FROM GENERALIZED CAUCHY-RIEMANN EQUATIONS TO LINEAR ALGEBRAS. (1) Ê dkij ^ = 0 (* = 1, 2,, (n2- «)),

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1 FROM GENERALIZED CAUCHY-RIEMANN EQUATIONS TO LINEAR ALGEBRAS JAMES A. WARD I a previous paper [l] the author gave a defiitio of aalytic fuctio i liear associative algebras with a idetity. With each such algebra there was associated a set of partial differetial equatios called geeralized Cauchy-Riema differetial equatios which serve as a criterio to determie whether a give fuctio is aalytic i that algebra. A simplificatio for the commutative case has bee give by Wager [2]. The purpose of this paper is to give sufficiet coditios (Theorem 2) that a set of equatios (1) Ê dkij ^ = 0 (* = 1, 2,, (2- «)),,-,i=i dxj where the du) are costats i a field F, determie a liear associative commutative algebra A over F for which equatios (1) are the geeralized Cauchy-Riema differetial equatios. This will eable us to fid solutios of such a set by meas of power series i the algebra. Let 1, e2,, e be a proper basis for a liear associative commutative algebra A with a idetity over the field F. Multiplicatio will be defied by (2) e.ey = X) Cijktk *=i (i, j = 1, 2,, ) where the && are i F. Deote by P< the matrix (Ci8T) where r is the row idex ad î the colum idex. If a = aiei+a2e2+ +a e is ay elemet of A, the a<r->airi-)-a2r2-\- +ap is a isomorphism kow as the first regular represetatio of A by matrices. Let U deote a system of fuctios y,(xi, x2,, x) of variables Xi, Xt,, «of F, ad let the y be aalytic i a simply-coected regio R of M-space. The r = ^"=1 yiu will be called a fuctio over A of the variable = ^"=1 *<e<. Sice the algebra is commutative we defie with Wager [2, p. 456] that r] = 23?=i y,e< be a aalytic fuctio of if the y are i U ad the Jacobia matrix (dyt/dx,) is i A, that is, if there exist fuctios Zi, z2,, z such that Received by the editors April 15, Licese or copyright restrictios may apply to redistributio; see

2 FROM CAUCHY-RIEMANN EQUATIONS TO LINEAR ALGEBRAS 457 (3) ( J = ziri + z2r2 + + zr. \dx,/ This implies a set of homogeeous relatios amog the dyr/dx, called the geeralized Cauchy-Riema differetial equatios of the algebra with basis ei, e2,, (. Lemma. // A is a liear associative commutative algebra of order with a idetity over F, the a ecessary coditio that rj = E"=i y,etbe aalytic is that the compoets of r\ satisfy a liearly idepedet system of differetial equatios (4) È im ^ = o (k = i, 2,, (y -»)), i,j=i dx where the dkii are costats i F depedig oly o the multiplicatio costats Cm of the algebra, ad such that (5) im = 0 (k = 1, 2,, (2 - )). <-i By the defiitio of derivative the matrix equatio (6) WiiCi + u2r2 + + ur = dya gives w2 equatios i the ukows, with coefficiets amog the set Cijh. From ay of the equatios (6) for which the determiat is ot zero, we ca solve for the w, ad substitutio i the remaiig equatios leads to 2 equatios of the form (4). Uder a chage of basis (7) i,- = E UA tijef (i=l,2,---,), ;-i dxj we have x = X)"=i *«*< ad y'j E"=i U yi- Therefore Let Dh - (dkrs) ad D't = (d'trs) ; the / ; dkij - Z-, «Auv ~ i,)=l OXj u, v=l C %v (8) Du = T~*DkT where T= (t ). Sice the trace of a matrix is ivariat uder similarity trasformatio, we see that E?=i ^*» is ivariat uder chage Licese or copyright restrictios may apply to redistributio; see

3 458 J. A. WARD [Jue of basis of the algebra A. Wager [2, equatios 3] has show that if ei is the idetity, the geeralized Cauchy-Riema differetial equatios may be writte oyr " dyi (9) - + Z Cur ~ = 0 (r, s = 1, 2,, ). dx, j_i dxi Sice ei is the idetity, Cur = S,r (Kroecker 5). For 5 = 1 we get equatios -\- 2-, cr = 0 (r = 1, 2,, ) dxi,«.1 dxi which are idetically zero. The 2 remaiig equatios are i the form (4). If, i (9), r = s^l we get _ dy. dyi» r djl dx, dxi j_2 âaci because G = l, hece ^?=i *,-,= =0. If r^s, s^l, Cr = 0 ad the equatios reduce to -^+ C,r^ = 0 dx. <=2 dzi so that dkii = 0 for every i. Hece i every case we have (10) dkii = 0 ad the lemma is proved. Theorem 1. If Ai = (aitr) (i = 1, 2,, ) is a set of by matrices over F such that (11) AiA^AjAi (*,i- 1,2,,*) aa if there is a p such that (12) aipr = 5ir (i, r = 1, 2,,»), Äew (13) j4*4j = X «iyt^t (* / = 1. 2,, ) «=i ad the Ai form a basis for a liear, commutative, associative algebra of order over F with a idetity elemet. Licese or copyright restrictios may apply to redistributio; see

4 1953] FROM CAUCHY-RIEMANN EQUATIONS TO LINEAR ALGEBRAS 459 By (12) the elemet of uth row ad pth colum of AiA; is 2-1 aitvßjvt = _, auusjt aut-1 <=-l By chage of otatio we obtai the elemet of the wth row ad the pth colum of A A, to be ajiu. Thus we have (14) aiju = aiiu (i, j, u = 1, 2,, «). Aother form of (11) is - (15) j aua]vt = _i Qjtußivt. t \ <=i Thus we have AiA = Í E aurait) = í E «y<rö,-.(j (by (15)) = ( 2~2 ajtra.it J = I Z ßtir«y.i J = ( E durat J (by (14)) (by (15)) (by (14)) = 2^atAt. From (12) we see that the At are liearly idepedet with respect to F. Equatios (12) ad (14) together show that A, = I. Hece the Ai form a basis for a algebra A whose costats of multiplicatio are Cijk = a</*, so that Ri = A,-. Let (16) /» - L im ~ (k=l,2,---,(2-»)),3=i dxj such that du are i a field F ad the /* are liearly idepedet with respect to F. If there is a p such that it is possible to solve for each dyi/dxj i the system (17) E im- - 0 (k = 1, 2,, (y -»)) Licese or copyright restrictios may apply to redistributio; see

5 460 J. A. WARD [Jue as a liear fuctio of dyi/dxp, dy2/dxp,, dy/dxp, the (17) may be put i a matrix form * idyr\ /v dy<\ X^f ï dyt V a dyi \dx,/ \ 1=1 d^j,/ (=i dxp t=i dxp by adjuctio of the idetities dyt dyi /- O OCp = /- O Xp ( =1,2,...,«). Theorem 2. Suppose the system of differetial equatios (17) has the property that for some fixed iteger p, it implies the set (18). Suppose further that the matrices A, = (a18r), i \,2,,, are commutative ad (19) E iku = 0 >'=i (k = 1, 2,, (2 - )). The there is a uiquely determied liear, commutative associative algebra A, over F, for which (17) is a set of geeralized Cauchy-Riema differetial equatios (i the sese of the lemma). Sice each equatio of (18) which is ot a idetity is a liear combiatio of the set (17), it follows from (19) that alpr = ôrt. Hece by Theorem 1 the Ai (i=\, 2,, ) form a basis for a algebra A with AP = I as the idetity elemet. It is see that (17) ad hece (16) is a set of geeralized Cauchy-Riema differetial equatios for the algebra. For = 2 we have the Corollary. A ecessary ad sufficiet coditio that the liearly idepedet equatios (20) E im = 0 (k= 1,2),-,3=i ax, determie a algebra A for which (20) is a set of geeralized Cauchy- Riema differetial equatios is that (21) it + im = 0 (k = 1, 2). The ecessity of (21) follows from the lemma. Sice (21) holds ad the matrix of the coefficiets of (20) is of rak 2, (20) may be put ito form (18). Therefore either Ai = I ad therefore is commutative with A2 or A2 = I ad is commutative with ^4i. Therefore the corollary follows from Theorem 2. Licese or copyright restrictios may apply to redistributio; see

6 19531 FROM CAUCHY-RIEMANN EQUATIONS TO LINEAR ALGEBRAS 461 Ketchum [3, pp. 646 ad 653] proved that every solutio yi. y»> ' ' '» y» of the geeralized Cauchy-Riema differetial equatios for a commutative algebra may be expressed i the form co v = X) y#i = H a A' i=l,=o where = /"=i *» i. Therefore if equatios (1) determie a algebra, the solutios of the equatios may be obtaied by use of power series i the algebra. Refereces 1. James A. Ward, A theory of aalytic fuctios i liear associative algebras, Duke Math. J. vol. 7 (1940) pp Raphael D. Wager, The geeralized Laplace equatios i a fuctio theory for commutative algebras, Duke Math. J. vol. 15 (1948) pp P. W. Ketchum, Aalytic fuctios of hypercomplex variables. Tras. Amer. Math. Soc. vol. 30 (1928) pp Uiversity of Ketucky Licese or copyright restrictios may apply to redistributio; see