n dimensional hyperbolic numbers

7/29/2019 n Dimensional Hyperbolic Numbers

1/22

n-DIMENSIONAL HYPER BOLIC COM PLEXNUMBERSPaul Fjels tad3P991 Dresden Ave.Northfield, MN USAe-mail: ][email protected] G. GalDepartment of MathematicsUniversity of OradeaStr. Armate i Romane 5 RO MA NI Ae-mail: [email protected](Received: December 10th, 19 97 , Accepted: January 23th, 1998)

A b s t r a c t a n d I n t r o d u c t i o nDirect product r ings have received relatively lit t le attention, perhaps becausethey are sometimes labeled "tr ivial" [8, p.6] . Nevertheless, the 2-dimensionaldirect p roduc t r ing of the reals, when expressed in the "hype rbolic basis", isanalogous in many ways to t he sys tem of complex numbers and also has aphysical interpretation. This prompted an exploratory foray into the worldof n-dimensional direct p roduct r ings of the reals to see how much can beext end ed from th e 2-dim ensional case (see, e.g. [3,4,5]). Sect ion 1 providesalgebraic notation, up to the point of defining polar coordinates. Section 2uses analysis to explore differentiability and conformality.1. AlgebraThe 2-dimensional hyperbolic complex numbers (also called double numbers[10] and perpl ex nu mb ers [5]) ate, o ne might say, "cousins" of the (elliptic)

Advances in Applied Clifford Algebras 8 No. 1, 47-68 (1998)


2/22

4 8 n - D i m e n s i o n a l H y p e r b o l i c C o m p l e x N u m b e r s P . F j e l s t a d a n d S . G . G a lc o m p l e x n u m b e r s . T h e y c o n si s t o f e l e m e n t s x + h y , x , y E R , h ~ R , h 2 = 1.T h e y p r o v i d e a r a t h e r n a t u r a l l a n g u a g e f o r d o i n g sp e c ia l r e la t i v i t y t h e o r y int w o d i m e n s i o n s [5]. A l g e b r a ic a l ly , t h e y f o r m a c o m m u t a t i v e r i n g w i t h e l e m e n t sx ( 1 + h ) b e i n g z e r o d i v i s o r s . T h e s e z e r o d i v i s o r s f o r m t h e l i g h t, c o n e i n t h ep h y s i c a l in t e r p r e t a t i o n a n d a r e a n a t u r a l o u t c o m e o f t h e fa c t t h a t t h i s n u m b e rs y s t e m i s a d i r e c t p r o d u c t r i n g o f t h e r e a l n u m b e r s , n a m e l y 2 ( + , . ) , w h e r ea d d i t i o n a n d m u l t i p l i c a t i o n a r e d o n e c o m p o n e n t w i s e a n d m u l t i p ly i n g b y a r e aln u m b e r d i s t r i b u t e s t h r o u g h t h e 2 - t u p l e . T o se e t h i s , s e t 1 = ( 1 , 1 ), h = ( 1 , - 1 ) .T h e nx + h y = ( 1 , 1 ) z + ( 1 , - 1 ) y = ( x , x ) + ( y , - y ) = ( x + y , x - y ) = ( x + y ) e o + ( X - y ) e i ,w h e r e e 0 = ( 1 , 0 ) , e l = ( 0 , 1 ) f o r m t h e n e w b a s i s , b o t h o f w h i c h a r e z e r od i v i so r s . T o e x p r e s s t h e m i n t e r m s o f 1, h , t h e h y p e r b o l i c b a s i s, u s e

1 - - - ( 1 , 1 ) = e 0 + e l , h = ( 1 , - 1 ) = e 0 - e t , t o g e tC o = ( 1 / 2 ) ( 1 + h ) , e , = ( 1 / 2 ) ( 1 - h ) .

T h e e 0, e l c o o r d i n a t e s y s t e m p r o v id e s m a n y a d v a n t a g e s i n m a k i n g c a l c u l a t i o n s( s e e , e . g . [ 4 ] ) .A v e r y n a t u r a l w a y t o e x t e n d t h e 2 - d ir n e n s i o n a l s y s t e m t o h i g h er d i m e n s i o n si s t o c o n s i d e r t h e d i r e c t p r o d u c t r i n g R ~ ( + , .) a n d t h e n c o n s t r u c t a b a s i s1, h i , " "" , h i , " " , h n - 1 , w h e r e h ~ = 1 , h i ~ .[ 91 f o r a ll i . T h e d i r e c t p r o d u c tb a s i s c o n s i s t s o f t h e u s u a l e i , 0 < i < n - 1 , w h e r e e l is t h e n - t u p l e w i t h t h ei t h e n t r y b e i n g 1 a n d a l l th e o t h e r s 0 . I t fo l lo w s t h a t

iA g e n e r a l n - t u p l e z = ( z o , ' " , z i , " ' , z n - 1 ) , z i E R , is t h e n r e p r e s e n t e d b yz = ~ / z i e i . I n p a r t i c u l a r ,

, ( 1 ) 1( z + z ' ) , = z i + 4 , ( z z ' ) , = z , z , , = - .i z iz is a z e r o d i v i s o r i f z i = 0 f o r o n e o r m o r e i . I f z i = r f o r a ll r , z b e h a v e sl ik e t h e r e a l n u m b e r r , so w e d o n o t d i s t i n g u i s h b e t w e e n z a n d r , i .e . w e w r i t er i = r . T w o s p e c i a l c a s e s a r e 1 i = 1 , 0 i = 0 . T h e m o d u l u s f i l ( z ) a n d n o r m ] z ]a r e d e f in e d b y

. ~ ( z ) = 1 - ~ z ~ , I z l = [ r o ( z ) ] ~ / ~ ,i


3/22

Advances in Applied Clifford Algebras 8, No. 1 (1998) 49w h e r e , f o r x E R , x 1 l a = s gn z q 1 /'~ . T h u s I z I = 0 f o r z a z e r o d i v i s o r , a n d I z 1is n e g a t i v e w h e n z i r 0 f o r a l l i a n d a n o d d n u m b e r o f z i ' s a te n e g a t i v e . A l s o

M ( z z ' ) = M ( z ) M ( z ' ) , I z z ' l = I z l l z ' l .T h e z e r o d i v i s o r s p a r t i t i o n t h e s p a c e o f n o n z e r o d iv i s o r s i n t o 2 ~ p a r t s . F o re a c h p a r t t h e r e i s e x a c t l y o n e z w h e r e z i - + 1 f o r a ll i. T h a t m e a n s z 2 = 1 ,a n d t h e s e z a t e t h e c a n d i d a t e s f o r t h e b a s i s m e n t i o n e d a b o v e . O n l y n a re t ob e c h o s e n , s o t h e r e i s m o r e t h a n o n e w a y t o d o t h i s . T h e s e 2 ~ e l e m e n t s a r e t h em e m b e r s o f t h e s e t { 1 , - 1 } '~ , a n d s i n c e { 1 , - 1 } ( . ) is a g r o u p , s o i s t h e d i re c tp r o d u c t { 1 , - 1 } ~ ( . ) . I f t h e n e le m e n t s c h o s e n fo r t h e b a s is f o r m a s u b g r o u p ,c e r t a i n h i c e p r o p e r t i e s f o ll o w , b u t s i n c e t h e s iz e o f a s u b g r o u p m u s t d i v i d e 2 ~( L a g r a n g e ' s T h e o r e m ) , t h i s is o n l y p o s s i b l e w h e n n i ts e l f i s a p o w e r o f 2 . T h e s ec a se s w i ll b e c o n s i d e r e d f i r st b e c a u s e o f t h e a d d i t i o n a l s y m m e t r i e s w h i c h e n s u ea n d b e c a u s e o f t h e i r r e l a t i o n t o C l i ff o r d a lg e b r a s .T h e C a s e o f D i m e n s i o n n = 2 vB a s i s R e l a t i o n sT o e x p lo i t t h e s y m m e t r i e s , s o m e u n c o n v e n t i o n a l n o t a t i o n w i l l b e i n t r o d u c e df o r t h e c a s e s n = 2 " , v a p o s i t i v e i n t e g e r . T h e K r o n e c k e r d e l t a f u n c t i o n ( ~ i j w i l lb e w r i t t e n i 9 1 w i t h 0 < i , j < n - 1 w h e r e 6 is a h n x n m a t r i x , a c t u a l l y t h ei d e n t i t y m a t r i x , a n d i 91 is t h e r o w ( q 1 9 1 1 6 1 = ( 0 , . . . , 1 , . . . , 0 ),s o t h e s y m b o l " i w i ll b e u s e d f o r t h e b a s i s e l e m e n t u s u a l l y s i g n i fi e d b y " e i " .F o r t h e d i r e c t p r o d u c i b a s i s ,

B ~ , ~ : { i ~ 1 0 < i < n - 1 } ,n o t e t h e d i s t i n c t i o n b e t w e e n i 6 j 6 a n d q 9 j 6 , t h e i n n e r o r s c a la r p r o d u c t .

T h e K r o n e c k e r p r o d u c t o f t w o d i r e c t p r o d u c t r i n g s | : R n x R TMp r o d u c e s a t h i r d d i r e c t p r o d u c t r in g , w h e r e~ R ~ ~ ,

z | z ' = ( z o , . . . , z ~ _ l ) | z ' = ( z o z ' , . . . , Z ~ _ l Z ' ) =( z o z 9 ' . . ' z '" ,ZoZm--1, " ,Zn--lZo,''',Zn--I m--l)

T h i s o p e r a t i o n is n o t c o m m u t a t i v e , b u t i t is a s s o c ia t iv e a n d h a s a " c o m p o n e n -t w i se " m u l t i p l i c a t i v e p r o p e r t y , f o r z , z' E [ w , w ~ E R m , . . . , v , v ~ E F 91


4/22

50 n-Dimensional Hyperbolic Complex Numbers P. Fjelstad and S. G. Gal

( z | | v = z | 1 7 4( z | 1 7 4 . . . | | | . . . | 1 6 2 = z z' | | . .. |

I n p a r t i c u l a r , f o r q 6 R '~ , j 6 E R mi 1 7 4 : ( ~ 1 6 1 : { ~ + j ~ ,

s o th a t , f o r q 6 R 2 , w h i c h m e a n s i , j , k 6 { 0 , 1 } ,i 6 | | k 6 -~- i 2 + j 6 | k ~ = i 4 + 2 j + k 6 = i j k 6 = a 6 ,

w h e r e a = i j k i s e x p r e s s e d i n b a s e 2 . H e n c e f o r t h s u b s c r i p t s a , b , c , d w i l l b ee x p r e s s e d i n b a s e 2 , s o a = a v - 1 . . . a l a o a n d

|a91 = a , _ , 6 | . . . @ a o 6 = 1 ~ ~ , 6 ,

iw h e r e t h e o r d e r o f t h e p r o d u c t i s i m p o r t a n t . T h e n b a s i s e l e m e n t s a 6 6 R na t e t h u s g e n e r a t e d f r o m o6 a n d 16 i n R 2. N o t e t h a t v is t h e n u m b e r o fd i g i t s in t h e s u b s c r i p t . F o r e c o n o m y , a t i l d e o v e r a b a s e 10 i n t e g e r m e a n st h a t i n t e g e r e x p r e s s e d i n b a s e 2 u s i n g v d ig i t s . F o r e x a m p l e , f o r v = 3 ,a 6 { 0 0 0 , 0 0 1 , . . - , 1 1 1 } = { 0 , i , . . . , 7 } .

T h e f o r e g o i n g s u g g e s t s a s y s t e m a t i c w a y t o se l e c t t h e e l e m e n t s f r o m { 1 , - 1} '~f o r t h e h y p e r b o l i c b a s i s . F o r v = 1 t h e b a s i s e l e m e n t s w e r e 1 , h , w h i c h w i l l n o wb e r e la b e l e d o H , 1 H , w h e r e

o H = ( 1 , 1 ) = 1 1, q - - ( 1 , - 1 ) = i i ,w i t h i : - 1 a n d th e p a r e n t h e s e s a n d c o m m a s a r e d r o p p e d f o r e c o n o m y . T h ee l e m e n t a H i s t h e n d e f i ne d a s t h e K r o n e c k e r p r o d u c t ,

|a H = 1 - I ~ , H -i

E x a m p l e .1 1 0 H = l H | 1 7 4 l i | 1 7 4 = l i | l l i i i i l l .

F o r ~, = 1 , 2 , 3 , t h e a H a r e


5/22

Advances in Appl ied Clifford Algebras 8, No. 1 (1998) 51o H = l l o o H = l l l l o oo H = 11 1 1 1 1 111 H = 1 i 0 I H = l l l i 0 0 1 H = l i l i l i l l

1 0 H = l l l l 0 1 0 H = 1 1 1 1 1 1 1 11 1 H = 1 1 1 1 0 1 1 H = 1 1 1 1 1 1 1 1

1 0 0 H = l l l l l l l l1 0 1 H = 9n 0 H = l l l l l l l l1 1 1 H = 1 1 1 1 1 1 1 1

I f o n e u s e s b a s e 2 s u b s c r i p t s b , i .e . a H b , t o d e s i gn a t e t h e b t h e le m e n t o f a H ,t h e na H = E a H bb 91

bH c a n t h u s b e c o n s i d e r e d a m a t r i x , n a m e l y t h e n x n a r r a y s d i s p la y e d a b o v e .F o r t h e hyperbol ic basis

B f t , ~ = { a H l a E { 0 , 1 } ~ } ,t h e r e i s t h e f o l lo w i n g L e m m a r e l a t i n g i t t o t h e d i r e c t p r o d u c t r in g { 0 , 1 } v( @ , |o f t h e 2 - e l e m e n t f i el d { 0 , 1 } ( @ , | w h e r e 1 @ 1 - 0 , ( o f t e n d e n o t e d Z 2 ( + , . )) .T h e m e m b e r s o f { 0 , 1 }~ a re ~ - t u p l e s o f 0 ' s a n d l ' s , w h i c h , u p o n d r o p p i n gp a r e n t h e s e s a n d c o m m a s w i l l b e i d e n t if i e d w i t h t h e b a s e 2 n u m b e r s a , b , s oa ~ b i s d e f i n e d , n a m e l y ( a @ b ) i = a i ~ b i . I n p a r t i c u l a r , a @ a = 0 .L e m m a 1 . 1BH,,~( . ) i s a g r o u p i s o m o r p h i c t o { 0 , 1 } ~ (@ ) .P r o o f : F i r s t n o t e t h a ta ~ H b , H = a , e b ~ H , s i n c e o H o H = o H , o H I H = 1 H o H - 1 H, 1 H 1 H = o H .T h i s p r o v i d e s a r u l e f o r m u l t i p l y i n g e l e m e n t s a H , n a m e l y

| @ @ @ |a H b H = ~ - ~ a , H l - I b , H = H a , H b , H = ~ a . ~ b . H = r I ( a ~ b ) , H = a@ bH .

i i i i iw h i c h p r o v i d e s t h e p r o o f .


6/22

52 n-Dimensional Hyperbolic Complex Nu mb ers P. Fjelstad and S. G. Ga lR e m a r k . I r w i ll b e c o n v e n i e n t t o c a l c u l a te th e v al u e o f a H b f r o m it s s u b -s c r i p t s . T o t h i s e n d , d e f i n e t h e i o t a f u n c t i o n ,

= 1 - 2 i 8 j = ; - 1 , i = j w i t h s p e c i a l i z a t i o n t oig j [ 1, i # j( 1 , a i = 091 = 1 ta , = - -1 , a i = 1

T h e n t ( a i 9 b i) = t ( a i ) t ( b i ) ,w h i c h s i m p l y s a ys t h a t t h e g r o u p { 0, 1 }( @ ) is i s o m o r p h i c t o { 1 , - 1 } ( . ) . A l s o,

~ , H b , = t ( a i b i ) , s i n c e 0 H 0 = I , 0 H 1 = 1 H 0 = 1 , 1 H 1 = - 1 .I r f o l t o w s t h a t

@ |E o m ~ = oL , = 1 - I o n " = [ I ( o , ~ 0 0 + o , ~ , , ~ ) =b i i

b i bSO,

(9a H b = H a , H b , - - - - H t ( a i b i ) = 9 1 = 9 1 (1 )i i i

w h e r e a . b i s t h e @ - i n n er p r o d u c t . S i n c e a e b = b e a , t h e m a t r i x H is s y m m e t r i c .L e m m a 1 .2T h e b a s i s B H , n i s o r t h o g o n a l , a n d a l s o n o r m a l in th e s e n s e t h a t l a H ] = 1 .P r o o f : F o r a = 0 , a , , b = 0 , f o r a ll b . F o r a # 0 , a . b = 0 f o r h a l f t h e b ' s ( 0 ' sa n d l ' s a r e d i s t r i b u t e d e v e n l y i n a ll d i g i t p l a c e s) a n d a 9 b = 1 f o r t h e o t h e rh a l f. T h u s t h e f o ll o w in g r es u l ts f o r s u m a n d p r o d u c t ,

{ ~ ~ 1 7 6 no H b = 0, a ~ , a l s o M ( o Z ) = o H b = l .b


7/22

Advances in Applied Clifford Algebras 8, No. 1 (1998) 53S i n c e M ( a H ) = 1 , c l e a r ly q I = 1. T h e i n n e r p r o d u c t o f e l e m e n t s i n B H , n is

~H * = ~HcbH~ = ~H = ~ebH~ = O, a b 'c c r

w h i c h s h o w s o r t h o g o n a l i t y . [ ]L e m m a 1 .3F o r z = ~ ~ z ~~ x = ~ r a n d z = z ,

1x~ = - ~ z ~ ~ H r zc = ~ x ~ ~ H ~ .n

a r

Proof : F i r s t , n o t e t h a t ,Z ( n , a = b~(H 2)b = ~H Hb = ~H~r = ~HcbHr = O, a r b ' s o

c c

H n = n6 , a n d H - 1 = 1 H .n

O n e c a n n o w e x p r e s s a ~ i n t e r m s o f t h e b a s i s B H ,,~ ,~ ~ = ~ ( H 2 ) = n , H ~ ~ H ,

c

w h i c h a l l o w s z t o b e e x p r e s s e d i n t h i s b a s i s ,( 2 )

a ~1 C C G C

T h u s , i n s e q u e n c e ,1 l z H ,z ~ = ~ z ~ " H ~ ' x = n z = x H , zr = ~ x a ~ H r ~ (3 )

a a

A G r a d e d A l g e b r aT h e g r o u p BH,,~( ' ) i s g e n e r a t e d b y a n y i n d e p e n d e n t s u b s e t o f v e l e m e n t s . Ac o n v e n i e n t s u b s e t t o p i c k c o n s i s ts o f t h o s e a H s u c h t h a t t h e d i g i t s o f a a r e a l l0 e x c e p t f o r o n e , s a y th e i t h o n e , w h i c h w i l l b e d e s i g n a t e d ( i) . T h u s t h e ( i ) Hf o r r o a ge ne r a t i ng s e t , w h e r e ,


8/22

54 n-Dimens ional Hyperbo l ic Comp lex Numb ers P . Fje l s t ad and S. G. Gal

(i) e {0, 1}", (i)j = ,6j, and ~H = I I ( ( ' )H ) ~ ' (4)i

E x a m p l e .10tH = 100e001H = lo0H 001H = (2) H (0) H = ((2)H ) 1 ( (1)H) ~ ((0)H ) 1

R e m a r k . Us ing e qua t ions ( 1) a nd ( 4 ) now, e qua t ion ( 2) c a n be f a c to r e d inthe fo l lowing way,

a~ = 1-n~ a H r : 1 ~ H91 H ((i) H)c' : r I ~(1 + 91c c i i *

where the fac tors 89 -4- (0 H ) are idem po ten ts .In te rm s of the gen era tors , one can th ink of R '~(+ , .) , n = 2 ~ , a s a gradedalgebra by pa r t i t ion ing the ~H in to g r a de s a c c o r d ing to to the nu mbe r o f d ig i tsin a wh ic h a r e l ' s , na me ly ,

6H, (i)H, (oH(j)H, (i)H(j)H(k)H, ... ,@-I)H...(1)H(o)H,whe r e the i , j , k. . . are d is t inc t for each grade . A sa graded a lgebra , th is takeson some s im ila r i ty to a Cl i f ford a lgebra , and in fac t , the bas is e lem ents of aCl i f ford a lgebra , Cp,q, a te a p r o je c t ive r e p r e s e n ta t ion o f BH,,~ . ) wi th u = p+ q .

L e t Bcp,q( . ) be the sys tem of bas is e lements wi th mult ip l ic a t ion . T his is no ta grou p b ecaus e m ulti plic atio n is not c losed, so le t /~c~.~ = {- AL A E Bc~,~ }a nd Bcp,q [J [~cr.~ = B~p,q. T h e n B~p,~(.) i s a group ana BH,r,(.) is a homo-mo rphi c image under the mapp ing f which forge ts abo ut m inus s igns. For 7 i ,0 < i < p + q - 1, the ge ne rat ors of B~~,~, let a"f : 1-L(')'i)a', a E {0, 1 } P+ q .T h e n f( ~ 7 ) = f ( - ~ 7 ) = aH. I n the l a ngua ge o f g r oup r e p r e s e n ta t ion t he o r yB~, . , ( . ) i s a double group r e la t ive to BH,,~(') and Bc,.~(') is a projective repre-senta~ion of BH,,~(').E x a m pl e . One ma n i f e s ta t ion o f the d ihe d r a l g r oup D2 i s D2( . ) ={E , C2~, C2v,C2~}(.), which descr ibes ro ta t io ns by 7r abo ut the z ,y , z axes , i .e .C2~C2~: = E , the ide n ti ty . The s e ope r a to r s c o r r e s pond to a m om e n t umope r a to r s in qua n tu m me c ha n ic s f o r in te g r a l sp in . Fo r ha l f in te g r a l s p in thesame ope ra tor needs twice the angle to reach E, i .e . C2~ * C~~ = - E . The setwo sys tems a re represented be low [1] .


9/22

Advances in Applied CUfford Algebras 8, No. 1 (1998)E C2~ C2y C2z E C~z C~y C2zC2~ E C2~ C~y C2~ - E C2~ -C 2 yC2y C2z E C2~ C2y -C 2~ - E C2~C2, C2y C2~ E C2z C2~ -C 2~ - E

55

D2(') D2(*)D2(*) is a projective representatio n of D2(.) and D~(*) = (D2 U D2)(*) is adouble group of D2(.). Further, D2(.) is isomorphic toBH,2(-) -- {ooH, olH, loH, 11H}(') = {1,(o)H, 0)H, (o)H0)H}(. ) and D2(*)corresponds to Bco.=(') = {1,70,71,7071} with D~(*) being isomorphic toB ~ o , = ( . ) .For G' the double group of G, some properties of G carry over to G'. Forexample, the irreducible representations of G are also irreducible representa-tions of G'. G' has in addition some spinor representations, namely one foreach regular elass of G [1, p.236]. The regular classes of G are classes of G'and the negatives of the elements of a regular class forro a class of their ownfor G'..The irregular classes of G double themselves by adding the negativesof its elements to forro a single class for G'.Polar Coordinates .In order to allow for the development of trigonometry and things like conformalmappings, we need to define some 'polar eoordinates'. The radial coordinatep(z) is the absolute value of the norm, and it is used to define the unir hyper-surface U,p ( z ) = I lz ll , u - - { z l p ( z ) = i } , ~ u = f z ~ e u l a e { i , - 1 } ~ , z ~ > 0, f o r a l l a } .The zero divisor hyperplanes parti tion U into 2 ~ discrete pieces, ~U, one foreach h E {1 ,- 1} n. Clearly U(.) is a group with 1U(.) a subgroup.To assign a unique angle to each element in U and to allow for additionof angles, let ~ (+ ) be an isomorphic copy of the group U(.). The elements ofate angles. The angle function c~ (usually written "arg") is an isomorphismfrom U(.) to ~(+), i.e. for z, z' E U,

~ ( z z ' ) = o ~ ( z ) + ~ ( ~ ' ) .In particular then, for 0 the additive identity of ~(+),

~ ( i ) = o .


10/22

86 n-Dimensional Hyperbolic Complex Numbers P. Fjelstad and S. G. GalFu r th e r , sp ec i fy a pa r t i cu la r e l em en t p E U , /-* :/= 1 , a s the uni! measure forangles a n d d e f in e an e l e m e n t 1 E r b y

~ ( ~ ) = i .T h e n l e t p b e t h e i n v er s e m a p p i n g o~ i b y m e a n s o f t h e f o l lo w i n g n o t a t i o n , f o r

# r = ~ - i ( r i . e. p= ( * ) = o~ - l ( oe ( z ) ) = z .I t f o l l o w s t h a t /_,0 = 1, / . i i : /.91a n d s i n c e t h e i n v e r s e o f a n i s o m o r p h i s m is a n i s o m o r p h i s m ,

# r 1 6 2 = # r 1 6 2Trigonometric [unctions c h a , f r o m 9 t o R , a t e d e fi n e d b y

c h : a ( z ) = z a , s o t h a t p ~ ( Z ) = z = ~ x ~ ~ H = ~ - ~ c h ~ ~ ( z ) : H , (5 )a a

w h i c h i s a g e n e r a l i z a t i o n o f t h e E u l e r f o r m u l a . F o r n = 2 , c h0 a n d c h l a r e t h ef u n c t i o n s c o s h a n d s i n h .For p(z ) ~k 0 ,p [ z l p ( z ) ] = p ( z ) / # ( e ( z ) ) = e ( z ) / p ( z ) = l ,

s o z / p ( z ) E U . T h e angle fun ctio n c~ i s t he n extended t o t h e s e z b y~ ( z ) = ~ [ z / p ( z ) ] .

T h u s , f o r p(z) 0,z = p ( z ) [ ~ / p ( z ) ] = p ( z ) ~ ~ = p ( z ) ~ ~ < ' ~ ,

w h i c h p r o v i d e s t h e ' p o l a r c o o r d i n a t e s ' .S p e c i f y i n g o~ a n d pC o n s i d e r e z==~ i n t e r m s o f t h e i n f i n i t e s er i e s

e Zo , k ( z ~ ~6) k z ~ '= i! - - 1 + ~ ~ ~ = ~ b a + ~ ~ ( - l + e " ~ soi=O i = 1 b


11/22

A d v a n c e s i n A p p l i e d C l i f f o rd A l g e b r a s 8 , N o . 1 ( 1 99 8 )

T h e n ,

a n d

1, b a

e ' : e Z o ' - - ' : y I e , - o ' : Z e ' ~ s o ( e ' )o : e ~ -,

5 7

b b( e O . . H ) b = { e O * ,= H b = 1e - O " , a H b - - 1

A s is k n o w n f r o m t h e n = 2 ca s e , e x p a n d i n g e ~ 176 i n a n i n s e r i e s r e s u l t sin

e ~ " ~ H = cosh 0a + s inh 0~ aH .T h e m o d u l u s o f e O" ~ H is

. A d ( e O . . H ) = H e O . .H b : e O . ~ ~ . H b = { e n ~ a ; ~1_b 1 , a 0T h us w e ha v e , f o r a 7~ 0 ,

( e~162 > 0 a n d p(e O~=H) : 1 , s o e o ' ~ H E 1 U .n - 1I t fo l l o w s t h a t 1 - L = i e ~ E ] U . F u r t h e r , e a c h e l e m e n t o f 1U g e ts a h a n g l e .

( 6 )

T h e o r e m 1 . 4F or e a c h z E I U t h e r e i s a un i qu e 0 = c ~( z) .P r o @ " F r o m

o n e g e t s

n - 1 E n - t 0 6 a H bZb = H eO~~Hb - - e . = ia=].

n - 1) - -~ o ~ o H b = I . ~ ~ . ( 7 )a - - - - . i

T h i s p r o m p t s d e f in i n g fo r 1U a n a n g l e s e t t ~ ( w h e r e t ~ 5 ( + ) w i ll b e a s u b g r o u po f ~ ( + ) ) b y t h e d i re c t p r o d u c t


12/22

5 8 n - D i m e n s i o n a l H y p e r b o l i c C o m p l e x N u m b e r s P. F j e l st a d a n d S . G . G a l

l(I)(nt_ ) : ] ~ n - l( _ [ _ ), w i t h 8 = ( 0 i , 0 ~ , " ' , O , ~ _ z ) E lq ~ .W e w i l l r e f r a i n f r o m p u t t i n g a t i l d e o n n - 1 . I n p a r t i c u l a r , f o r a n g l e s 0 a n d1 , 0~ = 0 , l a = 1 , f o r 1 < a < n - l . A l s o , f r o m w = l n z , s o z = e w , o n e h a sz b = ( e w ) b = e w~ a n d t h u s ( l n z ) b = w b = l n z b . F i n a l l y , b y r e m o v i n g t h e t o pr o w o f H t o g i r e a h ( n - 1 ) x n m a t r i x H , w h e r e ~ / /b = ~ H b , f o r 1 < a < n - 1 ,< b < n - 1 , o n e c a n w r i t e ( 7) a s

8 I I = I n z .F o r / : q t h e t r a n s p o s e o f / / a n d 1 _< a , b < n - 1 ,

n--I n--1a ( I: I ~ t T ) b : ~-'~~a / : / c c / S / [ - ~ '~ ~ a H c b H c : n a ~ b,

c=5 rT h u s , m u l t i p l y i n g b o t h s i de s o f ( 8 ) b y ( 1 / n ) f f I T ,

so Hf/ = n 6 .

n - 10 = l ( l n z ) / : / T a n d 0 a = -I ~ _ l n z b b H a .

n n b=5T h e e l e m e n t # i s n o w d e f i n e d b y f ir s t d e f in i n g0 a = l a = 1 ,

n--I n-I/ z o = r I e 8 ~ ~ H , ~ = H e ~ H .

a = i ~ = iF u r t h e r ,

# e a n d t h e n s e t t i n g 0 = 1 , i .e .

N o t e n o w t h a t = { /i #e [5 E { 1 , - 1 } " , 8 6 R n - 1 } . F u r t h e r , t o h a v e a n g l e sf o r a l l e l e m e n t s o f U , d e f i n e

x R n - l = { s o n o w ~ = { 1 , - 1 } " x R " - 1 ,w h e r e t h e o r d e r e d p a i r ( h, 0 ) i s w r i t t e n T h e a n g l e f u n c t i o n a f r o m U o n t o

i s n o w s p e c i f i e d b ya ( h ~ e ) = ~ 8 ,

( 8 )

n - - I n - - 1 n - - I]_91 = H eO ~H = I' ~1 .~ H e O : = H e O ' ~ H = I I e ( O = + O " ) ~ H = ] AO +O '

:=i ~=i :=1


13/22

Advances in Applied Clifford Algebras 8, No. 1 (1998) 59w h i c h is a o n e - to - o n e m a p p i n g . I t i s m a d e a n i s o m o r p h i s m b y d e f in i n g a d d i l i o no f a n g l e s b y

h o + ~ ,o ' = ~ ( h ~ ~ + ~ ( h ' ~ ~ = ~ ( h ~ ~ ~ = ~ ( ~ 1 7 6 = ~ ~ ,( 0 + 0 ' ) .R e m a r k s : 1) S in ce { 1 , - 1 } n ( . ) is a g r o u p a n d R ' ~ - t ( + ) is a g r o u p , t hi sl a s t r e s u lt s h o w s t h a t t h e g r o u p q ~ (+ ) is t h e d i re c t p r o d u c t o f / { , ~ - 1 ( + ) w i t h{1 , -1}~ ( . ) .2 ) F o r z E h U , o n e h a s z = h # e = p ( z ) # ~ ( * ), w i t h p ( z ) = 1 , ~ ( z ) = ~ 0 , s o

h # o = # ~ o .Fo r h = 1 , i t fo l lo ws t h a t 0 an d 1 8 a re i d e n t i f i ed . A l so , b e ca u s e h = = 1 , i tf o ll o w s t h a t f o r, z , z ' 6 o n e h a s z z ' 6 1 U .3 ) A c o n n e c t i o n b e t w e e n t h e f u n c t i o n s c h a a n d c o s h a n d s i n h n o w r e s u l t s b yc o m b i n i n g ( 5 ) a n d ( 6 ) .

n--1 n--I n--i

E c h a ~ O ~ H - - p ~o = h # o = h H e O " H = h 1-I (cosh co + s inh 0~ ~H ) .a=5 a=i a=i

T h e n u m b e r z = # ~0 c o r r e s p o n d s to a r o t a t i o n t h r o u g h t h e a n g l e o n t h eu n i t h y p e r s u r f a c e . F o r n = 2 , h = 1 , t h e s e a t e th e L o r e n t z t r a n s f o r m a t i o n s . A hi n v e s t ig a t i o n o f th e s e f o r s o m e o t h e r n m i g h t b e o f s o m e i n t e r e s t t o p u r s u e .T h e G e n e r a l C a s e o f nF o r n n o t a p o w e r o f 2 , t h e K r o n e c k e r p r o d u c t t e c h n i q u e w i ll n o t w o r k f o rs e l ec t in g t h e h y p e r b o l i c b as i s e l e m e n t s f r o m { 1 , - 1 } n . T h e r e a r e w a y s h o w e v e rt o d o i r s o t h e r e s u l t s h o l d f o r a ll n . O n e w a y i s t o u s e r o w s o f t h e i o t af u n c t i o n i 91 = 1 - 2 q e x c e p t t h e y d o n ' t f o r m a n i n d e p e n d e n t s e t o f b a s ise l e m e n t s , s o t h e t o p r o w i s c h a n g e d t o 1 , w h i c h i s d e s i r e d f o r o n e o f th e b a s i se l e m e n t s a n y w a y . H e r e s u b s c r i p t s w i ll n o t b e e x p r e s s e d i n b a s e 2 , a n d w i l l b ed e s i g n a t e d b y i , j , k . T h e b a s i s e l e m e n t s w i l l b e q w h e r e " h " , w h i c h s t a n d sf or " h y p e r b o l i c " , is s y m b o l i ca l ly n o t a s s y m m e t r i c a s t h e p r e v io u s " H " . T h e ya t e d e f i n e d b y l . 5

n - - I n - - i

o h = 1 = ~ _ , ~ ~ , , h = , ~ = F _ , " ~ ~ ~ = 1 - 2 , 6 , 1 < i < ~ - 1 . ( 9 )O 0

T h u s 0 h = 1 a n d q = q f o r 1 < i < n - 1 , a n d t h e m a t r i x h i s s y m m e t r i c .F or l < i , j < n - l , a n d i # j ,( q 2 = 1 , q j h = q o h . j h = n - 2 , i h . q = n - 4 , I oh I = 1, I j h I = - 1 .


14/22

6 0 n - D i m e n s i o n a l H y p e r b o l ic C o m p l e x N u m b e r s P . F j e l st a d a n d S . G . G a lT h e b a s i s B h , . = { q < i < n - 1 } l a ck s b o t h n o r m a l i t y a n d o r t h o g o n a l i t ya n d i s n o t a g r o u p w i t h r e s p e c t to m u l t i p l ic a t i o n .L e m m a 1 . 5F o r z = E o - i z i i 6 = E o - I z q 1 6 1 z , o n e h a s ,

n - iZ0 = E x i ;

0 o = ~ [ ( 3 - ~ ) z 0 +

. - 1Z j : E X i i t j : Z O - - 2 x i ,

o. - I 1z , ] , ~ ~ : ~ ( ~ 0 - z ~ ) ,

1P r o @ S o l v i n g f o r t h e i 91 i n ( 9 ) r e s u l t s i n

n - - 10 l ( 3 - n ) n t - E i h , , , : l ( 1 - q

1w h i c h , t o g e t h e r w i t h ( 9 ), r e s u l t s i n (1 0 ) a n d ( 1 1 ) .

1 _ __ j < n - 1 , ( 1 0 )

1 < j < n - 1 . ( 1 1 )

i < i < n - 1 , ( 1 2 )

T h e i n t r o d u c t i o n o f p o l a r c o o r d i n a t e s p r o c e e d s t h e s a m e a s b e f o r e up t ot h e d e t e r m i n a t i o n o f # . T h e n , f o r 1 < i < n - 1 ,

e O " h : 6 ' ~..,j=o ' ~ J i ~ ~ - I n - 1 { e _O , i = jj = 0 j = 0 6 0 ' ' / ~

T h e n o r m i n t h i s c a s e i s l e ~ = e ( l / n ) ( n - 2 ) O ' , s o t h a t l e ( 1 / " ) (~ - ' ~ ) ~ 1 7 6 I = 1 ,1 < i < n - 1 , a n d o n e d e f i n e s # o a s f o l l o w s a n d t h e n s e t s 0 = 1 t o d e t e r m i n e~91

n - - 1 n - - 1 n - - I

i = 1 i = 1 i = 1F o r z = # ~ E 1 U ,

{ e } E ~ - ' ~ j = 02 ~ - - I 3z~ = ~ - ( ( E , o , ) . o ~ ) J ~ o

I n s i m i l a r f a s h i o n o n e t h e n c o n t i n u e s o n t o d e f i n e t h e a n g l e a n d t h e t r i g o n o -m e t r i c f u n c t i o n s c h i .


15/22

A d v a r t c e s i n A p p l i e d C l i f f o r d A l g e b r a s 8 , N o . 1 ( 1 9 9 8) 6 12 . A n a l y s i sS o m e r e s u l t s i n [ 3,4 ,5 ] f o r t h e 2 - d i m e n s i o r t a l c a s e w i l l b e e x t e n d e d h e r e t o t h en - d i m e n s i o n a l c as e . I n p a r t i c u l a r , e x t e n s i o n o f t h e C a u c h y - R i e m a n n c o n d i t i o n sw i ll be g iv e n f o r t h e t h r e e d i f f e re n t b a s i s s y s t e m s i n t r o d u c e d i n t h e p r e v i o u ss e c t io n . T h e d e f i n i t io n s o f l i m i t , c o n t i n u it y , a n d d e r i v a t i v e a r e e x t e n d e d i n t h eo b v i o u s w a y . F o r e x a m p l e , t h e d e r i v a t i v e o f f a t ~ i s

f ' ( ~ ) = l i m f ( z ) - f (s

B e c a u s e o f t h e s u b s c r i p t n o t a t i o n u s e d i n th i s p a p e r , ~ r e p l ac e s t h e c u s t o m a r yZ0.T h e o r e m 2 . 1 ( C a u c h y - R i e m a n n c o n d it io n s f o r t he b a si s B H , n )Fo r n = 2 V, f : D ---* R '~, D C R ~ , f ( x ) = ~ ~ U ~ ( z S , . . . , z , , _ l ) a H , z =}-'~~~z ~ ~ H , ~: = ~ ~ 5 :~ =H , i f t h e d e r i v a t i v e f ' ( ~ ) e x i s t s a t ~:, t h e n t h e p a r t i a ld e r i v a t i v e s o f t h e f i r s t o r d e r f o r a l l a r g u m e n t s o f t h e Ua m u s t e x i s t a t 5: a n ds a t i s f y t h e f o l l o w i n g c o n d i t io n s t h e r e ,

O U ~ _ c g U ~ e b 0 < c < n - 1 , i < b < n - 1 . (1 3 )Oz 5 Ozb 'Proo f : T o u s e t h e c l a s si c al m e t h o d o f t a k i n g t h e l i m i t a l o n g e a c h c o o r d i n a t eax i s , l e t x - ~ = (zb - ~b ) b H , x b r ~ :b . T h e n M ( z - 5 :) ~ 91 and

f ( x ) - f ( ~ ) ~ ~ [ U a ( x s , " " , x n - 1 ) - U ~ ( ~ : 5 , ' ' " , ~ : n -1 )] a Hz - ~ (zb - 2.b)

S i n c e f ~ ( & ) i s a s s u m e d t o e x i s t , a n d 1 / b H = b H , i t fo l l o ws i n t h e l i m i t a s x - --* 5:t h a t OU~ OU= OU~r

a a r

w h e r e w e s e t a G b = c , g i v i n g a = c | b ( s i n c e b @ b = 0 ) . S i n c e t h i s h o l d sf o r e a c h b , i t fo I l ow s t h a t , f o r e ~ c h c , t h e n p a r t i a l d e r i v a t i v e s O U r0 < b < n - 1 , a r e a l l e q u a l .r e m a r k . A s n o t e d i n [6] f o r n = 2 , t h e co n d i t io n s o f t h e t h e o r e m a r e n o ts u f f i c i e n t f o r t h e d e r i v a t i v e t o e x i s t a t ~ : .T h e o r e m 2 . 2 ( C a u c h y - R i e m a n n c o n d it io n s f or t h e b a si s Bh,,~)


16/22

62 n-Dimensional Hyperbolic Complex Numbers P. Fjelstad and S. G. GalF o r n E N , f : D ---+ R '~ , D C R '~ , f ( x ) = E i U i( x o , . . ' , x ,~_ l )q x = ~~~ix iq5: = )-~~~~?~~h, i f t he der iva t ive of f ' ( fc) e x i s t s a t ~ , t h e n t h e p a r t i a l d e r i v a t i v e so h e f ir s t o r d e r f o r a l l a r g u m e n t s o f t h e Ui m u s t e x i s t a t ~ a n d s a t i s f y t h ef o l lo w i n g c o n d i t i o n s t h e r e .

O ( U o + u ~ ) O ( U o + u j )- - 3Ozo Ozj

OUi OU~Oxa - O x i ' i r j '

l < _ j < _ n - 1l < _ i , j < _ n - 1 (14)

For j = O, wi th oh = 1 , t he resu l t i s

wh i l e for j r O ,OUj OUo ~ OUi ~ -l OUi ~ OUis'(~)=~~-+~ ,:o?-g~J+.Z,~;~Jh + ~ h : ,~

i r i C j i ~ jE q u a t i n g f o r e a c h c o o r d i n a t e r e s u l ts i n t h e f o l lo w i n g s y s t e m o f n (n - 1) di ffer-

OUo ~ OUif ' ( Y:) = ~ zo - t- ~xo ih ,i=1

e n t i a l e q u a t i o n s .OUo _ OUjOzo OzjOUi OUiOxj OxjOUi n-1Ozo - ~ - -t ~ - o*r

OU_ _ n-10Ui+ O z j ~ O z i ' 1


17/22

A d v a n c e s i n A p p l i e d C l i f f o r d A l g e b r a s 8 , N o . 1 ( 1 9 9 8 ) 6 3F o r a s i m p l e r e q u i v a l e n t s y s t e m , a d d i n g f i r s t a n d t h i r d l i n e s g i v e s ,

O(Uo + u~) O (U o+ us )Ozo Oxj , l < _ j < _ n - l . [ ]

W h e n t h e b a s i s v e c t o r s a r e z e ro d i v i so r s , a s i n th e d i r e c t p r o d u c t b a s i s, a n -o t h e r a p p r o a c h m u s t b e u s e d , a n d b y d e m a n d i n g s o m e a d d i t io n a l s m o o t h n e s s ,a t h e o r e m w i t h n e c e s s a r y a n d s u f c o n d i t io n s r e s u l ts .T h e o r e m 2 . 3 ( C a u c h y - R i e m a n n c o n d it io n s f or t h e b a sis B 6 , n )F o r n E N , f : D -- +- R n , D C R '* , f ( z ) = 7] q P i ( z o , . . . , z , _ l ) q z = ~ q z i i 6 ,i = ~ i i i q w i t h t h e P i hav i ng a l l pa r t i a l de r i va t i ves o f t he f i r s t o r de r c on t i n -u o u s o n D , t h e d e r i v a t i v e f ' ( i ) ex i s t s a t e a ch ~ E D ( i . e . f i s ho l o m or p h i c i nD ) i f a n d o n l y i f t h e f o l lo w i n g c o n d i t i o n s a t e s a t i s fi e d ,

OPiO z j = 0 , i r O < i , j < _ n - 1 . (15)I n th i s c a s e , f o r z E D , f i s o f t h e f o r m ,

S ( z ) = ~ s 1 6 2i

w i t h S ~ c o n t i n u o u s o n e a c h zi .(16)

P r o @ F i r s t , a s s u m e f i s h o l o m o r p h i c o n D . F o r z , ~. E D , a n d M ( z - i ) =I -L(z i - i i ) 0 , i t fo l lows th a t

f ( z ) - f ( s _ ~ [ P i ( z o , ' " , z n - 1 ) - P i ( s 1 6 1z -- s i z i - s

A ppl y i ng t he m ean va l ue t heo r em ( s ee , e . g . [ 2 , p . 70 - 71 ] ) t o each P i , one has

p , ( z o , . . . , z ~ _ ~ ) - p , ( ~ o , . . . , ~ , _ ~ ) : ~ L ~ ( ~ o ( k ) . . . ,3

w h e r e e a c h z i (k ) is b e t w e e n z i and z i . S i nce t he l i m i t ex i s t s o f( f ( z ) - f ( ~ ) ) / ( z - ~ ) a s z ~ ~ , . M ( z - ~ ) ~ O , s o d o e s t h e l i m i t


18/22

64 n-Dimensional Hyperbolic Complex Nambers P. Fjelstad and S. G. Gal

l i m P i ( z o , ' . - , z , ~ - l ) - P i ( i o , . ' . , ~ ,~ - 1 ) =

l im ( z o ( k ) , . . . , z , ~ _ l (k ) ) z iz , ~ 2 , i Z i - - Z i.7

B y c o n t i n u i t y , O P i l O z j ( z o ( k ) , - . , Z n - l ( k ) ) g o e s t o O P i l O z j ( ~ . o , . . . , z ' , ~ - a ) int h e l i m i t , a n d w h e n i t i s d i f f e r e n t f r o m 0 , t h e i n d e p e n d e n c e o f z i a n d z j , f o ri r j , m e a n s t h a t t h e l im i t o f ( z i - ~ . j ) / ( z i - i i ) a s z - - * i c a n b e a n y r e a lv a l u e o n e d e s i r e s . T h u s f o r t h e l i m i t t o b e u n i q u e , a s a s s u m e d , ( 1 5 ) m u s t h o l d .C o n v e r s e l y , s u p p o s e t h a t ( 1 5 ) h o l d s a t e a c h i E D . T h e n P i = f i ( z i ) w i t h l ic o n t i n u o u s a t e a c h z i , s o t h a t ,

l i m S ( z ) - f ( 5 ) Z l im [ f i ( z i ) - f i ( 2 i ) J i 5 Z f i= i ) i ~ . ~.~( ,-i )#o i 2,4(,- i)r i

R e m a r k s : 1 ) B e c a u s e o f t h e r e s t r i c t i o n t h a t z - ~ n o t b e a z e r o d i v i s o r i nt h e l i m i t p r o c e s s , o n e c a n , f o r a g i v e n f , a r b i t r a r i l y r e d e f i n e f o n t h e l i n e sz j = i i w i t h o u t a f f e c t in g th e e x i s t e n c e o f t h e d e r i v a t i v e o f f a t ~ . O n e c a nt h u s c o n s t r u c t " p a t h o l o g i c a l " f u n c t i o n s h a v i n g d e r i v a t i v e s a t 5 , b u t w i t h t h e( O P i / O z j ) ( s f o r a tl i, j , b e i n g a r b i t r a r y , i n c l u d i n g b e i n g n o n e x i s t e n t .2 ) R ea so ni ng e xa c t ly a s in the ca se n - 2 ( s ee [4] ) , i t fo l low s tha t i r f : D ---+ R '~i s o f t h e f o r r o ( 1 6 ), t h e n t h e n a t u r a l d o m a i n o f d e f i n it i o n o f f is t h e s m a l l e s tn - d i m e n s i o n a l b o x o f t h e f o r m { z - ~ o - 1 z i i 6 1 a i < z i < b i } i n w h i c h Dc a n b e i n s c r ib e & T h e r e f o r e t h e n a t u r a l d o m a i n s o f d e f i n i ti o n fo r h o l o m o r p h i cf u n c t i o n s a r e j u s t s u c h b o x e s w i t h z e r o d i v i s o r f a ce s .3 ) I t is w e l l k n o w n t h a t f o r n - - 2 a n d t h e h o l o m o r p h i c f u n c t i o n f ( z ) =~ : U a ( z o , X l ) a H t h ~ t 0 2 U a / O x 2 = O 2 U ~ , / O x ~ , a - 0 , 1 , w h i c h i s t h e w a v ee q u a t i o n . F o r n > 2 , i r f o l l o w s f r o m ( 1 3 ) , f o r c = 0 , a n d t h e n f o r c = a @ b ( s oa - - c @ b a n d b = c ~ a ) , t h a t

au . aub ag: OU~r OU~~~ aubO z : a m b ' O z b O z b - O z : - " O z:

T h e n , o f c o u r s e ,

s o o n e c a n w r i t e ,02 U , 02 Ub 02 U:O z O - O~,~Ozb - O z b 2 '


19/22

Advances in Appl ied Clifford Algebras 8, No. 1 (1998) 65

0 2 U o 0 2 U o O 2 U ~0 z 5 2 - O x f 2 . . . . . O z n _ 1 2 , 0 < a < n - 1 . (1 7)A h a l t e r n a t i v e m e t h o d is t o u s e t h e f o r m ( 1 6) f o r a h o l o m o r p h i c f u n c t i o n a n dt h e n , u s i ng ( 3 ) , t r a n s f o r m i t b y t h e m a t r i x H t o a f u n c t i o n~-]o U o ( z S , . . . , z ~ _ l ) o H , s o t h a t

Ozc : bH c.U o = In Z f ~ ( z ~ ) ~ H o , w h e r e z~ = E z ~ ~ H ~ ' s o O xbC a

T h e n ,O U oO Xb - - - f ~ ( z r ~ H o .n ~ ~Ho, O~b~ n

C C

Since (b Hc ) 2 = 1 , i t fo l lows th a t 0 2 U a / O x b 2 i s t h e s a m e f o r a l l b , g i v i n g ( 1 7 ) .S i n c e (2 ) a n d ( 3 ) a t e l i n e a r t r a n s f o r m a t i o n s a s a b o v e , t h e f o r r o ( 1 6 ), w h e n

c r a n s f o r m e d t o a f u n c t i o n ~ i U i ( x o , . " , z n -1 ) q w i ll g i r e t h e s a m e r e s u l t a sa b o v e f o r t h e U i, n a m e l y ,

0 2 U i 0 2 U i 02 Ui. . . . . . 0 < i < n - 1 .O z o 2 O z 12 O z , _ 1 2 'F o r t h e c a s e n = 3 , t h i s s y s t e m o f e q u a t i o n s w a s s t u d i e d i n [ 9].

I n w h a t f o ll o w s , s o m e s i m p l e a p p l i c a t i o n s o f t h e d e r i v a t i v e w i l l b e g i v e n .T h e o r e m 2 .4G i v e n n e N , a b o x B = { z = E q < z i < b i } , f : B ~ R n ,f ( z ) = ~ ' ~ ~ iP i ( z o , . . . , z ~ - l ) q w i t h t h e P i h a v i n g c o n t i n u o u s p a r t i a l d e r i v a -t i ve s o f f i rs t o r d e r a n d f b e i n g h o l o m o r p h i c o n B . I f A 4 ( f ' ( z ) ) ~ 0 f o r z E B ,t h e n f i s o n e - t o - o n e o n B ( i. e. u n i v a l e n t o n B ) .P r o @ B y T h e o r e m 2 .3 , f i s o f t h e f o r m f ( z ) = ~ - ~ ~ q z e B , w h e r et h e f [ ( z i ) a t e c o n t i n u o u s o n [a i , b i ] . A s s o c i a t e f w i t h t h e v e c t o r i a l f u n c t i o nP = ( P o , " ' , P n - 1 ) , w h e r e P i : bi] --+ 1:91P i ( z o , " ' , z , , - 1 ) = f i ( z i ) , s u c ht h a t f f ( P ) ( z ) = H i f = . A / [ ( f' (z ) ~s 0 ( h e r e J ( P ) ( z ) is t h e J a c o b i a n o f Pa t z ) . A s a c o n s e q u e n c e o f a w e l l- k n o w n t h e o r e m o f v e c t o r a n a l y s i s ( s ee , e. g .[2, p .97 ] ) i t fo l lows th a t P : x i [ a / , bi] --~ R "~ i s o n e - t o - o n e , w h i c h m e a n s t h a t fi s o n e - t o - o n e . 0

S ince z = p ( z ) I z ~ ( ~ ) w h e n . M ( z ) :/= 0 , t h e i d e a o f a c o n f o r m a l m a p p i n g c a nb e a p p l i e d . T h i s w i l l b e d o n e u s i n g b a s i s v e c t o r s q b u t w o r k s e q u a l l y w e l l


20/22

66 n-Dimen sionaI Hyperbo lic Complex Numbe rs P. Fjelstad 91 S. G. Gelusing basis vectors aH. An a rc 7 ( t ) = EiTi(t)q t 9 [a,b], is differenliablei f 7 ' (~) exis ts and is cont in uous . I t i s regular i f 3d (7 ' ( t ) ) 7s 0 , which meansa(7 ' ( t ) ) i s de f ined . A regu la r d i f fe ren t i ab le a rc i s smoo~h and has a t angen twhos e direct ion is det erm ine d by the angle o~(7 '( t ) ) . A func t ion f : D --*/~n,D C R '~, i 9 D, is hyperbolic conformal a t ~ i f fo r each pa i r o f smo oth a rcs7, r [a, b] --~ D, w ith 7 ( t ) = t i ( i ) = 5 , the fol lowing holds

~ ( 7 ' ( i ) ) - ~ ( ~ ' ( ~ ) ) = ~ ( ( y o 7 ) ' ( ~ ) ) - ~ ( ( f o ~ ) ' ( / ) ) .

T h e o r e m 2 . 5Gi ve n f : D ~ /~~, z 9 D. If f ' ( z ) exis ts a t ~? and M (f ' ( ~ )) 7s 0 , the n f isconformal a t ~ .Pro@ L et 7,fl be a pa ir o f smo oth a rcs a s in the above de f in i t ion . S ince.M (f ' (2 )) 7s O, a( f ' (~? ) ) i s de f ined . By hypo th es i s i t fo llows tha t the re ex i s t s an e i g h b o r h o o d o f l , V ( t ) , such that for a l l t 9 V ( t ) \ { t } w e h a v e M ( 7 ( t ) -7( 91 7s O, so we can wri te,

f ( 7( t ) ) - f (3 ' ( / ) ) _ f (q ' (g)) - f ( 7( ~) ) 3 '( 91 - 3 ' (~)t - i 7(t ) -- 7( i) t --

and s imi la r ly fo r t i. Pass ing to the l imi t and fo l lowing s tan dard a rg um ent s weg e t t h e t h e o r e m .I I . em ark s : 1 ) For n = 2, Th eo re m 2 .5 was s ta ted in [3].2 ) V er y s i m pl e e x a m p l e s o f h y p e r b o l i c c o n f o r m a l m a p p i n g s a r e f ( z ) = e ~ a n df ( z ) = a ,+b where M ( c z + d) 7 O, Ad (a d - bc) 7s O. Indeed , fo r z = ~i z i i6,cz+ da = ~i a i i6, etc.,

az + b ~ ai zi + bie ~ = E e ~ ' i S , a n d ~ z ~ d - ~ cizi +d-----~iqi

f r o m w h i c h o n e e a s il y d e r iv e s t h a t M ( f ' ( z ) ) r 0 .In the fol lowing a cons ider at io n of the in teg ral wi l l be given. As in [4], we

sa y th a t f : D ---* R n is pseudoholomorphic on D if f is of the f or mf ( z ) = ~ 2 S , ( z , ) , r ~ = E z ,, 9 D,

i i

where the li are con t inuous on the i r dom ains o f de f in i t ion .


21/22

Advances in Applied Clifford Algebras 8, No. 1 (1998) 67R e m a r k s : 1 ) O b v i o u s l y , i f f i s h o l o m o r p h i c t h e n f i s p s e u d o h o l o m o r p h i c .2 ) R e a s o n i n g e x a c t l y a s in [4] i t f o l l o w s t h a t t h e n a t u r a l d o m a i n s o f d e f i n i t i o nf o r p s e u d o h o l o m o r p h i c f u n c t i o n s a r e th e n - d i m e n s i o n a l b o x e s d e f i ne d e a r l i e r .

L e t f : D - -* R ~ , D C R n , a n d l e t 3' : l a , b ] - -~ D b e a n a r c i n D . C o n s i d e r i n ga d i v i s io n d : 7 ( a ) = z 0 , z l , . - . , zm = 7 ( b ) a n d t h e p o i n t s ~ ~ ' b e t w e e n ' z k a n dz ~ + l o n 7 , w e c a n d e f i n e th e i n t e g r a l i n th e s t a n d a r d w a y a s

f f ( z ) d z =n-1

l i m ~"~~f(~k)(Z~+ l - - Zk) ,v(d)---*O = 0

( w h e r e v ( d ) i s t h e n o r m o f d ) .B y s i m p l e s t a n d a r d c a l c u l a t i o n s , a s i n [4 ], i t f o l lo w s t h a t i f f i s i n t e g r a b l e o n7 , f ( z ) = ) ~4 P i ( z o , " " , zn - 1 ) q z : ~ i z i i 6 , then

/ f ( z ) d z = ~q f T P i ( z o , " ' , z n - 1 ) d z i i&T h e o r e m 2 . 6I r f : D ~ R ~ is p s e u d o h o l o m o r p h i c o n D a n d 7 : l a , b ] - -* D i s c l o s e d ( i . e .7 ( a ) = 7 ( b ) ) , t h e n f 7 f ( z ) d z = O.P ro@" I f 7 ( a ) = ~- , i a i i~ , 3 ' ( 5 ) = ~ i b i i t h e n w e e a s il y g e t

y ( z ) d z = I , ( z , ) d z , , e = O,t

s i n c e a i = b i f o r 0 < i < n - 1 , w h i c h p r o v e s t h e t h e o r e m .R e m a r k . W o r k i n g i n t h e b a s i s B ~,n m a k e s i t e a s y t o e x t e n d t o t h e n -d i m e n s i o n a l c a s e m a n y o t h e r r e s u l t s i n [ 4, 6, 7] . Fo r e x a m p l e , r e a s o n i n g e x a c t l ya s in [ 7 , s e c t i o n 3 ] , w e g e t t h e f o l l o w i n g .


22/22

68 n-Dimensional Hyperbolic Comp lex Num bers P. Fjelstad and S. G. GalT h e o r e m 2 . 7L e t f : R ~ [ 0, 1 ] ~ R ~ b e p s e u d o h o l o m o r p h i c o n t h e b o x R ~ [ 0 , 1]= { z = ~ i z ii

n dimensional hyperbolic numbers

Documents