk (a) = γk (b), 0 ≤ k ≤ n − 1 and therefore Z

Z P (x0 )dx0 =

b

Z

a

γ∗

Z

γ0 (b)

Rk (x0 )dx0 =

Z

[xk P (x0 )dx0 + P (x0 )dxk ] = γ∗

Rk (u)du = 0, γ0 (a)

γ∗ 0

P (u)du = 0, γ0 (a)

Z

Z

γ0 (b)

P [γ0 (t)]d[γ0 (t)] =

Z

γk (b)·P [γ0 (b)]

d[xk P (x0 )] = γ∗

du = 0, γk (a)·P [γ0 (a)]

which proves the theorem. [] Remark. In the case n = 2, Theorem 2.7 was proved in [12]. Also, other results in [11-16] can be extended for the case of n-dimensional dual complex numbers.

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Paul Fjelstad and Sorin. G. Gal

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