By Alexander M. Kytmanov, Simona G. Myslivets
The monograph is dedicated to crucial representations for holomorphic services in numerous complicated variables, resembling Bochner-Martinelli, Cauchy-Fantappiè, Koppelman, multidimensional logarithmic residue etc., and their boundary homes. The functions 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 e-book investigates features with the one-dimensional estate of holomorphic extension alongside advanced strains, and comprises the issues of receiving multidimensional boundary analogs of the Morera theorem. This ebook is a useful source for experts in advanced research, theoretical physics, in addition to graduate and postgraduate scholars with an realizing of ordinary college classes in advanced, genuine and useful research, in addition to algebra and geometry.
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Additional resources for Multidimensional Integral Representations: Problems of Analytic Continuation
Example text
Z/ˇ@D D n 2n N 2 j zj @ k j grad . z/ is a solid angle of the tangent cone to the surface @D at z. z/ is defined and different from zero. 2 Z p:v: @D for z 2 @D. U. z/ 24 2 Properties of the Bochner–Martinelli Integral and the Logarithmic Residue. . Proof By definition Z Z p:v: @D U. z;"/ U. z;"/ U. z;"// U. z;"/ U. z; "/. z; "/) is opposite to that of @D. z; "/ while the form U. z;"// U. 2 i/n "2n U. z;"/ n X . 1/k 1 . z;"/ kD1 U. 1 it follows that the restriction of the forms equals n X . 1/k 1 .
Thus, the form of the integral I2 is invariant under unitary transformation. 0/. @zk kD1 k z˙ k / n X @ . Nm @ Nm zN˙ m/ mD1 D D i . 2 n z˙ n / n 1 i X @ N m. 2 mD1 @ Nm n X @ . 3 Boundary Behavior of Derivatives of the Bochner–Martinelli Integral 49 We split the integral I2 over the surface into three integrals: I20 I2˙ D inŠ Z n inŠ D˙ n 2 ! n 1 X @ @ N f . w// 2'yn C m yn jw @ Nn mD1 @ Nm B0 Z B0 f . u2n C ' 2 C y2n ˙ 2'yn / @ Nn n 1 X @ N C m. @ Nm ! ˙2'yn zC j 4 2n d 0; mD1 where d 0 is the image of d under the mapping w !
1). z/jp d ˇZ ˇ ˇ f. U. ; z / ˇ ˇU. ; zC / ˇ U. ; z /ˇ Z @D ˇp ˇ ˇ U. ; z / ˇ ˇ jf . z/jp ˇU. ; zC / ˇ U. ; z /ˇ by Jensen’s inequality (see, for example [31, Sect. z/j ˇU. ; zC / jf . / Z We estimated the integral @D ˇ U. p : ˇ ˇ ˇU. ; zC / U. z/ @D Ä C1 jf . z/jp jU. ; zC / Z d . / @D jf . / U. 2 Jump Theorems for the Bochner–Martinelli Integral 37 If z 2 B. ; r/ \ @D, then ˇ ˇ N ˇ m zNm ˇ ˇj zC j2n Nm j D j Nm ˇ zNm ˇˇ ˇ z j2n ˇ ˇ zNm jˇj zC j X1 ˇ 2n z jˇ j jD0 1 j zC jjC1 j Ä z j2n kn j 6"n n; jw "j2 C "2 while ˇ ˇ ˇ ˇ ˇj " m zC j2n C " j ˇ ˇ ˇ ˇÄ z j2n ˇ kn jw m 2" n: j2 C " 2 Then Z @D\B.
