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The elementary function theory of an hypercomplex variable and the theory of conformal mapping in the… Fox, Geoffrey Eric Norman 1949

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Op-' •THE ELEMENTARY FUNCTION THEORY OF AN HYPERC OMPLEX VARIABLE AND THE THEORY OF. .CONFORMAL MAPPING IN THE HYPERBOLIC PLANE py G e o f f r e y E r i c Norman Fox A t h e s i s submitted i n p a r t i a l f u l f i l m e n t o f requirements f o r the degree o f MASTER OF ARTS In the department of MATHEMATICS .THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1949 . ABSTRACT OP THESIS The present t h e s i s i s based on a paper by Bencivenga . In t h i s paper the author develops a theory of f u n c t i o n f o r the dual and b i r e a l v a r i a b l e s • He c o n s t r u c t s the " r e t t o " and " h y p e r b o l i c " planes f o r the geometric r e p r e s e n t a t i o n of the du a l and b i r e a l v a r i a b l e s , r e s p e c t i v e l y , and e s t a b l i s h e s a type of conformal mapping ibf these planes i n t o themselves by means of d i f f e r e n t i a b l e f u n c t i o n s of the v a r i a b l e . F u r t h e r , i n each of these planes he proves the analogue f o r the Cauchy i n t e g r a l theorem of the complex plane . F i n a l l y he shows that f u n c t i o n s of the dual and b i r e a l v a r i a b l e which possess a l l d e r i v a t i v e s at a g i v e n p o i n t of the plane may be expanded i n a T a y l o r s e r i e s about that p o i n t . In the f i r s t c h a p t e r we give a summary of t h i s paper . Bencivenga 1s dual and b i r e a l number systems , and a l s o the complex number system * are two-dimensional cases of the fl - dimensional a s s o c i a t i v e , commutative l i n e a r a l g e b r a w i t h u n i t element • In chapter I I we g e n e r a l i z e Bencivenga's f u n c t i o n theory to f u n c t i o n s over the above mentioned l i n e a r . An important c l a s s of r e s u l t s from the theory "of f u n c t i o n s of a complex v a r i a b l e ape not g e n e r a l i z a b l e , s i n c e they depend on the f i e l d p r o p e r t i e s p e c u l i a r t o the complex a l g e b r a . In chapter I I I we undertake a d e t a i l e d study of the hyper-- b o l i c plane w i t h p a r t i c u l a r r e f e r e n c e t o the conformal prop-e r t i e s of d i f f e r e n t i a b l e f u n c t i o n s of the b i r e a l v a r i a b l e , as a s p e c i a l case of conformal t r a n s f o r m a t i o n of the h y p e r b o l i c plane , we study the b i l i n e a r t r a n s f o r m a t i o n • We f i n d t h a t the r e c t a g u l a r hyperbola i s the g e o m g e t r i c a l form which i s i n v a r i a n t under t h i s t r a n s f o r m a t i o n of the h y p e r b o l i c plane . S i n g u l a r i t i e s p l a y a l a r g e r r o l e i n t h i s theory than i n the case of the analgous t r a n s f o r m a t i o n theory of the complex plane . TABLE OP CONTENTS Section Topic Page Chapter I . Introduction 1.1 The dual number system 1 1.2 Representation of dual numbers in the retto plane 2 1.3 Elementary operation formulae 3 1.4 Functions of the dual variable 4 1.5 Conformal representation 7 1.6 The bireal number system 7 1.7 Representation of b i rea l numbers in the hyperbolic plane 8 1.8 Elementary operations on the b i rea l numbers 10 1.9 Functions of the b i rea l variable 12 Chapter II . Function Theory of a Hypercomplex Variable 2.1 Class i f ica t ion of l inear algebras over the real numbers 15 2.2 Hypercomplex algebra 19 2.3 Matrix representation of a hypercomplex number 21 2.4 Functions , continuity , d i f fe ren t i ab i l i ty and convergence 22 2.5 Generalized Cauchy-Riemann equations 24 2.6 Analytic functions 28 2.7 On the relat ion of differentiable to analytic functions 31 2.8 T a y l o r S e r i e s 2.9 Line - i n t e g r a l s 2.10 Conformal mapping Chapter III. Conformal R e p r e s e n t a t i o n i n the H y p e r b o l i c Plane. 3.1Geometry of the h y p e r b o l i c plane 46 3.2 Length of an h y p e r b o l i c l i n e - segment 47 3.3 Rectangular hyperbola 51 3.4 H y p e r b o l i c o r t h o g o n a l i t y 55 3.5 A n a l y t i c r e l a t i o n s of b i r e a l numbers 62 3.6 B i l i n e a r t r a n s f o r m a t i o n 72 3.7 B i l i n e a r t r a n s f o r m a t i o n ibf the r e c t a n g u l a r h y p e r b o l a 77 3.8 B i l i n e a r e quivalence 82 3.9 I n t e r l o c k e d systems 85 3.10 Inverse p o i n t s 88 3.11 Example of an i n t e r l o c k e d p e n c i l 90 CHAPTER I I n t r o d u c t i o n In a paper e n t i t l e d " S u l l a Rappresentazione Geometrica D e l l e A l g e b r e Doppie Dotate D i Modulo" * , U. Bencivenga has g i v e n a geometric r e p r e s e n t a t i o n and f u n c t i o n theory f o r d u a l and b i r e a l numbers . I t i s the purpose of the present t h e s i s t o i n v e s t i g a t e the f u n c t i o n theory f o r a more g e n e r a l form of hypercomplex v a r i a b l e , and t o develop the theory of conformal mapping i n the plane of the b i r e a l v a r i a b l e . We w i l l give f i r s t a summary of the r e s u l t s of Bencivenga's work which w i l l be r e q u i r e d i n l a t e r chapters • I.I The Dual Number System . The d u a l numbers are d e f i n e d by 2 = x + ^ W , where and ^- are r e a l and I , W are b a s i s elements which have the m u l t i p l i c a t i o n t a b l e : 1 w I 1 w w w 0 * A t t i . Accad. S c i . N a p o l i Ser(3) V.2, No.7 (1946) RETTO PLANE F i g u r e I Chapter I These numbers form a commutative l i n e a r a l g e b r a over the r e a l numbers • Moreover , the a l g e b r a of d u a l numbers i s isomorphic w i t h that of the 3 L K Hi m a t r i c e s : Z -= 7C t The modulus of i s d e f i n e d by - . 1 * 1 . * 3 O 7C 1.2 R e p r e s e n t a t i o n of Dual Numbers i n the "Retto" Plane The r e t to plane c o n s i s t s of a l l p o i n t s of the c a r t e s i a n plane w i t h the d i s t a n c e PC^,/? ) between any two p o i n t s IK*',?') and / ?<X, t fO d e f i n e d by ) « /*<-**/ Thi s metric i s symmetric and s a t i s f i e s the t r i a n g l e i n e q u a l i t y . However , C (^ } = ° d o e s n o t i m P l y t n a t • In f i g u r e I , the v e c t o r O P d e f i n e s a r e t t i l i n e a r angle whose magnitude i s g i v e n by twice the a r e a of t r i a n g l e LoA , whose a l g e b r a i c s i g n i s p o s i t i v e , and which bears the sub-- s c r i p t 2 . Magnitudes of angles i n o t h e r quadrants are d e t e r --mined i n the same manner , w i t h a l g e b r a i c s i g n s and s u b s c r i p t s a c c o r d i n g to f i g u r e I . A d d i t i o n of r e t t i l i n e a r angles i s de-- f i n e d by (<t> + V ) . where s u b s c r i p t s are taken modulo 2 . R e t t i l i n e a r s i n e and c o s i n e are d e f i n e d by x'ffiA. <ps -= (-<) ) 3 and s a t i s f y the a d d i t i o n formulae : C^ + ^a) - 4 C<raA ^  • The dua l number 2T — TC -h^ W i s rep r e s e n t e d i n the r e t t o plane by the p o i n t , where X and ^ are sign e d E u c l i d e a n d i s t a n c e s from the ^ - and X - a x e s , r e s p e c t i v e l y . I f |z.J = a » a*n(z.) = cj>s ; then X = /I , ^ =• ; and 1.3 Elementary Operation Formulae Bencivenga e s t a b l i s h e s the f o l l o w i n g set of m u l t i p l i c a t i o n , d i v i s i o n , and power formulae 2. F o r an i n t e g e r ffl > o > 3. F o r r a t i o n a l ^ ° : where f x - J^*" ^ m 0 d 2 • From t h i s e q u a t i o n f o r X , i t f o l l o w s that f o r numbers 4 l y i n g i n quadrants designated by the s u b s c r i p t I , even roots do not e x i s t w i t h i n the d u a l system . 4. F o r i r r a t i o n a l ? 0 : has a merely formal s i g n i f i c a n c e 5. F o r (X ir O •— - — (/teA.- \A/OWI$S) 6. Formulae 2 , 3 , 4 , 5 may be i n c o r p o r a t e d i n the general formula : v a l i d f o r a l l r e a l ~)C . 1.4 Functions of the Dual V a r i a b l e . Fun c t i o n s over the du a l numbers are d e f i n e d i n the u s u a l manner : a f u n c t i o n F (?0 i s d e f i n e d over a set of du a l numbers when a method i s g i v e n f o r u n i q u e l y determining a second d u a l number to correspond , as linage^ to any g i v e n member of the set . The mapping i s e x p r e s s i b l e : FtfO = + F2 (*,{,). w , where ^ , £ are r e a l f u n c t i o n s of i< and ^ . By d e f i n i t i o n , F(z) i s d i f f e r e n t i a b l e at Z i f there e x i s t a 4(z) * <*0 )^ + \C such that 5 assuming the e x i s t e n c e and c o n t i n u i t y of the f i r s t p a r t i a l s of ff" and £ w i t h r e s p e c t t o X and ^ . From t h i s d e f i n i -t i o n Bencivenga d e r i v e s the "Cauchy - Riemann" equations : 9 f _ ( ) which , w i t h the e x i s t e n c e and c o n t i n u i t y of the p a r t i a l s , are necessary and s u f f i c i e n t c o n d i t i o n s f o r the e x i s t e n c e of the d e r i v a t i v e F(z) of Ffr) at a p o i n t Z . The r e a l components of a d i f f e r e n t i a b l e f u n c t i o n F&) are of the form and the d e r i v a t i v e i s g i v e n : A power s e r i e s ^ w i l l d e f i n e a f u n c t i o n of z i f the corresponding r e a l ( component ) s e r i e s both converge . W r i t i n g / L - CL' + JQ W ~ > w e r e q u i r e the convergence That i s , '* converges w i t h i n some open r e g i o n of the 6 r e t t o plane : < Ci , which i s hounded by the "modulus curves" X-(I , X - - f\ . Bencivenga e s t a b l i s h e s the T a y l o r expansion about a p o i n t 2 = Z„ ef*l> ( 1-43 ) f o r f u n c t i o n s p o s s e s s i n g a l l d e r i v a t i v e s at Z # . The expans-i o n i s v a l i d f o r a l l p o i n t s z w i t h i n the r e g i o n of converg--ence of the r i g h t member of ( 1.45 ) . The l i n e - i n t e g r a l F(z)o1z over a p a t h C i n the r e t t o plane i s d e f i n e d i n the u s u a l manner : Take a decompo--s i t i o n of the curve segment : 2. ~z, - z. -Z„z, and a point §*. on each segment Zt_, } z . 44 Form the sum cr Refine the decomposition CT by i n c r e a s i n g /vi and a l l o w i n g Z^ . t o approach $ i n the E u c l i d e a n sense • Denoting t i e refinement : J c r j —> O , the i n t e g r a l i s d e f i n e d : Bencivenga shows that i f F(z) i s continuous on C the i n t e g r a l e x i s t s , and i s g i v e n by the formula : , 2 2 Z , c 2 C 2 7 Bencivenga next proves the Cauchy i n t e g r a l theorem f o r t h i s case : I f F<£) , i s d i f f e r e n t i a b l e on the c l o s e d contour and at a l l p o i n t s of the i n t e r i o r bounded by C > then •z Consequently , w i t h i n any such r e g i o n [ ffe} Jz d e f i n e s J V a d i f f e r e n t i a b l e f u n c t i o n ICZ) which i s independent of the path V , and has the d e r i v a t i v e : - F(z), 1.5 Conformal R e p r e s e n t a t i o n The author proves that a d i f f e r e n t i a b l e f u n c t i o n of a dual v a r i a b l e maps the r e t t o plane i n t o i t s e l f so that r e t t i l i n e a r angles are preserved . The mapping w i l l , i n g e n e r a l , f a i l t o be conformal at those p o i n t s f o r which the d i f f e r e n t i a b i l i t y of t h e f u n c t i o n f a i l s 1.6 The B i r e a l Number System . The b i r e a l numbers are d e f i n e d 2 - 5 f + % u , where •)( , tj, are r e a l and b a s i s elements I , U have m u l t i p l i c a t i o n t a b l e 1 U I 1 u u u I 8 The b i r e a l system i s a commutative l i n e a r a l g e b r a over the r e a l s , isomorphic w i t h the m a t r i x a l g e b r a : 2 - X t J M < ^ The modulus of 2 i s d e f i n e d by 1 1*1 = 1 1 V •a * 1 x - 3 The author c o n s t r u c t s a theory of the b i r e a l v a r i a b l e and i t s f u n c t i o n s s i m i l a r to that f o r the d u a l v a r i a b l e ; the d i f f e r -e n c e s i n the p a r a l l e l t h e o r i e s are c o n f i n e d t o d e t a i l s of proof and f o r m u l a t i o n • 1.7 R e p r e s e n t a t i o n of B i r e a l Numbers i n the H y p e r b o l i c Plane . The h y p e r b o l i c plane c o n s i s t s of a l l p o i n t s of the c a r t e s i a n plane w i t h the d i s t a n c e P C f t , ^ ) between any two p o i n t s ft ( * • J]> ) a n d E ( *> * 9«) d e f i n e d by F o r t h i s m etric (° (t*, fj ) =0 d o e s n o t imply P, ~ fl > and , furthermore , the t r i a n g l e i n e q u a l i t y f a i l s . A l l p o i n t s on the r e c t a n g u l a r hyperbolas ~)C - ^ = I / are u n i t d i s t a n c e from the o r i g i n • H y p e r b o l i c angles are d e f i n e d i n terms of the u n i t modulus curve X - £j « + | . I n o r d e r that an h y p e r b o l i c angle s h a l l d e f i n e a unique v e c t o r o r i g i n a t i n g at the o r i g i n , i t i s HYPERBOLIC PLANE F i g u r e 2 Chapter I 9 necessary to specify angles by magnitude , algebraic sign ,and subscript : Quadrants of the hyperbolic plane ( figure 2 ) are signed according to the same scheme as for the retto plane. Quadrants ^ 0 K } K ° ^ ' , l ^ ' o t f ' and ^ ' ^ ^ are distinguished by subscripts 1 , 2 , 3 , 4 respectively . In figure 2 , O P defines hyperbolic angle <i> , where (p is positive and equal in magnitude to twice area bounded by X-axis , O P , and X*- = / ; OQ defines vf^ where - vf is equal to twice area bounded by ^ - axis , O 0. , and ")< - ^  r - | • Similarly every other vector origin--ating at the origin defines a unique hyperbolic angle • Addition of hyperbolic angles Is defined by the equation s 3 a r I j_ 3 *f| From ^ , / l determine f as follows : Find S in column I , and ft in row 4 ; find £" at the inter--section of the row and column so determined • For example , if S-Z ,0.-3 then t -The sine and cosine functions for this plane are defined as follows : &nA $t - <Vrx4 cf> , /<Ln*il 4*, <<l*W? <p > where At&4 , $ are the ordinary hyperbolic sine and cosine . Functions of angles in other quadrants are defined according to scheme : 10 i ~ 'i - /uA<P, - A * H ( 1.72 ) With d e f i n i t i o n s ( I.71 ) the a d d i t i o n formulae : are s a t i s f i e d by every p a i r ^ » ^ • Furthermore , i f \Z\ - CI > c*r*(z) =• (p^ » then ( 1.73 ) •z •= 3 " » /i ( ^ % + U A*^- i ) > where ^ = X-r- i s represented i n the h y p e r b o l i c plane by p o i n t (yc, , >t and *j. being the signed E u c l i d e a n d i s t a n c e s of the p o i n t from the ^- and X - axes , r e s p e c t i v e l y . 1.8 Elementary Operations on the B i r e a l Numbers . The formulae of s e c t i o n ( 1.3 ) have exact analogues i n the b i r e a l system : 1. a, (^4S + " . a* uaJL^) 2. F o r an i n t e g e r fft- "Z O : II 3&- For* r a t i o n a l y > O where • ^ • ( ' ^ ) K , <r ( f,x)=-^ f f a ) = S'** + + s to f terms , the add-- i t i o n b e i n g c a r r i e d out a c c o r d i n g t o the above m a t r i x r u l e for s u b s c r i p t s 4. F o r i r r a t i o n a l /* > 0 : [a(<c*Jk$ + u -= [**A(ft) + has a merely formal s i g n i f i c a n c e 5. F o r (I O a 6. Formulae 2 , 3 , 4 , 5 may be i n c o r p o r a t e d i n the g e n e r a l formula : f o r a l l r e a l X . F o r some purposes i t i s convenient t o r e p l a c e the b a s i s j , (A by the e q u i v a l e n t b a s i s 1^ V \ y * I - K + K The , ^ m u l t i p l i c a t i o n t a b l e i s Figure 3 Chapter I 12 0 < 0 i s expressed i n the /, , ** system : ( 1.75 ) 1.9 Functions of the B i r e a l V a r i a b l e . A power s e r i e s i n the b i r e a l v a r i a b l e 0 0 ~ = Z te< +4r< X*« •••'<) 2 d e f i n e s a f u n c t i o n o f Z i f the r e a l s e r i e s UP mfta Z both converge . I f the r a d i i of convergence of ** /) f , J 2*t are /f , r e s p e c t i v e l y , ** then the r e g i o n of convergence f o r the b i r e a l s e r i e s n^Z i s the i n t e r i o r of a r e c t a n g l e i n the h y p e r b o l i c p l a n e ( f i g u r e 3 i . F u n c t i o n s of a b i r e a l v a r i a b l e and d i f f e r e n t i a b i l i t y of such f u n c t i o n s are d e f i n e d i n e x a c t l y the same manner as f o r the dual v a r i a b l e ( s e c t i o n 1.4 ) . The necessary and s u f f i c i e n t c o n d i t i o n s f o r d i f f e r e n t i a b i l i t y at a p o i n t i n t h i s case are : (I) the e x i s t e n c e and c o n t i n u i t y of the f i r s t p a r t i a l d e r i v -- a t i v e s w i t h res p e c t to X and ^ of the f u n c t i o n at the 13 p o i n t i n q u e s t i o n . (2) that the Cauchy - Riemann equations he s a t i s f i e d at the p o i n t , by the f u n c t i o n The d e r i v a t i v e , i f i t e x i s t s , i s g i v e n by the formula = 9T( Q^' ( 1.92) I f we t r a n s f o r m to the K » K. a l g e b r a , a d i f f e r e n t i a b l e f u n c t i o n takes the form ( 1.93 ) •fa) = {co^ + t « ) •< where Z • + t Vx , The author e s t a b l i s h e s t h e - T a y l o r expansion : K - o •* ! v a l i d f o r some r e c t a n g u l a r r e g i o n about a p o i n t at which a l l d e r i v a t i v e s of F(£) e x i s t The l i n e - i n t e g r a l hOO<*z of a f u n c t i o n of a b i r e a l v a r i a b l e i s d e f i n e d i n the same manner as f o r the d u a l v a r i a b l e ( S e c t i o n I . 4 ) - . The author shows that i f i s continuous on C > the i n t e g r a l may be decomposed : 14 ( 1.95 ) The Cauchy i n t e g r a l theorem i f FdO i s d i f f e r e n t i a b l e on C and at every p o i n t w i t h i n r e g i o n bounded by C • n i s proved by the author f o r b i r e a l f u n c t i o n s F i n a l l y , Bencivenga proves the conformal p r o p e r t y of d i f f -e r e n t i a t e b i r e a l f u n c t i o n s : " A d i f f e r e n t i a b l e f u n c t i o n of the b i r e a l v a r i a b l e maps the h y p e r b o l i c plane i n t o i t s e l f w i t h the p r e s e r v a t i o n of h y p e r b o l i c angles The mapping w i l l , i n g e n e r a l , f a i l to be conformal at those p o i n t s at which the d i f f e r e n t i a b i l i t y of the f u n c t i o n f a i l s • 15 CHAPTER I I F u n c t i o n Theory of a Hypercomplex V a r i a b l e In t h i s c h a p t e r we w i l l develop the f u n c t i o n theory , analogous to that of Bencivenga , f o r any l i n e a r a l g e b r a over the r e a l numbers which i s a s s o c i a t i v e , commutative , and poss-e s s e s a u n i t element . We a h a l l see that the g e n e r a l i z a t i o n s of d i f f e r e n t i a b i l i t y , T a y l o r development of f u n c t i o n s , the Cauchy i n t e g r a l theorem , and conformal r e p r e s e n t a t i o n are , consequences of the f a c t that the a l g e b r a forms a commutative r i n g w i t h u n i t element*on the other hand , we s h a l l f i n d that tthere i s another c l a s s of r e s u l t s In the theory of f u n c t i o n s of a complex v a r i a b l e which cannot be g e n e r a l i z e d , These are consequences of the f i e l d p r o p e r t i e s of the. a l g e b r a of complex numbers , and t h e r e f o r e p e r t a i n only t o the theory of f u n c t i n i B of a complex v a r i a b l e 2.1 C l a s s i f i c a t i o n of L i n e a r Algebras over the Real Numbers which are A s s o c i a t i v e , Commutative and Possess a Uni t Element . Theorem 2.II The only independent b i n a r y a s s o c i a t i v e comm-u t a t i v e l i n e a r algebras w i t h u n i t element over the r e a l numbers are the complex , d u a l and b i r e a l number; systems, any o t h e r b i n a r y form i s e x p r e s s i b l e i n terms of one of these independent forms . Proof : The g e n e r a l b i n a r y form i s g i v e n by ^ = ,f« + * 16 where "><> , ><2 are r e a l , and the has i s /, £ has the m u l t i p l i c a t i o n t a b l e - / c / i t I o< , ^ being r e a l numbers . From the m u l t i p l i c a t i o n t a b l e we see that 2 i s a root o f a q u a d r a t i c e q u a t i o n w i t h r e a l coe-- f f i c i e n t s , namely which may be w r i t t e n - * = O , ( 2 .II ) /3 +• *f < say Let (ft , £ ^ » denote the a l g e b r a w i t h b a s i s * ?Z > > C > where ( 1 ) I f Y > o , l e t U -The n . 0 i s e q u i v a l e n t to ( l , U ) , which i s the b i r e a l a l g e b r a s i n c e U sr / . ( i i ) I f Y - O , l e t M/ = £ - X ' Since M/l = 0 * Q- > *) i a e q u i v a l e n t t o the dua l 17 a l g e b r a (l , 14/) . f -( 111 ) I f T< C , l e t C =• — Then * * = - / , and t h e r e f o r e (? > i s e q u i v a l e n t to the complex a l g e b r a , t J . Theorem 2.12 Of the three b i n a r y algebras , the complex a l g e b r a alone forms a f i e l d . The ot h e r two form merely r i n g s w i t h u n i t element • Proof : The b i n a r y a l g e b r a (it?*) forms a f i e l d i f and only i f t* O l s i r r e d u c i b l e i n the t e a l f i e l d ^ . This c o n d i t i o n i s s a t i s f i e d i f and only i f p + *f < < C , i n which case the a l g e b r a (j , f ) i s e q u i v a l e n t t o the com--plex a l g e b r a C ) Then*. & ( * y =• * ( 0 -In the cases ( i ) and ( i i ) of the theorem 2.II , where , £ ^  i s e q u i v a l e n t t o £\ ,t$fand 0 * ^) r e s P e c t i v e l y » f - f i ? - * - 0 i a r e d u c i b l e . Hence , and t h e r e f o r e the e q u i v a l e n t - A (fO and «4(W) are not f i e l d s . By ^ (?< O we willmean the a l g e b r a obtained by ad-- j o i n i n g the elements f, , . . . . ,£to the r e a l f i e l d . Thus , f o r example , A ( O w i l l denote the complex number f i e l d s i n c e i t i s obtained by a d j o i n i n g the element C t o the r e a l number f i e l d . 18 In f a c t , / V * * * i s a d i v i s o r o f zero i n the h i r e a l a l g e --bra , and W i s a d i v i s o r o f zero i n the d u a l a l g e b r a j s i n c e (j+U ) ( | - U ) r r O , , 7_ and Theorem 2,13 The complex a l g e b r a i s the only a s s o c i a t i v e , commutative l i n e a r a l g e b r a w i t h u n i t element over the r e a l numbers , which forms a f i e l d «. Proof : Suppose the a l g e b r a , Su, ... , £ ) , where f, - I , over the r e a l f i e l d A , forms the a d j u n c t -- i o n f i e l d 4 (?, , €L , .... , ^ ) - A ( £ , f 3 , , Any polynomial i n 2Z w i t h r e a l c o e f f i c i e n t s i n A i s f a c t o r a b l e i n the complex f i e l d AO- ) . Therefore A ( ft ) C A ^ ) • Any polynomial f w i t h c o e f f i c i e n t s i n A Cc ) i s f a c t o r -s-able i n . Therefore A (<f, ) C 4 6 3 > So that f , ) C ^ f c ' ) -C o n t i n u i n g the argument , we have f i n a l l y A (** , f , , , <: Hence i f ( f, > fA ^ . . . . . , f „ ) forms a f i e l d , i t i s a sub-- f i e l d of the complex f i e l d o r the complex f i e l d i t s e l f , But i f Y , S are r e a l and not zero then (jj c ) and{V, are e q u i v a l e n t bases over the r e a l numbers . Hence A ( t ) C 4 ( , f, , 1 9 So that A ( £ a ?t } ^ ) _ ^(c ) , P r o v i n g that any a l g e b r a (?i ?z } > ?~ ") which forms a f i e l d over the r e a l numbers must form the com-p l e x f i e l d . 2.2 Hypercomplex A l g e b r a Let ^ *"2 , w i t h ?, — I , be the b a s i s of an a s s o c i a t i v e , commutatitve l i n e a r a l g e b r a over the r e a l s . Then an element of the a l g e b r a may be w r i t t e n i n the form X = 21 C > where the are r e a l numbers . The hypercomplex a l g e b r a so d e f i n e d possesses the u n i t element - I . Moreover , l e t the m u l t i p l i c a t i o n t a b l e f o r the b a s i s elements be g i v e n by f f - r r - c l r- • c •» I T h i s t a b l e i s t h e r e f o r e d e f i n e d by the ('H + O r e a l constants Since m u l t i p l i c a t i o n i s a s s o c i a t i v e , we have O^ p = ^ t ^ and so C-.l c ~l — — t 1 •=— t o 20 The necessary and s u f f i c i e n t c o n d i t i o n s f o r a s s o c i a t i v e m u l t i p l i c a t i o n are t h e r e f o r e 3 ' c ~( ( 2.21 ) c -I The equations ( 2.21 ) impose c o n d i t i o n s on the constants . As examples of a l g e b r a s of more than two dimensions s a t i s f y i n g these c o n d i t i o n s , we note the f o l l o w i n g : ( i ) The a l g e b r a r a r 3 ( i i ) The c l a s s of l i n e a r a l g e b r a s f o r which the b a s i s f/ £^ . j Z» forms an a b e l i a n group ( i i i ) Take an i r r e d u c i b l e p o l ynomial /°(?c} of degree /n. , w i t h c o e f f i c i e n t s i n the r e a l f i e l d /± . B7 r i n g adjunc-t i o n , a d j o i n the /Vt r o o t s 6, 0* , Q to form the l i n e a r a l g e b r a A &i }ex. , } w i t h b a s i s 21 2.3 M a t r i x R e p r e s e n t a t i o n of a Hyperciomplex Number .. The hypercomplex number _ ^ v T ? C " ' " m i l l d e f i n e a unique x matrix 00 whose ft pow v e c t o r i s formed from the r e a l c o e f f i c i e n t s o f Thus /V1<^) jf^Z C*„ *,j • ( 2.31 ) I t i s w e l l known that there e x i s t s an isomorphism under a d d i -t i o n and m u l t i p l i c a t i o n d e f i n e d by x < > M <50 • The modulus f u n c t i o n of >C i s the determinant | M 00 I O F /H (*) ' t h e a c t u a l modulus being g i v e n by M = / I I wool | T h i s r e p r e s e n t a t i o n w i l l enable us t o study the hypercomplex v a r i a b l e through the p r o p e r t i e s of the corres p o n d i n g system of m a t r i c e s a f a c t which w i l l be e x p l o i t e d i n the theory of conformal mapping 22 2,4 F u n c t i o n s , C o n t i n u i t y , D i f f e r e n t i a b i l i t y and Convergence. In the f o l l o w i n g development o f the theory of a f u n c t i o n of a hypercomplex v a r i a b l e , two p r o p e r t i e s of the hypercomplex numbers are of fundamental importance : ( i ) The hypercomplex numbers form a commutative r i n g w i t h u n i t element . ( i i ) The base f i e l d of the hypercomplex system i s the f i e l d o f r e a l numbers . A f u n c t i o n - f ^ } of the hypercomplex v a r i a b l e ">( = ^ X„ fu i s d e f i n e d t o be a s i n g l e v alued mapping of the space (~>c, , , < ) i n t o i t s e l f . I t can be expressed i n the form f<* ) = Z i( *>>••> *-) I > where the •£ ( *, , , ) are r e a l f u n c t i o n s of the v v a r i a b l e s TC*J i s s a i d t o be continuous at ->C = "7 'X f, i f each of the r e a l f u n c t i o n s continuous i n the r e a l v a r i a b l e s j - - - j rt" * • I f each component -f* of 4 V * " ) possesses a l l f i r s t p a r -- t i a l . d e r i v a t i v e s at ~)( (* then 6*-) possesses a d i f f e r e n t i a l at x and we can w r i t e 2 3 J. = Z. •?< f o r * = I f -P(?0 possesses-a d i f f e r e n t i a l at tl''* , and there e x i s t s a d i f f e r e n t i a l c o e f f i c i e n t 4 60 such that ol -pC'O < 2 » 4 1 ) a t }C •» x''5 » then -^C*) i s s a i d t o he d i f f e r e n t i a b l e C.) at X A power s e r i e s Z = 2 ( Z<\....,/» - ^ PC d e f i n e s a hypercomplex number (?0 at X i f each of the component r e a l s e r i e s converges at ?C . In t h i s case we say that nK X converges at X . ^ * /)K X converges over a r e g i o n i f i t converges at K-o every p o i n t of the r e g i o n j and converges u n i f o r m l y over the r e g i o n i f each component r e a l s e r i e s converges u n i f i r m l y over the same r e g i o n . W i t h i n i t s r e g i o n of u n i f o r m convergence, the s e r i e s A X* £ ~ * d e f i n e s a continuous f u n c t i o n o f X 24 I f l i m i t -5>o and /. - 2L ^ then we s h a l l say that 2.5 G e n e r a l i z e d Cauchy - Rieraann Equations We s h a l l now g e n e r a l i z e the Cauchy-Riemann equations of complex v a r i a b l e f u n c t i o n theory . We prove the f o l l o w i n g theorem : Theorem 2»5I The necessary and s u f f i c i e n t c o n d i t i o n s that -f(K) be d i f f e r e n t i a b l e at X « "x' are : ( i ) that the f i r s t p a r t i a l d e r i v a t i v e s of the § w i t h r e s --pect t o the v a r i a b l e s ?(, -• • , e x i s t and be con--tinuous at x = >t y ( i i ) that the "Cauchy-Riemann" equations — •= 7 C 7T- > < ' 2 - 5 1 > be s a t i s f i e d at X s X . Proof : i s d i f f e r e n t i a b l e at X ^ X * i f there e x i s t s a 25 f u n c t i o n such that W r i t i n g t h i s e q u a t i o n i n expanded form : E q u a t i n g the >f components E q u a t i n g c o e f f i c i e n t s of the independent 0 / * ^ = J ^ . ( 2.52 ) Since f; = / } €L - C, ?u -So that C — ~ a the Kronecker d e l t a S e t t i n g L - I i n ( 2.52 ) and a p p l y i n g t h i s r e s u l t : Combining ( 2.52 ) and ( 2.53 ) we have the Cauchy-Riemann 26 equations ( 2.51 ) C o r o l l a r y ( i ) I f -ffe) i s d i f f e r e n t i a b l e at ?C , then ( by e q u a t i o n (2.55) ) the d e r i v a t i v e of -ftf) i s g i v e n by the formula : C o r o l l a r y (11) I f ffr) i s d i f f e r e n t i a b l e at >C ,and a l l second p a r t i a l s of the w i t h r e s p e c t t o x, f - • • , e x i s t at , then the second d e r i v a t i v e of -£(x) w i t h res p e c t t o % e x i s t s at It , and i s g i v e n by the formula : To prove t h i s i t i s only necessary t o show that , under the hypothesis , f ^ -P 6 0 - Z or" ^  s a t i s f i e s the Cauchy^Riemanni equations (2.51) , and then t o apply e q u a t i o n ( 2.54 ) to £ & ) . We must show that which i s e q u i v a l e n t t o ^ 27 which i s merely the r e s u l t o f p a r t i a l d i f f e r e n t i a t i o n w i t h respect to X, of the Cauchy-Riemann equations f o r "T^) . T h i s r e s u l t i s immediately g e n e r a l i z a b l e t o C o r o l l a r y ( H i ) I f •fC'O i s d i f f e r e n t i a b l e at X * and a l l (» ft p a r t i a l s of the + up t o the /M o r d e r e x i s t at X , ft (*s ^ then the /Wl d e r i v a t i v e of T&<) w i t h r e s p e c t t o >C e x i s t s at X and i s given : + <*) = L *' ' < 2^5) C o r o l l a r y ( l v ) Assuming the e x i s t e n c e of the h i g h e r p a r t i a l d e r i v a t i v e s i n v o l v e d , a l l h i g h e r d e r i v a t i v e s of the w i t h r e s p e c t to X, > , of a d i f f e r e n t i a b l e f u n c t i o n •PtfO * a r e e x p r e s s i b l e i n terms of p a r t i a l d e r i v a t i v e s of the same or d e r w i t h respect t o X, F o r , d i f f e r e n t i a t i n g 20? f / H w i t h respect t o *>Cj, , we o b t a i n 28 D i f f e r e n t i a t i n g t h i s w i t h res p e c t to ?C : z * ' * and c o n t i n u i n g i n t h i s way , any /ni o r d e r p a r t i a l d e r i v a -- t i v e of + K i s e x p r e s s i b l e i n terms of the /YH order d e r i v a t i v e s of the w i t h respect t o X, . 2.6 A n a l y t i c F u n c t i o n s We w i l l say that +(*0 i s a n a l y t i c at X - i f / „ / / / a l l d e r i v a t i v e s : e x i s t at ">< = X Theorem 2.61 The necessary and s u f f i c i e n t c o n d i t i o n s f o r (fO t o be a n a l y t i c at ~>C are : ( i ) that -ft? c) be d i f f e r e n t i a b l e at X } ( i i ) that each component $ , possesses a l l p a r t i a l d e r i v a t i v e s w i t h res p e c t t o ~>t x at ?C Proof : By theorem 2.51 , c o r o l l a r y ( i i i ) the c o n d i t i o n s ik of the theorem guarantee the e x i s t e n c e of a l l /W o r d e r d e r i v a t i v e s w i t h respect t o ?C : 29 V = / ' The c o n d i t i o n s are t h e r e f o r e s u f f i c i e n t . JAM") I f -j- e x i s t s , then , by the l a s t mentioned equation, each e x i s t s f o r ^ » Then hy theorem 2.51 , c o r o l l a r y ( i v ) a l l /m order p a r t i a l s w i t h r e s p e c t t o the v a r i a b l e s X } must e x i s t . Hence the c o n d i t i o n s of the theorem are necessary • Theorem 2.62 I f -f$<) i s d i f f e r e n t i a b l e at X , then so i s >C {(>c) . P r o o f : Let F60 = >c f(ie) • Then A F • rr (*-f A x ) -f C** . where — O Since , by hypothesis , fifc) i s d i f f e r e n t i a b l e at X 30 where ' Therefore F"'(><) - • C o r o l l a r y ( 1 ) Every p o l y n o m i a l i n tC i s a n a l y t i c . z F o r , ">C i t s e l f i s a n a l y t i c , so by the theorem ?c , 3 K X } x > are a n a l y t i c • Any hyper-complex constant Ct i s a n a l y t i c , so by the theorem d n i s a n a l y t i c . . S ince a f i n i t e sum s a t i s f i e s the Cauchy-Riemann equations i f each component f u n c t i o n does , any p o l y n o m i a l i n X Is d i f f e r e n t i a b l e . Since the d e r i v a t i v e of a polynom-i a l i s a g a i n a polynomial , a l l d e r i v a t i v e s of a p o l y n o m i a l e x i s t , so that every p o l y n o m i a l i s a n a l y t i c . C o r o l l a r y ( i i ) W i t h i n i t s r e g i o n of u n i f o r m convergence, the s e r i e s ^ i s an a n a l y t i c f u n c t i o n • F o r , each term of the s e r i e s s a t i s f i e s the Cauchy-Riemann equations , hence the s e r i e s i t s e l f s a t i s f i e s these equations • Since the r e a l s e r i e s converge u n i f o r m l y , a l l t h e i r p a r t i a l s w i t h respect t o } , X^ e x i s t 31 2.7 On the R e l a t i o n of D i f f e r e n t i a b l e t o A n a l y t i c F u n c t i o n s • Every d i f f e r e n t i a b l e f u n c t i o n of a complex v a r i a b l e i s ana-l y t i c at the p o i n t i n q u e s t i o n , T h i s r e s u l t of the theory of f u n c t i o n s of a complex v a r i a b l e i s a consequence of the Cauchy i n t e g r a l formula , which i n t u r n r e s t s on the f i e l d p r o p e r t i e s of complex a l g e b r a . I f the hypercomplex v a r i a b l e i s o t h e r than the complex v a r i a b l e , then a f u n c t i o n may be d i f f e r e n t i a b l e at a p o i n t and yet f a i l t o b e a n a l y t i c at the same p o i n t . We give the f o l l o w i n g example of $his s i t u a t i o n where the v a r i a b l e i s b i r e a l : Let and Now d e f i n e R?o - F (*>,**) + " s a t i s f i e s the Cauchy - Riemann equations 32 21 = 21 at every p o i n t 6f the h y p e r b o l i c plane . Since a l l the f i r s t p a r t i a l s of f and w i t h respect to and ~><z e x i s t and are continuous at every p o i n t of the plane , F f x ) i s d i f f e r e n t i a b l e at $very p o i n t of the plane , by theorem (2.51 ) The d e r i v a t i v e of F ( * ) i s by theorem ( 2.51 ) , c o r o l l a r y (I), But the second d e r i v a t i v e F (x) f a i l s t o e x i s t on the l i n e IC, + ->(x ~ / Henoe on t h i s l i n e , i s d i f f e r e n t i a b l e but not a n a l y t i c The i d e n t i t y of d i f f e r e n t i a b l e and a n a l y t i c f u n c t i o n s does not n e c e s s a r i l y h o l d f o r an a l g e b r a o t h e r than the complex a l g e b r a . In the theory of f u n c t i o n s of a complex v a r i a b l e , t h i s i d e n t i t y belongs to the c l a s s of r e s u l t s which are d e r i v -e d from the f i e l d p r o p e r t i e s of the a l g e b r a . 33 2.8 T a y l o r S e r i e s Theorem 2.81 I f •f'tft) i s a n a l y t i c at the p o i n t x « A( , then the expansion i s v a l i d f o r some r e g i o n about X To prove t h i s , w r i t e : A t . L<$0 - ^+") = Z $(*•**•> 5 ' ( 2.82 ) 4*4 S O ^ ( * ) w i l l he an a n a l y t i c f u n c t i o n over a c e r t a i n r e g i o n about X - O , w i t h i n which the s e r i e s ( 2.83 ) i s uniform-l y convergent . The proof w i l l c o n s i s t i n the i d e n t i f i c a t i o n of L60 w i t h R(x) over t h i s r e g i o n . Since n KPO i s a n a l y t i c at X a O , we'have by ( 2.83 ) : Since i s a n a l y t i c at X = O , we apply theorem(2.5I ), c o r o l l a r y ( i i i ) t o ( 2.82 ) : 34 • r / 'PA', ( 2.85 ) Since = 2 -T CX^. , , AL^c) C i s a n a l y t i c at X = O : From the equations ( 2.84 ) , ( 2 .85 ) , ( 2.86), t h e r e f o r e : L (O) ^ R (O) . < 2.87 ) D i f f e r e n t i a t i n g (2.82 ) and ( 2 . 8 3 ) by r u l e of theorem ( 2.51 ) , c o r o l l a r y ( i i i ) ,we have : 0*> f , 35 and so by ( 2.87 ) we have Tl 7 TIP - , ( 2.88 ) By theorem (2.51 ) , c o r o l l a r y ( i v ) , a l l p a r t l a l s , of a l l orders , of the X are e x p r e s s i b l e i n terms of p a r t i a l s of the 4? w i t h r e s p e c t to X, j hence by ( 2.88 ) a l l par-- t i a l d e r i v a t i v e s , of a l l orders , of the X and IP are V V equal, at X - O A l s o L(0) -= R(p^) i m p l i e s that ^ = at ?( =• O Hence by the theory of ttaal f u n c t i o n s : •••> * ~ ) * fij O w i t h i n the r e g i o n of u n i f o r m convergence of ^ . Therefore l < ? 0 = ( 2 , 8 9 } over the i n t e r s e c t i o n of a l l the regions of convergence of the ^ ( t^ ^ . That i s , ( 2.81 ) holds over some r e g i o n about X 36 2.9 Line Integrals The l ine Integral of a function -^(^) over a curve C i n space ^Kt } ic^ ) i s defined i n usual manner Let curve C 0 6 defined by the parametric equations : Let X > A be i n i t i a l and terminal points of C Make a decomposition of C by subdivisions at points X i X j X s ^ and take in ter--mediate points = £ (ir-.) j*f) <*r; ftr-* /*•) c„; such that X ~ % ~>t 0r- X t > ?/ The l ine- integral is defined : -> c« ^ ( 2-$I ) *« X-x —>^ 37 Since assuming that i s continuous on C t we may w r i t e ( 2.92 ) Theorem 2.91 G e n e r a l i z e d Gauchy I n t e g r a l Theorem : Let „ \ (>C) he a n a l y t i c w i t h i n the r e g i o n 0) Let C he a simple c l o s e d curve w i t h i n t h i s r e g i o n . Then <f A?&) dye O • ( 2.93 ) C To prove t h i s , we decompose the i n t e g r a l i n t o i t s r e a l com-ponents By ( 2.92 ) we must prove \ 2 - & }<* '>->'".( 2.93 ) 38 Since , by hypothesis , a l l p a r t i a l d e r i v a t i v e s w i t h r e s p e c t t o t ---- ; X*„, of the $ e x i s t and are continuous over the r e g i o n i n which C i s embedded , the necessary and s u f f i c i e n t c o n d i t i o n s f o r ( 2.93 ) are : A p p l y i n g the Cauchy -Riemann c o n d i t i o n s to ( 2.94 ) , we o b t a i n : s «• < _ 7 c< ^ But equations ( 2.94 ) h o l d i f • ( 2.94 ) 2 e <! =r 7 C ( 2.95 ) But equations ( 2.95 ) are merely the a s s o c i a t i v i t y c o n d i t i o n s ( 2.21 ) • T h i s proves equations ( 2.93 ) as a consequence of 39 the Cauchy - Riemann c o n d i t i o n s and the a s s o c i a t i v i t y o f the a l g e b r a , and hence e q u a t i o n ( 2.93 ) of the theorem • 2.10 Conformal Mapping In t h i s s e c t i o n we seek a g e n e r a l i z a t i o n of the n o t i o n of conformal mapping which has been e s t a b l i s h e d f o r d i f f e r e n t -- i a b l e f u n c t i o n s of cooplex , d u a l and b i r e a l v a r i a b l e s • The angle between l i n e - e l e m e n t s ci* 3 i n the complex plane i s d e f i n e d by i t s co s i n e f u n c t i o n as f o l l o w s -1\ t\ • Let ^ - % & ) 0 6 a d i f f e r e n t i a b l e f u n c t i o n of the comple v a r i a b l e X t and Then the law of conformal mapping f o r a f u n c t i o n of a comple v a r i a b l e s t a t e s t h a t at every p o i n t X 0 f o r which ^6l) l s d i f f e r e n t i a b l e and 40 F o r the b i r e a l v a r i a b l e , the h y p e r b o l i c angle between the l i n e - elements i n the h y p e r b o l i c plane i s de-- f i n e d by the h y p e r b o l i c c o s i n e f u n c t i o n : ± and the law of c o n f o r m a l i t y , proved by Bencivenga , may be expressed as f o l l o w s : At every p o i n t ">C f o r which ^ •= i s d i f f e r e n t ' - i a b l e and J j ' f c ) f ^ O , F i n a l l y , the r i g h t - c o s i n e f u n c t i o n f o r elements i n the r e t t o plane i s g i v e n </x, dx, dx, dvA o dnt and the law of conformal mapping s t a t e s that at every p o i n t 41 7C f o r which tj = ^dO i s d i f f e r e n t i a b l e and Let M denote the m a t r i x ( 2.31 ) c o r r e s p o n d i n g t o the hypercomplex number > &nd l e t J/H^o| be i t s determinant . Let be the m a t r i x obtained from by r e p l a c i n g the f i r s t row v e c t o r of by the f i r s t row v e c t o r o f • With t h i s n o t a t i o n , the angle f u n c t i o n i n the above three cases i s e x p r e s s i b l e by the s i n g l e formula : and the law of conformal mapping reads : At every p o i n t ">C f o r which *j =• ^ 0 i s d i f f e r e n t -- i a b l e and I u I J- Q \ 0 \ ' , we have In the above cases the f u n c t i o n Jl.^d'Xj i s symmetric , i . e . Jl (4*, S*) *= Jl(S*>*tx)-42 We now seek a formula f o r -H- (dtj g w h e r e the -K v a r i a b l e ^ - 2 * K ^ / i s the g e n e r a l hyperbomplex v a r i a b l e . The r e q u i r e d e x p r e s s i o n must reduce t o the above forms f o r the cases that ?C i s the complex , b i r e a l o r dual v a r i a b l e • I t i s a l s o d e s i r a b l e that i t remain symmetric • As above , l e t M be the matrix { iS.Sl .) correspond-- i n g to ->/ *y X F HA \ jzx J u and let M(j*J be the matrlx obtained from M/*i-u\ *IlLm y by r e p l a c i n g i t s f i r s t row by that of A f u n c t i o n f u l f i l l i n g the r e q u i r e d c o n d i t i o n s i s : |MC^)I- i M < ? * ) i \> ( s ; I O i ) Theorem 2 , 1 0 1 General Law of Conformal Mapping : I f the f u n c t i o n ijj •= o f t l i e v a r i a b l e x = 2 ** < i s d i f f e r e n t i a b l e at >C =r * then ( 2 . 1 0 2 ) 43 We prove t h i s by r e d u c i n g the e x p r e s s i o n f o r jTLfdy, ^ t o that f o r JL (W*, t u s i n g the hypothesis that has a d e r i v a t i v e whose modulus does not v a n i s h at the p o i n t Since c^ (5<) i s d i f f e r e n t i a b l e at X = X , we may write ^ 3 - J j ' ^ O ^ * at t h i s p o i n t . I t f o l l o w s from the isomorphism ... X <: > that iog ) M u l t i p l i c a t i o n i s commutative f o r the a l g e b r a and t h e r e f o r e , by the isomorphism , f o r the matrices . Therefore fl(Y<*)) ' ( 2 . 1 0 4 ) A l s o i t f o l l o w s that s i n c e both members of ( 2.105 ) are obtained fromthe correspond - i n g members of ( 2.104 ) by an e q u i v a l e n t replacement of the f i r s t row v e c t o r . 44 We have v, • ^ l l « < M - l « 6 ' ) | l « < M l | 4 Since then , by hypothesis , Hence the above equation reduces at ~)( "= to which i s the equation ( 2.102 ) , required by the theorem.. 45 The theorems which have been g e n e r a l i z e d i n t h i s c h apter r e q u i r e as hypothesis merely the commutative r i n g p r o p e r t i e s of the a l g e b r a and the e x i s t e n c e of a u n i t element • In se c -t i o n ( 2.7 ) we have encountered one p r o p e r t y which i s not g e n e r a l i z a b l e , namely the i d e n t i t y of d i f f e r e n t i a b l e and ana-- l y t i c f u n c t i o n s . A l l r e s u l t s of complex v a r i a b l e f u n c t i o n theory r e q u i r i n g f i e l d p r o p e r t i e s as hypothesis w i l l not be g e n e r a l i z a b l e t o r i n g algebras • To t h i s c l a s s belong the "residue theorems" and the whole theory of p o i n t s i n g u l a r i t i e s i n the theory of f u n c t i o n s of a complex v a r i a b l e • CHAPTER I I I Conformal R e p r e s e n t a t i o n i n the H y p e r b o l i c Plane . Bencivenga has shown that a f u n c t i o n o f a b i r e a l v a r i a b l e maps the h y p e r b o l i c plane i n t o i t s e l f i n such a manner , that at those p o i n t s f o r which the d e r i v a t i v e of the f u n c t i o n e x i s t s and i t s modulus does not v a n i s h , h y p e r b o l i c angles are p r e -served i n the mapping . In t h i s s e c t i o n we study the conformal mapping of the h y p e r b o l i c plane In more d e t a i l , and , i n par-t i c u l a r , we attempt a systematic treatment of the b i l i n e a r t r a n s f o r m a t i o n of the h y p e r b o l i c plane . 3.1 Geometry of the H y p e r b o l i c Plane . The p o i n t ("X*>^^)of the h y p e r b o l i c plane represents the b i -- r e a l number Z = X + <jj u . Many of the E u c l i d e a n theorems of the complex plane have analogues i n the h y p e r b o l i c plane . In t h i s correspondence , E u c l i d e a n d i s t a n c e + villi be r e p l a c e d by hyper-- b o l i c d i s t a n c e p c i r c u l a r angles by h y p e r b o l i c angles , and the c i r c l e ^o)" 1 , + (j^- ^fo)X - <^ °y t n e r e c t a n g u l a r hyperbola 4 7 3.2 Length, of a H y p e r b o l i c Line Segment . The l e n g t h of a h y p e r b o l i c l i n e segment of a curve C may be d e f i n e d as f o l l o w s : Let the parametric equations of a curve C be and l e t the parameter f i n c r e a s e monotonely from £ t o t • Then Z - 7T (£) « +• y&) • w i l l a l s o be an equ a t i o n of C , Now make a decomposition 0~* of the curve by l e t t i n g £~ take the set of values t. * t, < <, < t - i 1 , and l e t \CT~ I denote the /VHtfX ^ t" - ) » Form the C5~ - sum S - ^ I - Z < & 0 I • ( 3.21 ) Then the h y p e r b o l i c l e n g t h of the l i n e segment i s d e f i n e d t o be -<r - ( 3.22 ) i f t h i s e x i s t s independently of the decomposition <J~ Theorem 3.21 The h y p e r b o l i c l e n g t h of the segment of the curve X - } g , % (ft fo¥ t.4t4 t, i s g i v e n by 4 8 p r o v i d e d ( i ) and ) possess continuous f i r s t d e r -i v a t i v e s and ( i i ) f o r t * t * t, . P r o o f : A p o i n t f o r which "X'<?".) ~tfti') — O i s c a l l e d a s i n g u l a r p o i n t • Suppose that c o n d i t i o n s ( i ) and ( i i ) are s a t i s f i e d . Then and , a p p l y i n g the law of the mean , t h i s i s equal t o where Now w r i t e A (t) = x*) - ]fo Then | z ( i y | ? | ( > 4 9 where £ tends t o zero u n i f o r m l y over the c l o s e d I n t e r v a l , [t , C,] as C " t_, ~>-O , and hence ( 3. 23) s i n c e does not v a n i s h i n the i m t e r v a l , and s i n c e i t i s continuous , i t i s hounded away from zero . Now £* tends u n i f o r m l y to zero , and so we have , by ( 3.21 ) , ( 3.22 ) and ( 3.23 ) , r _ ( ]\Mt)\ <** t. X as s t a t e d by the theorem . C o r o l l a r y As neighbouring p o i n t s approach c o i n c i d e n c e i n a n o n - s i n g u l a r r e g i o n of the curve , the r a t i o of a r c - l e n g t h to c h o r d - l e n g t h ( both i n h y p e r b o l i c m e t r i c ) tends t o u n i t y • 50 P r o o f : The h y p e r b o l i c l e n g t h of the chord i s where Thus = JUXb - fCf)\ where £ tends u n i f o r m l y t o xero as t - t0 tends t o zero . The h y p e r b o l i c a r c l e n g t h i s where £ £ * t , by the law of the mean f o r i n t e g r a l s . There f o r e £ J Z I ^ | F i g u r e I Chapter I I I . . 51 3.5 Rectangular Hyperbola . The curve (oC-->(,)\_- <^0y~ - <x~ $ t o g e t h e r w i t h i t s conjugate 6<" v O 10*"" Play i n the hyper-b o l i c plane the r o l e which the c i r c l e p l a y s i n the complex plane . We r e f e r t o ( ^ • j ' f l ' O a s t i i e c e n * r e and to ct as the r a d i u s of the hyperbola . The r a d i u s i s the constant hyper-- b o l i c d i s t a n c e of any p o i n t on the h y p e r b o l a from the centre . Since a l l hyperbola's e n t e r i n g i n t o t h i s s u b j e c t are r e c t a n g -u l a r , w i t h axes p a r a l l e l t o the c o o r d i n a t e axes , we r e f e r to a r e c t a n g u l a r hyperbola of t h i s type simply as an"hyperbola" . Theorem 5.51 An h y p e r b o l i c a r c of h y p e r b o l i c l e n g t h £ subtends an h y p e r b o l i c angle of magnitude at the c e n t r e of the hyperbola of r a d i u s /X P r o o f : Let the p o i n t 'P(n>%) o n t n e hyperbola ( f i g . 1 ) determine the r a d i u s v e c t o r O "P making an angle <p w i t h the X - a x i s , and l e t the n e i g h b o u r i n g p o i n t p (->C-u>ttj 3*^3^) 0 X 1 t h e b y p e r b o l a determine^ the r a d i u s v e c t o r O P ' making an angle $ -f a 4 w i t h the x - a x i s . Since and , s i n c e we are working i n quadrant | , then 52 X A j - ( 3 # 3 1 ) a" By the c o r o l l a r y t o theorem ( 3.21 ) , the h y p e r b o l i c l e n g t h A S of the h y p e r b o l i c element of a r c P"P i s a s y m p t o t i c a l l y e q u i v a l e n t to the chord l e n g t h of p P ( i n the h y p e r b o l i c metric ) as tends t o zero : Therefore ( 3.31 ) becomes D i f f e r e n t i a t i n g the equation of the h y p e r b o l a w i t h ^ constant we get and hence ( 3.32 ) 53 ( 3.32 ) then becomes /UMJ\ A 4 j B u t * ^ U * = *f + <±±f+. and so Moreover , s i n c e AS ~ d S > we have C* *f ~ ;> / 2 and t h e r e f o r e where £ i s the a r c l e n g t h subtending at the centre • Theorem 3.52 Sine law f o r t r i a n g l e s . Let s i d e s CLt d d of a t r i a n g l e have h y p e r b o l i c lengths ft (X Q r e s p e c t i v e l y , and l e t the i n t e r i o r angle d e f i n e d by .si a. d- be denoted by \v " ' 4 ^ Figure 2 Chapter I I I 54 Then : \sU %J _ M < £ ' I _ N C I T h i s law i s proved by f i r s t d e v e l o p i n g the formula f o r the area of the t r i a n g l e . Let O Tf ( f i g . 2 ) be of l e n g t h fj and d e f i n e angle <p and M e t 0 1^  be of l e n g t h and d e f i n e angle 0t The angle measured from OV, t o Ol^ i s Ot — (j£ Then where /\ i s a r e a of the t r i a n g l e O P, ^ Prom the a r e a formula , A i n/-> i ' n ,ij i * X RCJ. ^ | , the s i n e law f o l l o w s immediately , on e q u a t i n g the three express--ions f o r area : 55 Therefore % p. e 3,4 H y p e r b o l i c O r t h o g o n a l i t y We must give three d e f i n i t i o n s . V e c t o r s ; To each number X+- ^ u •= /7 corresponds a v e c t o r o r i g i n a t i n g at the o r i g i n of the hyper-- b o l i o plane , and d e f i n e d completely by the modulus 0. and angle <PS . F u r t h e r , each v e c t o r o r i g i n a t i n g at the o r i g i n determines a unique angle <P D i a g o n a l L i n e s : The asymptotes of any r e c t a n g u l a r hyper-m i l l be s a i d t o - b o l a : <§<-*.•)*- (a-v"*1"- a ' c o n s t i t u t e a p a i r of d i a g o n a l l i n e s i n the h y p e r b o l i c plane . Thus to every d i s t i n c t p o i n t of the plane corresponds one p a i r of d i a g o n a l l i n e s . H y p e r b o l i c O r t h o g o n a l i t y : Two v e c t o r s i n the h y p e r b o l i c plane are mutually orthogonal., i f the h y p e r b o l i c tangent of the angle between them i s i n f i n i t e • Theorem 3.41 Two v e c t o r s , c o r r e s p o n d i n g t o angles <fy &t r e s p e c t i v e l y , are mutually orthogonal." i f and only i f ^ ' M^ 6t ~ I • 56 Proof : The c o n d i t i o n jtfrj[ ( fs- &t) = 0° may he w r i t t e n - 0° ' ( 3.41 ) I f the numerator i s f i n i t e and not zero , the l a s t e q u a t i o n i s e q u i v a l e n t t o : -^j , ^ % jta*l 6t - / . I f the numerator i s i n f i n i t e then at l e a s t one of the compon--ents of t h i s sum i s i n f i n i t e . Suppose XOMIL <t>s — Oo • Then, i f JOMA. & t O ) the denominator i s a l s o i n f i -n i t e so that the q u o t i e n t i s not i n f i n i t e , as r e q u i r e d Hence i t i s necessary t h a t %cuJ\ 6^ = O , and we may a s s i g n the value | t o the indeterminate form o: 6t "= &° > o - I I f the numerator i s O then ^cut/v <P — JXLJ{ O s t } l Q t X, + r a t ( 4nA <PS + u ^ 4 4e ) = ^ I 5 7 E q u a t i o n ( 3.41 ) co u l d be s a t i s f i e d only i f . fa*A <fc- &4 £ = / and then , only i f we a s s i g n the-value t o the i n d e t e r -Ci -minant form - , O The equations of t h i s s p e c i a l case , gi v e E i t h e r o r i _ £ - -I Hence both v e c t o r s l i e on $he same d i a g o n a l l i n e through the o r i g i n , and have the same sense . By a u n i t v e c t o r a s s o c i a t e d w i t h a g i v e n v e c t o r we mean the v e c t o r of u n i t h y p e r b o l i c l e n g t h d e f i n e d by the same hyper -- b o l i c angle Theorem ( 3.42 ) : Two v e c t o r s are mutually orthogonal i f and only i f t h e i r u n i t v e c t o r s are mutually r e f l e c t i o n s of one another iLn one or oth e r of the d i a g o n a l l i n e s through the 58 o r i g i n F o r i f we w r i t e * * flt(4M*s + U*~4 the c o n d i t i o n Jj&Jk <J£ JtoJk &t = / gives or which expresses the symmetry w i t h respect t o one of the d i a -g o n a l l i n e s through the o r i g i n , as s t a t e d i n the theorem • Theorem 5>43 : Cosine law f o r t r i a n g l e s : Def i n e a " l e n g t h f u n c t i o n " of l i n e segment j o i n i n g C*«, $ ) then i f s i d e s ^, f &x ^ of a t r i a n g l e have lengths ^ fx 3^ r e s p e c t i v e l y , and l e n g t h f u n c t i o n s P,1 > y r e s p e c t i v e l y , and V J ^ i s angle i n c l u d e d by O, dx ; 59 P r o o f : F o r the t r i a n g l e of f i g . 2 : = ML (T, ~ Z T> _ — z. Then s i n c e and s i n c e — - z nz _ a c e , we have « \, ~r va. 3 C o r o l l a r y : I f s i d e s tf, ^ 4 are mutually orthogonal, then F o r Since the t r i a n g l e i s d e f i n e d by t h r e e p o i n t s i n the f i n i t e plane , f^A. cannot he i n f i n i t e because of the s i n e law ( theorem 3.32 ) . Hence XVaJL ^ = O \ \ \ \ \ \ V / / s 1 / /A / / \ / / \ / j j ) ' \ \ / J y /' N VP / \ / / \ / / \ * F i g u r e 3 Chapter I I I 60 By the angle between two curves i n t e r s e c t i n g at a p o i n t we mean the h y p e r b o l i c angle between the r e s p e c t i v e tangents t o the curves at the p o i n t of i n t e r s e c t i o n • Theorem 3.44 : The r a d i u s v e c t o r of an hyp e r b o l a i n t e r -s e c t s the hyp e r b o l a o r t h o g o n a l l y • Note t h a t i f <PS i s angle determined by-the v e c t o r from o r i g i n t o p o i n t (X, «^  ") , then : * ->(. Hence the h y p e r b o l i c tangent of the angle (jb of a v e c t o r i s merely i t s " s l o p e " as understood i n E u c l i d e a n plane geometry. The slope of tangent at O*'^) o n * h e h y p e r b o l a i s g iven by where Q i s angle made by the tangent and the p o s i t i v e tC - a x i s . But the slope o f the r a d i u s v e c t o r t o ft, ^) o n the hyperbola i s : jtuJk & = Hence jfcjt <fc . taAQ ~ / •* } which proves the o r t h o g o n a l i t y s t a t e d i n the theorem . In the f o l l o w i n g theorem we d i s t i n g u i s h a p o s i t i v e sense V0 L f ? o m a negative sense L a l o n g a l i n e i n the plane ( F i g . 3 ) . We a s s i g n a p o s i t i v e s i g n t o sense l^L 61 when we a s s i g n an angle <^  t o 1^  L » and thus regard i t as a v e c t o r . Theorem 3.45 : Let a p e n c i l of l i n e s through any p o i n t 7^  of plane , whose v e c t o r s l i e i n one quadrant ( bounded by d i a -g o n a l l i n e s ) , cut an hyp e r b o l a i n p o i n t s *p } ^ . Then the product . i s constant over the membeis of the p e n c i l . The t r a n s i t i o n from one quadrant t o an ad-j a c e n t quadrant r e s u l t s i n a mere s i g n chance i n the product . Proof : Let oe a p o i n t on f [ . Let fL d e f i n e angle ^ and l e t % be d i r e c t e d h y p e r b o l i c distance of T from p Then s (3-U) S u b s t i t u t i n g the expr e s s i o n s ( 3.42 ) i n X - ^ = fl we o b t a i n : (6^^-ajf^fa But /CeflA ^  - ^ = i " J depending on the quadrant of 4 So that i f the roots are f , then C f4. i ( x> £ - fl1) which i s constant over a p e n c i l of l i n e s *pi. l y i n g i n one 62 quadrant . Since j _ | m ( and product £ depends on s i g n of <j> - /3**4 <fis the value of . Q changes s i g n on t r a n s i t i o n from one quadrant t o an adjacent quadrant • 3.5 A n a l y t i c fielations of B i r e a l V a r i a b l e s Theorem 3.51 E u l e r Theorem : A b i r e a l v a r i a b l e i s ex-p r e s s i b l e e x p o n e n t i a l l y i n terms of i t s modulus and amplitude: x+g« = a(/}fi&<Ps+ uAi^t;) - a*"** where the factor Jl obeys the r u l e s of a n ; e x p o n e n t i a l f u n c t i o n . S e t t i n g ?< •= w x 3 . * „ X , 1 " *! * l 4 + u $ /ZM 4 + U T ( 3.51 ) Wr i t i n g J, _ + ( ^  d e f i n i t i o n ) and a p p l y i n g r e l a t i o n s 63 9. -_ . - ^ 4> JL ' , We o b t a i n : A " % «<P ( 3.52 ) where ^ = ( ^ ^ _ u | ^ » - 1 ^ ^ = - e x The r e l a t i o n s o£\ u/hJify imply the e x p o n e n t i a l r u l e s : u(*s- ( 3.53 ) — r= J L 64 Prom ( 3.51 -}and ( 3.52 ) : 7$ = M - 7&*4 - u J Z ( 3 , 5 4 } CONVERGENCE OF POWER SERIES . By convergence of an h y p e r b o l i c s e r i e s we mean convergence of both r e a l s e r i e s . A s e r i e s of h y p e r b o l i c terms : (to i s not dominated by the absolute s e r i e s 21 a s * n t n e analogous case of a complex s e r i e s , because | * * < £ | * / . , I * / F o r the same reason , f o r the T a y l o r expansion Co of a b i r e a l v a r i a b l e Z = X-f , there e x i s t s no r a d i u s of convergence , As Bencivenga shows , the r e g i o n of convergence i s bounded not by an hyperbola , but by a r e c t a n --gl e . Figure 4 Chapter I I I 65 Theorem 3.52 : F o r a r e g i o n of the plane d e f i n e d by \a*n (SO / 5 _T2_ the T a y l o r expansion + ( i ^  •= *7 Z. ^- mi / H i t has a r a d i u s of convergence -^^ 2-) We prove t h i s g e o m e t r i c a l l y : The s e r i e s converges w i t h i n some r e c t a n g u l a r r e g i o n A 3 C D w i t h s i d e s p a r a l l e l t o d i a g o n a l l i n e s through the o r i g i n O which i s at the i n t e r -- s e c t i o n of d i a g o n a l s A C and 3 O ( F i g . 4 ) , as Bencivenga shows . In f i g u r e 4 l e t the unshaded p a r t s of the plane represent a r e g i o n J ^ v r t ( z ) | _fi_ Of the two p o i n t s ' • , a t t l i e i n t e r s e c t i o n of boundary l i n e s of r e g i o n |<&t*t(-z)| ^ J7- w i t h r e c t a n g l e of con-v e r g e n c e , l e t one of them , say be c l o s e s t t o Q , i n the h y p e r b o l i c metric • Then "p determines a unique hyperbola "X " $ ~ which passes through \~ and such that I, i s e i t h e r on the h y p e r b o l a or l i e s t o the s i d e remote from the o r i g i n . S i m i l a r l y determine the hyperbolas of r a d i i ^ ^ r e s p e c t i v e l y i n the o t h e r three quad-r a n t s . Any p o i n t Z of the r e g i o n / | «^ _/2. and such that | z | < 6X af aH ) l l e g i n s i d e t h e r e c t a n g l e of convergence . The r e q u i r e d r a d i u s of convergence corresponding t o i s : *C-/2.) ^ /Kt^t (a, ax at * ) 66 We now prove that an a n a l y t i c f u n c t i o n of a h i r e a l v a r i a b l e maps the h y p e r b o l i c plane i n t o i t s e l f c o n f o r m a l l y , by apply-- i n g the T a y l o r expansion and the E u l e r theorem . We employ the f o l l o w i n g n o t a t i o n : Let 2 = H + — fLJL means fl o = o means <p o JIM* 1 - 0 means X — > ° a n d ^ —> ° Theorem 3 . 5 3 : The f u n c t i o n LO = A?(£") of the b i r e a l v a r -- i a b l e Z maps the Z -plane c o n f o r m a l l y i n t o the ^ -plane at every p o i n t Z at which i s a n a l y t i c and . At a l l such p o i n t s the mapping i i biunique and the m a g n i f i c a t i o n and mapping angle are | f (z) | } 0U*t ^ - f & O ) r e s p e c t i v e l y . Proof : Expanding i n T a y l o r s e r i e s about z 0 we have eO-e*), - f\(-z-7a') + R ~) W r i t i n g Z - 2 e = rt£ fy * <x4 , c O - c o , , 67 This becomes «„ ^ ^ ^ cn4 + S i n c e , by hypothesis , Ck - \t\ \ = |-p ( S j , ) | 0 Pje = « m £ * ~ * " " J We now impose the c o n d i t i o n , that = | + l * M. ( 3 . 5 5 ) With the r e s t r i c t i o n (3.55 ) the s e r i e s *C<M) «  fi ± < L A 5 V - . . -has a r a d i u s of convergence ^ . Then , s i n c e the terms of the s e r i e s have a common f a c t o r (\ , where ty(*>*) ^ O 68 That i s , if* approaches zero u n i f o r m l y w i t h respect to ( 3.56 ) i m p l i e s : p = an. ( i + /*(«>*)) where the r e a l f u n c t i o n s each tend to zero , u n i f o r m l y w i t h r e s p e c t to <f ,as ft approaches At (\ = 0 we have , ( 3.58 ) So that the m a g n i f i c a t i o n i s A s and the angle of the mapping i s e(p - ^'J** Write CO = 6^") i n the form : Equations ^ , £ ^ ^ ^ w £ ^ are u n i q u e l y s o l u b l e f o r X , H . i f 6 9 1I> Of, 4 o A p p l y i n g t&e Cauchy - Riemann equations, t h i s c o n d i t i o n reads 9« Z 9 ^ 'die Hence the mapping i s biunique at p o i n t s Z f o r which -j^O i s d i f f e r e n t i a b l e and ("P'^)! ^ ° I t remains t o remove the r e s t r i c t i o n ( 3.55 ) : Since -A- may be chosen as l a r g e as we p l e a s e , we can chose i t t o exceed any g i v e n f i n i t e ^ O V K ( Z , - 7 „ ' ) | Therefore the theorem i s proved f o r any f i n i t e value of dUvt (9r-"Z0 ) | , and i t remains to $rreat the case ; = <*> 9 Slnce j '^C^I ^  ^ the 7 0 mapping i s biunique and hence totbe mapping ~Z — > cOt , ~z — > ^ there corresponds the unique i n v e r s e mappings cO — > ~Z0 » cO — 2 . In t h i s case | 7 . ^ | = oo i m p l i e s | - G« , f o r i f | CM (cO-coOl were f i n i t e then \ G+« -*) \ i n would a l s o he f i n i t e • T h e r e f o r e A t h i s case a l s o , the mapping i s conformal , so that r e s t r i c t i o n ( 3.55 ) has been removed , to complete the proof • Poi n t at I n f i n i t y Let -z - •>< +• <3 u By 2 0 °r Jh"*? -O we mean •>( —>o and ^ —> o By ~Z —y 0 0 or Jh+* ~2 oo we mean ">C ->> GQ or (and) £j—> co We regard Co as a s i n g l e p o i n t added t o the f i n i t e h yperbolic plane ; any v a r i a b l e "Z ^ "X + % approaches t h i s p o i n t at i n f i n i t y as e i t h e r o r *fi ( o r both t o g e t h e r ) tend t o i n f i n i t y on the r e a l l i n e . Theorem 3.54 : Assuming that as a v a r i a b l e ~Z — X+ % C tends t o zero o r i n f i n i t y , i t does so a l o n g a curve , the 71 slope of whose tangent tends to a l i m i t ( f i n i t e o r i n f i n i t e ), then Proof : — - "T - 6 — • ( 3.59 ) By hypothesis — tends t o a l i m i t ( f i n i t e or i n f i n i t e ): JL, | = L or ^ s ^ C * - " * " * ) > where €* — > ° S u b s t i t u t i n g i n ( '3.59 ) : J L „ I [ LV L - .a) I f L > I , ; — ^> ~z ( 3.510 ) "J L J I f X » but not ^ , tends t o i n f i n i t y then L i s i n f i n i t e j i f ^ tends to i n f i n i t y then both components of r i g h t member * By - X ( 4 ^ ( u *J X-t+^u. we mean — , ^ 7 2 of (3.510) tend to zero . Hence tends^o zero as "Z tends to i n f i n i t y . As ^ tends to zero at l e a s t the second compon--ent of r i g h t member of ( 3.510 ) tends to i n f i n i t y . T h i s proves the theorem f o r L ^ ' • « L = ±I ± = J- [ Ail - -J- u l • • ^ _L r _ - L M 7 so that "j? ~H I * Zt ± & \ ( 3 . 5 1 1 ) as «J — ^ O both components of r i g h t member of ( 3.511 ) tend t o i n f i n i t y . Th i s completes the proof , s i n c e statement of theoremrules out case "2 -5> 0 0 when L — I 3.6 B i l i n e a r T r a n s f o r m a t i o n As a s p e c i a l case of the conformal t r a n s f o r m a t i o n of the h y p e r b o l i c plafae we s h a l l d i s c u s s i n d e t a i l the b i l i n e a r t r a n -s f o r m a t i o n : ol Z + P CO = — where <, p y r, & Tf -Z + C are b i r e a l constants s u b j e c t t o the c o n d i t i o n 7^ 0 ( 3.61 ) 73 Theorem 3.61 : I f the s i n g l e p o i n t at i n f i n i t y i s added to the f i n i t e h y p e r b o l i c plane t o give the complete h y p e r b o l i c plane , then the b i l i n e a r t r a n s f o r m a t i o n CO « I9  maps the complete h y p e r b o l i c Z -plane , w i t h p o i n t s f o r which |V z + t \ = a } r-z+S^o excluded , b l u n i q u e l y on the complete CO -plane , w i t h p o i n t s f o r which -oi \ •= 0 ) ~fU) — o(j?0 excluded . Proof : I f (C_ — : . e x i s t s then by c o n d i t i o n ( 3.61 )• ^ f a i l s t o e x i s t i f JV7 4 f I -=r O > *>h*t is if (Y^  4? | ^  |(%*«*X*+yO + ^ 4 f» I 74 Now a c o n i c A n't t ae?c + ^ F 3 + c = o degenerate i f a E F - O Therefore, s i n c e . a. t-= o ( 3.62 ) i s the e q u a t i o n of a p a i r of d i a g o n a l l i n e s i n t e r - s e c t i n g at the p o i n t f TTJ tt - tt p r o v i d e d that V | — *--z + £ = C? 75 p r o v i d e d t h a t | f \ -^ fc O That i s , 4- T = O a"t the i n t e r s e c t i o n of l i n e s ( 3.62 ) , i n the case that \~1 [ -fc O The i n v e r s e of CO ss . L_! , i f i t e x i s t s , i s g i v e n by •y -z + f ^ - ( 3.63 ) Case |Y| <9 i . e . tends t o Hence the one to one correspondence : -Z = — < > co ^ oo T and s i m i l a r l y : u ) s ^ < > 2 - «• oo Case* | V | •= 0 : By c o n d i t i o n ( 3.61 ) |*(| * 0 and | f | ^ O 76 The p a i r of excluded l i n e s ( 3.62 ) then reduce t o a s i n g l e s t r a i g h t l i n e as f o l l o w s : Set ^ = ^ * 0 i n ( 3.62 ) : * +3 + j ~ * & • Then Y z + % = ^ ' g * ( i - a ^ o where we have assumed Y" "J^= O • F o r T f « C * s i n c e (tf| -f 0 and |£| ^  O by (3.61 ), the mapping i s d e f i n e d over the f i n i t e z -plane , and over the f i n i t e CO -plane , f o r every p o i n t • Hence f o r \f\ O w e a s s i g n the correspondence "Z s Co >^ CO - GO • We w a l l the p a i r of d i a g o n a l l i n e s f o r which ^. j j _ 0 the s i n g u l a r l i n e s of the Z -plane , and s i m i l a r l y | Y ^ J — < | r=. O d e f i n e s the s i n g u l a r l i n e s of the W -plane . A l l p o i n t s of the s i n g u l a r l i n e s , except t h e i r i n t e r s e c t i o n ( i n the f i n i t e plane , o r at i n f i n i t y , i f the s i n g u l a r l i n e s reduce t o a s i n g l e l i n e ) , are excluded from the mapping . In the case that "Y — 0 , t h a i * i s no s i n g u l a r l i n e i n e i t h e r of the f i n i t e Z or f i n i t e co -planes • 77 We may then t h i n k of the s i n g u l a r l i n e s as r e d u c i n g t o the s i n g l e pibint at i n f i n i t y • 3.7 B i l i n e a r T r a n s f o r m a t i o n of the Rectangular Hyperbola • Let A,(ltC be b i r e a l constants and X « *,+ > X = , a b i r e a l v a r i a b l e and i t s conjugate • Then ( A + A )^ X x + B x t ft x + c + c - o ( 3 ^ ? 1 i s the e q u a t i o n of a true or degenerate r e c t a n g u l a r hyperbola w i t h axes p a r a l l e l t o X- and axes , or of a s i n g l e s t r a i g h t l i n e ( which w i l l be c l a s s e d as an h y p e r b o l a ) • F o r , w r i t i n g /) » A f \ e t c . , ( 3.71 ) may be w r i t t e n : V 2A, / V 3/9, / i f /}, ^  , o r f 3 | > t + Ra V 2 f <f( « 0 ^ i f A, - 0 -Theorem 3.71 - b o l i c plane : '3,-I3a She b i l i n e a r t r a n s f o r a t i o n of the hyper-78 transforms an hyperbola i n the X -plane i n t o an hy p e r b o l a i n the ^ -plane and c o n v e r s e l y , where "hyperbola" denotes a r e c t a n g u l a r hyperbola w i t h axes p a r a l l e l t o c o o r d i n a t e axes or a s t r a i g h t l i n e . To prove t h i s we apply the b i l i n e a r t r a n s f o r m a t i o n t o ( 3.71 ): Since f o r any two b i r e a l numbers <k and X • Then _ S u b s t i t u t i n g these e x p r e s s i o n s f o r x and 5E i n ( 3.71 ) tbs r e s u l t i s : f [J| (/uM • pre +S*6Sf*)Js which i s of same form as ( 3.71 ) • ( 3.72 ) 79 Theorem 3.72 ; A b i l i n e a r t r a n a f o r m a t i o n , which maps a f i n i t e p o i n t of an hy p e r b o l a i n t o the p o i n t at i n f i n i t y , transforms the hyperbola i n t o a s t r a i g h t l i n e . Proof : . Let *1, *1i % H.+ be images of x, xt x, it r e s p e c t i v e l y . Then the c r o s s - r a t i o : i s i n v a r i a n t under the t r a n s f o r m a t i o n • Thus the b i l i n e a r trare-- f o rmation i s determined by making ^ correspond t o ? ( i ^ X 3 r e s p e c t i v e l y : then the image ^ , of any f o u r t h p o i n t ?c i s gi v e n by ) ( 3,- fr) ^r--x ^  v y O Now l e t ^J3 —> co ( i n the sense of theorem (3.54) ) i , e . f o r ^ •* GO : 80 or _ V 3 + P X -where <* •= X 3 f x, -C - I ( V O - ^ C v * 0 Prom these we see that 4 ^ - ->^x3 rf *r = "tf3 r-7 7 ir =• x, -zr-7 • ( $ . 7 3 ) The t r a n s f o r m of + + ^ ^ + c + C = 0 i s a s t r a i g h t l i n e i f and only i f ( hy ( 2,72 ) ) : which , on a p p l i c a t i o n of ( 3.73 ) , reduces t o which i s merely the statement that l i e s on the o r i g i n a l h y p erbola , C o r o l l a r y ( 1 ) Given a p e n c i l o f hyperbolas each member of which passes through d i s t i n c t p o i n t s P and ^ ; then a 81 b i l i n e a r t r a n s f o r m a t i o n mapping Q i n t o the p o i n t at i n f i n -- i t y maps the p e n c i l i n t o a p e n c i l of s t r a i g h t l i n e s a l l pass-- i n g through p , the image of I . ( the p o i n t s P and <3 w i l l be r e f e r e d t o as the p o l e s of the p e n c i l - ) C o r o l l a r y ( 2 ) Given a p e n c i l o f hyperbolas each member of which touches at P , then a b i l i n e a r t r a n s f o r m a t i o n , mapp-- i n g P i n t o the p o i n t at i n f i n i t y , maps the p e n c i l i n t o a p e n c i l of p a r a l l e l s t r a i g h t l i n e s C o r o l l a r y ( 3 ) Every p e n c i l H of hyperbolas through two p o i n t s P , ( d i s t i n c t o r c o i n c i d e n t , one or both of which may be at i n f i n i t y ) determines a unique p e n c i l hC orthogonal t o i t . P e n c i l s H * K a r e orthogonal conjugates i n the sense that e i t h e r p e n c i l determines the o t h e r u n i q u e l y . Any two members of e i t h e r p e n c i l determines the system H3 *f u n i q u e l y . Proof: A b i l i n e a r t r a n s f o r m a t i o n T" mapping <p i n t o the p o i n t at i n f i n i t y transforms \\ intd> a p e n c i l o f l i n e s i n t e r -- s e c t i n g i n Q/ , the image of ^ . There e x i s t s a , .': unique system K of hyperbolas c o n c e n t r i c at Q* . The in v e r s e t r a n s f o r m a t i o n T"' maps K ' i n t o the p e n c i l K which i s orthogonal to H ( B y theorem ( 3.44 ) and conform-- a l i t y ) . F i n a l l y , any two members of a s t r a i g h t l i n e p e n c i l through Q' determine the l i n e - p e n c i l , and hence the eq u i v a l e n t h y p e r b o l i c p e n c i l . 82 3,8 B i l i n e a r E q u i v a l e n c e . Any two systems , each of which i s the image of the o t h e r under some b i l i n e a r t r a n s f o r m a t i o n and i t s i n v e r s e , w i l l be s a i d t o be b i l i n e a r l y e q u i v a l e n t . T h i s r e l a t i o n i s r e f l e x i v e , symmetric and t r a n s i t i v e and thus a proper equivalence r e l a t i o n • Theorem 3,81 : Any two h y p e r b o l i c p e n c i l s ( one o r both of which may be a l i n e - p e n c i l through a p o i n t i n the f i n i t e plane^ each having two d i s t i n c t p o l e s , are b i l i n e a r l y e q u i v a l e n t . Proof : Let p e n c i l H have d i s t i n c t p o l e s P and Q There e x i s t s a t r a n s f o r m a t i o n mapping Q i n t o i n f i n i t y and "P i n t o P' , the i n t e r s e c t i o n p o i n t of l i n e p e n c i l ft' ( theorem (3,72) , c o r o l l a r y ( 1 ) ) Let second p e n c i l y( have d i s t i n c t p o l e s ft , £ . There e x i s t s mapping i n t o i n f i n i t y and 5 i n t o p ' . The product ' H \ maps i n t o k\ • Theorem 5.82 : Every h y p e r b o l i c p e n c i l w i t h two d i s t i n c t poles i s b i l i n e a r l y e q u i v a l e n t t o a c o n c e n t r i c system of hyper-bolas • Proof : The given p e n c i l H w i t h two d i s t i n c t p o l e s P and Q. determines a unique orthogonal h y p e r b o l i c p e n c i l y{ ( theorem (3.72) c o r o l l a r y (3) ) . Since the pole s of H are d i s t i n c t , so are the p o l e s o f K » f o r otherwise the p e n c i l and conjugate c o u l d be mapped i n t o a system of p a r a l l e l s t r a i g h t l i n e s w i t h a second system of l i n e s ( h y p e r b o l i c a l l y ) 83 orthogonal t o i t , which c o u l d then be mapped i n t o two tangen-t i a l p e n c i l s , each having common p o i n t of tangency at some p o i n t T ( theorem (3.72) , c o r o l l a r y (2) ). Then p and Gl would both map i n t o T by-the product t r a n s f o r m a t i o n , c o n t r a r y t o the p r o p e r t i e s of the b i l i n e a r t r a n s f o r m a t i o n . Therefore may be mapped i n t o a l i n e p e n c i l p a s s i n g through a p o i n t f{ , the ce n t r e o f the c o n c e n t r i c system H' • C o r o l l a r y : I f the h y p e r b o l i c p e n c i l H has two po l e s so a l s o has i t s orthogonal conjugate . Theorem 3.85 ; Let the b i l i n e a r t r a n s f o r m a t i o n T map an h y p e r b o l i c p e n c i l w i t h d i s t i n c t p o l e s p , (3 i n the -plane i n t o a c o n c e n t r i c system i n the a' -plane . Then *P » ^? l i e o n t h e s i n g u l a r l i n e s of the X -plane w i t h respect t o "7" . F o r i f P , ^ had images P' , Q' i n the X'-plane, I , Q would have t o be common t o a l l members of the con-- c e n t r i c p e n c i l . But members of a c o n c e n t r i c p e n c i l have no common p o i n t s , hence p and 0 must be p o i n t s which have no images under ~T" • li'+U Example : ")C •» transforms the c o n c e n t r i c system X' xf" - - (X ? - &o < a * i n t o p e n c i l / x , . , 84 whose p o l e s JL _ _L 2 2 CA l i e on the l i n g u l a r l i n e s X/ -Theorem 3.84 Every h y p e r b o l i c p e n c i l , the members of which are a l l tangent at a p o i n t T * i s b i l i n e a r l y e q u i v -a l e n t t o a p e n c i l of p a r a l l e l s t r a i g h t l i n e s . Hence every two t a n g e n t i a l p e n c i l s ( i . e . p e n c i l s f o r which the p o l e s c o i n c i d e ) are b i l i n e a r l y e q u i v a l e n t • The f i r s t statement i s merely c o r o l l a r y (2) of theorem(3.72^ The second statement f o l l o w s as i n theorem (3.81 ) , s i n c e a g i v e n t a n g e n t i a l p e n c i l may be mapped i n t o a g i v e n p e n c i l of p a r a l l e l l i n e s by determining the t r a n s f o r m a t i o n such t h a t the p o i n t of tangency maps i n t o i n f i n i t y and any two o t h e r p o i n t s on same hyperbola map i n t o two p o i n t s on the same l i n e of the g i v e n p a r a l l e l l i n e - p e n c i l • Theorem 3.85 : Every b i l i n e a r t r a n s f o r m a t i o n maps diagonal l i n e s i n t o d i a g o n a l l i n e s . I t i s obvious from the conformal p r o p e r t y t h a t t h i s must be tr u e under the mapping of any d i f f e r e n t i a b l e f u n c t i o n . 85 3«9 I n t e r l o c k e d Systems Two hyperbolas whose axes are mutually orthogonal are con-- c e n t r i c o r they I n t e r s e c t • However , two hyperbolas whose axes are p a r a l l e l may be so s i t u a t e d that they are n e i t h e r c o n c e n t r i c , nor do they i n t e r s e c t • Consider the two hyper--bolas w i t h p a r a l l e l axes - a" S o l v i n g f o r 20 The d i s c r i m i n a n t vanishes f o r J7. -f <f ( P - «f ) <X •= O < Write then JL > and l a s t e q u a t i o n reduces t o 0 + "X & of which ro o t s are F i g u r e 5 Chapter I I I 86 Hence the d i s c r i m i n a n t vanishes f o r : P 1 , = < ? - ^ ) * or ^-f^ <*+*y> In each case the l e f t member i s the square of the d i s t a n c e between centres (assuming o( jfi ) and the r i g h t member i s the square of the d i f f e r e n c e o r square of the sum of the r a d i i . From the geometry , the hyperbolas i n t e r s e c t ( i . e . d i s c r i m i n a n t > O ) f o r l a r g e values o f «f ^ - JS9" . I t f o l l o w s that the d i s c r i m i n a n t i s < 0 , that i s the hyper-bolas have no p o i n t i n common f o r < W ) r < c(z- p 1 - < One of the hypyfrbolas then has p o s i t i o n w i t h r e s p e c t t o the ot h e r as i l l u s t r a t e d i n F i g . 5 . The ce n t r e O of 3 must l i e w i t h i n the shaded area . Two such hyperbolas w i l l be s a i d t o be " i n t e r l o c k e d " . Theorem 3.91 : A b i l i n e a r t r a n s f o r m a t i o n maps an i n t e r -l ocked system i n t o an i n t e r l o c k e d system . Proof : Let a b i l i n e a r t r a n s f o r m a t i o n be a p p l i e d t o a system of two i n t e r l o c k e d hyperbolas /I and R> . Let the asymp-t o t e s be <v ,. X r e s p e c t i v e l y , and denote the co r r e s p o n d i n g image f i g u r e s by the corresponding primed l e t t e r s . I f l\' f$' i n t e r s e c t at P' , then a l s o / ) ' i n t e r s e c t s <x' and H>' i n t e r s e c t s JL at p ( s i n c e i n t e r s e c t i o n can occur only on s i n g u l a r l i n e s ) This means that one d i a g o n a l of OL' c o i n -- c i d e s w i t h one d i a g o n a l of A ( d i a g o n a l l i n e s map 87 i n t o d i a g o n a l l i n e s by theorem (3.85) ). T h i s means that one d i a g o n a l from (X and a p a r a l l e l d i a g o n a l from Jr are b o t h excluded from the one t o one mapping , c o n t r a r y t o the p r o p e r t i e s of the b i l i n e a r t r a n s f o r m a t i o n . Hence / j ' , f*/ do not i n t e r s e c t Theorem 3.92 : The orthogonal t r a j e c t o r i e s of an i n t e r l o c k --ed system form an i n t e r l o c k e d system . F o r suppose a p a i r of t r a j e c t o r i e s i n t e r s e c t e d at d i s t i n c t p o i n t s P and Q . Apply a t r a n s f o r m a t i o n mapping p i n t o i n f i n i t y , then the o r i g i n a l system w i l l be transformed t o a system c o n c e n t r i c at Q ' . The c o n c e n t r i c system w i l l i n ; t u r n map i n t o a p e n c i l with, two d i s t i n c t pibles (theorem (3.82)) which i s c o n t r a r y t o theorem (3.91 ) <. Suppose a p a i r of t r a j e c t o r i e s tangent at "J" . These t r a j e c t -o r i e s w i l l map i n t o p a r a l l e l l i n e s , hence the o r i g i n a l system w i l l map i n t o a p e n c i l of p a r a l l e l l i n e s '. This l a t t e r system w i l l map,in t u r n , i n t o a t a n g e n t i a l p e n c i l , c o n t r a r y t o theorem ( 3.91 ) . SUMMARY OF HYPERBOLIC PENCILS : There are three systems of hy^ - p e r b o l i c p e n c i l s • ( i ) C o n c e n t r i c system , w i t h b i l i n e a r l y e q u i v a l e n t forms : p e n c i l w i t h d i s t i n c t p o l e s , l i n e p e n c i l through a f i n i t e poirib. ( i i ) T a n g e n t i a l system , w i t h p e n c i l of p a r a l l e l l i n e s , b i -- l i n e a r l y e q u i v a l e n t t o i t . ( i i i ) I n t e r l o c k e d system . Each i s a c l o s e d system under the b i l i n e a r t r a n s f o r m a t i o n , 8 8 Orthogonal t r a j e c t o r i e s o f each system belong t o the same system . 3.1Q Inverse P o i n t s Let a t r a n s v e r s a l from any p o i n t Q0 of the plane cut an hyperbola i n p o i n t s 0 ( , . Denoting the h y p e r b o l i c d i s t a n c e between <s?0 and 0t by \Q Q J ( always a p o s i t i v e r e a l number ) we have shown t h a t I©. «, I - K<?J i s the same f o r a l l t r a n s v e r s a l s from Qo ( theorem 3.45 ). In the s p e c i a l case that Q% , c o i n c i d e at T" , Q0~V i s tangent t o the hyp e r b o l a at T" > and product . F o r a l l t r a n s v e r s a l s from Q : i s (0o | • ^ jPoTl2" ( 3.101 ) Theorem 3.101 : F o r every p o i n t *P of the h y p e r b o l i c plane there e x i s t s an i n v e r s e p o i n t w i t h r e s p e c t t o a g i v e n hyperbola • "P and i t s i n v e r s e P l i e on the same radius v e c t o r such that the product of t h e i r h y p e r b o l i c d i s t a n c e s from the centre i s the square of the r a d i u s o f the hyp e r b o l a • I f P i s the i n v e r s e of p , then P i s the i n v e r s e of P'. A b i l i n e a r t r a n s f o r m a t i o n maps a p a i r o f i n v e r s e p o i n t s i n t o a p a i r o f i n v e r s e p o i n t s • F i g u r e 6 Chapter I I I 89 Proof : C o n s i d e r an hyperbola A of r a d i u s <X , ( f i g . 6 ), and p l a c e the o r i g i n O at the c e n t r e of /{ . Let If be any p o i n t of quadrant occupied by a branch of /4 .Through If draw an hyperbola I3( orthogonal t o c u t t i n g O If ( produced i f necessary ) at /f ' , and f\ at . Since 0 "7J" i s tangent t o Qt at "Tf : |oT? 1 - |oT>'| = loxT -Thus T{ determines p f ' , and c o n v e r s e l y , independently of any p a r t i c u l a r orthogonal t r a j e c t o r y |3( : a second hyper-- b o l a , through I, and orthogonal to /J , w i l l i n t e r s e c t (3> at the same p o i n t ?f' Under any b i l i n e a r t r a n s f o r m a t i o n , the t r a n s f o r m of an hy-- p e r b o l a and two orthogonal t r a n s v e r s a l s i s a g a i n an hyperbola and two orthogonal t r a n s v e r s a l s . That i s , the image p o i n t s of 1^  » If' w i l l a g a i n be r e l a t e d t o one another as i n v e r s e p o i n t s r e l a t i v e t o the t r a n s f o r m of ft • Now c o n s i d e r , l y i n g i n a quadrant not occupied by a branch of . Through If draw hy p e r b o l a f$ , ortho-g o n a l to f\ c u t t i n g OP at *p ' and /) at f From the geometry of the s i t u a t i o n , the two branches oE /3 z l i e on p p p o s i t e s i d e s of the a x i s of (\ , so that O l£ $ CT^' a r e o p p o s i t i v e l y d i r e c t e d . Again from theorem (3 .45 ) , 9 0 are i n v e r s e p o i n t s w i t h r e s p e c t t o /\ t s i n c e one determines the o t h e r independently of the orthogonal that the images of "[^ ' under any b i l i n e a r t r a n s f o r m -a t i o n are a g a i n r e l a t e d as i n v e r s e p o i n t s w i t h r e s p e c t t o t r a n s f o r m of f\ . C o r o l l a r y : Since the centre and the p o i n t at i n f i n i t y form a p a i r of i n v e r s e p o i n t s w i t h r e s p e c t to- an hyp e r b o l a , a bi-rv. - l i n e a r t r a n s f o r m a t i o n maps the ce n t r e o f any hyperbola i n t o the c e n t r e of i t s t r a n s f o r m i f and only i f i t maps the p o i n t at i n f i n i t y i n t o i t s e l f . 3.11 Example of an I n t e r l o c k e d P e n c i l We apply the theory of i n v e r s e p o i n t s to the problem of determ i n i n g an i n t e r l o c k e d h y p e r b o l i c p e n c i l and i t s orthogonal conjugate from two g i v e n members of the p e n c i l : The p a i r : have no p o i n t i n common , that i s they are i n t e r l o c k e d . Let hype r b o l a (3 . The argument of pre v i o u s paragraph shows ( 3.111 ) ( 3.112 ) 0 J be any p o i n t , not the o r i g i n , on the ")( - a x i s . i s the i n v e r s e o f "P w i t h r e s p e c t 91 to h y perbola ( 3.111 ) . Therefore every hyperbola through "p and "P Is orthogonal t o ( 3.111 ) . i s the i n v e r s e of "p w i t h r e s p e c t t o ( 3.112 ) , assuming that | o oi- | • The three p o i n t s "P, *P "R^ d e f i n e an hyperbola : which i s orthogonal to both ( 3.111 ) and ( 3.112 ) . E v a l u a t i n g °> ^ i n terms of the c o o r d i n a t e s of P *P *P t n e orthogonal t r a j e c t o r y i s : Hence the p e n c i l orthogonal t o ( 3.111 ) and ( 3.112 ) i s x ' - j N ^ - ^ - <7 ( 3 # 1 1 5 ) i n parameter K The d i f f e r e n t i a l equation of f a m i l y ( 3,113 ) i s * 3 > 1 - 2 * ( j * : 0 y - o • ( 3 . i i 4 ) The d i f f e r e n t i a l e q u a t i o n of the f a m i l y of orthogonal t r a j e c t -i n ( 3.114 ) : - o r i e s i s obtained by r e p l a c i n g H "« T" to1 X * "T 0 X rf''! 92 o ? or which , on i n t e g r a t i o n , gi v e s : i . e . X + ( £ f 4 ) ^ 2 ^ - / = C? F i n a l l y , set — /wt - t j ( 3.115 ) i s the i n t e r l o c k e d p e n c i l d e f i n e d hy ( 3.111 ) and ( 3.112 ) which correspond t o yvvt = o > /M =• 3i r e s p e c t i v e l y . 

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