By Alexander M. Kytmanov, Simona G. Myslivets
The monograph is dedicated to vital representations for holomorphic capabilities in numerous complicated variables, equivalent to Bochner-Martinelli, Cauchy-Fantappiè, Koppelman, multidimensional logarithmic residue etc., and their boundary houses. The purposes thought of are difficulties of analytic continuation of features from the boundary of a bounded area in C^n. not like the well known Hartogs-Bochner theorem, this booklet investigates capabilities with the one-dimensional estate of holomorphic extension alongside complicated traces, and comprises the issues of receiving multidimensional boundary analogs of the Morera theorem. This ebook is a worthy source for experts in complicated research, theoretical physics, in addition to graduate and postgraduate scholars with an realizing of ordinary college classes in advanced, genuine and practical research, in addition to algebra and geometry.
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Additional info for Multidimensional Integral Representations: Problems of Analytic Continuation
Example text
2) bjQzC j : cos ˇ We fix a ball B0 in the plane T with the center at 0 and of radius " such that 1. B0 W; 2. w/ z˙ j for w 2 B0 , where C is the constant independent of the point z0 D 0. z0 ; "/ \ @D. f . z0 //U. z0 //U. f . U. U. ; zC / U. f . U. ; zC / U. ; z // D 0: U. U. f . / U. ; z //; we can take the limit inside, since z0 … @D n . It remains to consider this integral over the set . 2). 3) a1 zQC j in view of where d depends only on a, b, C, C1 , and ˇ. In precisely the same way, ˇ ˇ ˇ ˇ ˇj zNC k zC j2n j ˇ ˇ zNk jzk j d1 jQzj jzC ˇ k j C Ä : Ä ˇ z j2n ˇ j zC j2n j z j2n jw a1 zQC j2n Finally, d Ä d2 dS, where dS is the surface area element on the surface T, and d2 is independent of z0 .
Obviously the integral I1 over the surface @D n can be made as small as desired as z˙ ! 0. We now show that the form of the integral I2 is not affected by the unitary transformation. Indeed, the distance does not change, so the functions , d , and 48 j 2 Properties of the Bochner–Martinelli Integral and the Logarithmic Residue. . zj do not change either. Consider the expression n X @ . , by z0k D n X ajk zj ; k D 1; : : : ; n; jD1 and the inverse transformation is given by the matrix B D kbjk knj;kD1 .
2])). 2) in K to map , using closedness of the form U. / and passing over to the limit as j j ! 0 we obtain the required assertion. t u Let D be a bounded domain in Cn , n > 1, with a connected piecewise-smooth boundary, and let D . 1 ; : : : ; n / be a map consisting of holomorphic functions j , defined in some neighborhood KD D fw W w D z; z; 2 Dg and having a unique zero as the origin of multiplicity . D/, then ( Z Z @D f . / U. N ^ U. z/ D @D f . / U. D/). The vector z in the form U. z// is fixed.
