4 LOUIS NIRENBER G the differentia l operato r P involve s only ordinar y differentia l equations , th e syste m (1.1 ) and th e operato r b/by n . Th e syste m o f equation s (1.2 ) t o determin e th e integra l o r char - acteristic curve s is however nonlinear—despit e th e fac t tha t w e started wit h a linear operato r P actin g o n w . Now look a t firs t orde r operator s (1.1 ) wher e th e coefficients , a s well a s w , ar e per- mitted t o b e complex-valued . An y suc h operato r ma y b e writte n a s P = Px + iP 2 where th e P x an d P 2 ar e vector field s i.e . operator s wit h rea l coefficients . I f P v an d P2 ar e linearl y dependen t everywhere , say both ar e rea l multiple s o f a n operato r P 3 , the n P i s a (complex) multipl e o f P 3 an d it s study i s reduced t o th e stud y o f P 3 whic h we have describe d above . Thu s th e nex t generi c cas e t o b e studie d i s where P x an d P 2 ar e linearly independen t everywhere . * The mos t familia r exampl e o f thi s is the Cauchy-Rieman n operato r i n R 2 , wit h coordinates (x, y): Here z = x + iy, an d i t i s convenient t o denot e P b y 3/9F , an d l A(b/bx - id/by) b y b/bz, sinc e th e differentia l o f a function w(x, y) the n take s th e for m bw bw dw - dw dw = dx + - dy = —— dz + - dz. dx by dz oz Analytic functio n theor y i s concerned wit h th e stud y o f function s w = u + iv whic h satisfy th e homogeneou s Cauchy-Rieman n equations : 9w n bu bv bv bv =0 , i.e . - = 0 , + =0 . bz bx by by bx The classical literatur e o f analytic , or holomorphic , function s doe s not tak e u p th e inhomo - geneous equatio n (1.5) bw/bF=f. However i n th e moder n treatmen t o f th e theory , result s abou t th e inhomogeneou s equatio n have proved t o b e very usefu l i n dealin g with analyti c functions . Fo r "nice " function s / , a solution o f (1.5 ) i n a domain O i s given by (her e f = £ + i-q) « « ) - - i / / ! a - « * : * riif " z this formul a yield s man y propertie s o f solutions . W e recommend to th e reade r wh o i s inter- ested i n seein g how a study o f (1.5 ) yield s result s o n analyti c function s t o loo k a t Chapte r 1 of Hormande r [13] . Consider no w th e genera l operator , fo r n = 2 , P=PX +/P 2
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