多复变数函数论的几个问题(一)——多个复变数的幂级数的收敛点集

用 C~n 表示 n 个独立的复变数 z_1,…,z_n 的空间。该空间的点记为 z=(z_1,…,z_n)。本文给出了多个复变数 z_1,…,z_n 的幂级数sum frum m_1,…,m_n=0 to ∞=0 am_1…m_n(z_1-z_1~((0))~m_1…(z_n-z_n~((0)))~m (A)的收敛点集的解析式。其中 z~((0))、z∈C~n,z~((n))是定点,z 是变点,am_1…m 不依赖于     

多复变数函数论的几个问题(二)——多个复变数的幂级数的收敛区域

沿用文献[1]中的记号,本文给出了多个复变数 z_1,…,z_n 的幂级数sum frum m_1,…,m_n=0 to ∞=0 am_1·m_n(z_1-z_1~((0))~m_1…(z_n-z_n~((0)))~m_n (A)(zI—zi。’)优l…(z。～zj。’)“ (4)的收敛区域的解析式。一、记号和定理的陈述定义给定实或复变数(t_1,t_2,…,t_(?))的实或复函数     

Bochner-Martinelli积分表示的一些应用

C~n空间中有以下著名的Bochner-Martinelli积分表示: 定理1.1 设D是复变数z_1,…,z_n空间C~n的有界域,其边界D是C~2类2n-1维光滑可定向流形,设f(z)是在区域D全纯在D连续的函数(记为f(z)∈A(D)),那末     

多复变数函数论的几个问题(五)——多个复变数的幂级数的一致收敛性及其他

在《问题五》中讨论了级数(A)、级数(B)和函数序列的一致收敛性以及多重数值级数和多重函数项级数的收敛性判别法。在《问题六》中讨论了一个复变数的幂级数的阿贝耳第二定理和托贝尔定理在多个复变数的幂级数中的推广。以上许多结论对于维数低于2n的各种区域亦成立。     

多复变数函数论的几个问题(七)——一点注记

在文献中有如下定理.定理:二重幂级数sum(C_kl(w-a_l)~k(z-a_l)~l)from k,l=0 to ∞(1.42)的收敛点集或者归结为一点(级数的中心),或者构成这样一个(4维)星形区域,它可与通过级数的中心平行于坐标平面的平面上的2维收敛区域以及位于这个区域的边界上的收敛点相邻接.在其证明中说到"点(t_0c_1, t_0c_2)总是收敛点集的内点"本文的目的是指出这种论断是没有根据的.事实上,该定理的证明中所构造的仅是星形点集,还不是星形区域.     

多复变数函数论的几个问题(四)—准多圆柱

Consider the power series of several complex variables z_1m,…,z_nIn A. in 1962, the domain of the convergence of the series (A) was discussed and the concepts of "domains of the complete n-circular type" (the 2n-dimension) and "conjugate convergent radii" etc. were used. However, the author did not point out how to compute each conjugate radius, only the expression connecting each conjugate convergent radius was given and it is not correct [See《Some Problems of the Theory of Functions of Several Complex Variables (Ⅰ)》, The Convergent Set of the Power Series of Several Complex Variables, Section 13, 《Journal of South China Institutc of Technology》No.1, 1978, PP.82-101] and (Ⅵ) respectively.In 1972, C. pointed out that the convergent set of the series (A) is the same as its absolute convergent set [See 1973, 22, 10 141,1972, 185-195.]. But how large is such a point set? In Problem(I), not only its size had been computed, but also incidentally we obtained the conclusion that the convergent set of the series (A) is the same as its absolute conveagent set CSee remark 6.1].In Problem (Ⅰ), we first computed the convergent set which is also the absolute convergent set of the series of functions of several complex variablesp_0(z_1,…,z_n)+P_1)(z_1,…z_n)+…+P_m(z_1,…,z_n)+…(B)[See section 3], and incidentally we derived Cauchy-Hadamard formula for computing the convergent radius of the power series of a single complex variable. Secondly, we introduced the transformation of models [See section 1] thus the required results were obtained. The concepts of the convergent measure, convergent set and convergent limit etc. were used above.In Problem (Ⅱ)[The Convergent Domain of the Power Series of Several Complex Variables, See 《Journal of South Chiua Institute of Technology》No. 1; 1978, PP. 102- 120], we constructed the convergent domains whieh are also the absolute convergent domains of various dimensions by means of the convergent set which is also the absolute convergent set of the series (A) [See theory 1.8]. The concepts of the convergent radius and convergent domain were used here.In Problem (Ⅲ), the concept of the convergent bound was used, and we pointed out that there is at least one singular manifold and it is at least (n-1) -dimensional on the (2n-1)-dimensional convergent bound, and we gave the equation satisfied by singular manifolds; Two examples were given; in one of which the unique (n-1)—dimensional manifold was computed; and in the other, the only n number of (2n-2) - dimensional manifold were obtained [in the book of, A. cited above, in the discussion on the domain of the complete n-circular type (the 2n-dimension), it was only pointed out that the series (A) has at least one singular point on the bound of the domain, and the author did not point out the way to find the singular point].In Problem (IV), we discussed a wider kind of domains which were named the 2n-dimensional pseudomulticylinder, the 2n-dimensional convergent domain of the series (A) is its special case [See theory 4].A number of the above conclusions also hold for problems of various domains where the dimension number is less than 2n.     

多复变数典型域的Cauchy型积分(Ⅱ)

一个复变数的Cauchy型积分,不仅对于函数论本身,而且对于奇异积分方程以及边值问题,都是非常重要的。关于多个复变数Cauchy型积分的研究,至今还很少。作者之一曾对矩阵双曲空间的Cauchy型积分的研究发表过一个摘要。本文考虑多复变数z=(z_1,…,z_n)空间内的超球zz′<1上的Cauchy型积分。设φ(u)是在uu′=1上的连续函数,此处u=(u_1…u_n)。显然,Cauchy型积分     

典型域内部解析映照的一个定理——献给我最亲爱的朋友

§1.设是n个复变数z=(z_1,…,z_n)空间的一有界单叶域,命L~2()代表所有在解析的其绝对值平方在可积的函数所成的集合,已知(见Bergmann在L~2()中存在一组完整的正交就范函数系数φ_0(z),φ_1(z),φ_2(z),…,φ_v(z),…,并且级数     

圆型域中解析函数的积分表示

设域■为n个复变数z=(z_1,z_2,…,z_n)空间中包有原点的圆型有界单连通域,并且对原点而言是星状的;又设?的特征流形是圆型、紧致的。本文证明了,凡?内的解析函数f(z),如果属于Hardy族H_1,则必可表为Cauchy 积分和Schwarz 积分,而且f(z)属于H_1是f(z)可表为poisson 积分的充分必要条件。本文同时给出了这些结果在正规族方面的某些应用。     

勒襄特级数的亚倍尔型及刀培型定理

本文给出了勒襄特(Legendre)级数sum from n=0 to ∞a_nP_n(z)在收敛椭园E_p上一点z_0=cosh(μ iβ_0)收敛的充分必要条件为级数sum from n=0 to ∞δ_ne~(nβ0~i)收敛,其中δ_n=n~(-(1/2))e~(nμ)a_n。本文证明了勒襄特级数的亚倍尔(Abel)型定理:若级数sum from n=0 to ∞a_nP_n(z)的收斂椭园为E_μ,z_0=cosh(μ iβ_0),且sum from n=0 to ∞a_nP_n(z_0)收斂,则sum from n=0 to ∞a_nP_n(z)=sum from n=0 to ∞a_nP_n(z_0),这里z→z_0是在E_μ内沿与E_μ正交的双曲线H_(β_0)进行。本文还证明了勒襄特级数的刀培(Tauber)型定理:设级数sum from n=0 to ∞a_nP_n(z)的收斂椭园为E_μ,z_0=cosh(μ iβ_0)为E_μ上一定点,令δ_n=n~(-(1/2))e~(nμ)a_n,如果δ_n=o(1/n),且sum from n=0 to ∞a_nP_n(z)=S,这里z→z_0是在E_μ内沿H_(β_0)进行,sum from n=0 to ∞a_nP_n(z_0)收敛,其和为S。     

几种收敛速率较高的迭代求解程序

为求解方程f(x)=0,我们提出了下列二种迭代程序:x_n~(1)=ω(x_(n-1)~((m-1)),x_(n-1)~(m),x_(n-1)~(m)),x_n~(2)=ω(x_(n-1)~((m-1)),x_(n-1)~(m),x_m~(1)),x_n~(3)=ω(x_(n-1)~((m-1)),x_(n-1)~(m),x_n~(2),x_n~(m)=ω(x_(n-1)~((m-1)),x_(n-1)~(m),x_n~((m-1))),(?)n∈N_0和z_(n 1)=ω(x_n,y_n,x_n),y_(n 1)=ω(x_n,z_(n 1),z_(n 1)),x_(n 1)=ω(x_n,z_(n 1),y_(n 1)),其中ω(x,y,z)=z-f(z)/f(x,y),f(x,y)=f(x)-f(y)/(x-y),它们的收敛阶分别为m (m~2 4)~(1/2)/2和2 3~(1/2)。本文分别建立了程序(I_m)和程序(Ⅱ)的收敛性定理,并就两个定理作了六点注记。文中还给出了一个数值例子     

关于Fourier-Laplace级数绝对收敛性的注记

讨论了n 维球面上某些可微函数类的Fourier Laplace级数的绝对收敛性 ,其中指出 :设f是Hrp(Ωn)上 2 ( [n4 ] 1)次连续可微函数 ,则级数∑∞k =0 Ykf(n)一致收敛到f 参 5     

以{P}∪ F4m+2为不动点集的光滑对合

设（M^n,T)是n维光滑闭流形Mn上以{p}∪F^4m 2为不动点集的对合，其中F^4m 2-CP(2m 1),确定了流形M^n的维数并给出（M^n,T)的等价协边类，即[M^n,t]2=[CP(2m 2),τo]2,且n-4m 4.     

复变数对数函数一个性质的说明

针对复变函数教材中复变数对数函数的一个性质Lnz~(1/n)≠1/nLnz(n1)提出质疑,通过证明得到Lnz~(1/n)=1/nLnz(n≥1)的结论.     

用从属多项式映照链逼近多复变数单叶解析映照

§1.引言設f(z)=sum from n=1 to ∞(a(n)z~n)是单位圓盘E={z:|z|<1}中的解析函数,如果f(z)把E一一地映成凸域,則称f(z)是E上的凸映照.称多項式序列V_n(z)=n/(n 1)a_1z n(n-1)/((n 1)(n 2))a_2z~2 … n(n-1)…/((n 1)(n 2)…(2n))a_nz~n,(n=1,2,…)为f(z)的de la Vall(?)e Poussin平均。1958年,G.P(?)lya和I.J.Schoenberg証明了这样的結論:如果f(z)是E上的凸映照,那么;(1).V_n(z)也都是E上的凸映照;(2).矿V_n(z)在E上收斂于f(z);(3).V_n(z)在E上从属于f(z),即V_n(z)相似文献   

多變數幂級數的單變數化變换系統的正常性

1.问题的说明一个形式上的m个複變數z_1,z_2,…,z_m的幂级数F=F(z_1z_2,…,z_m)=sum from i_1,i_2,…i_m=0 to ∝ a_(i_1,i_2,…,i_m) z_1~(i_1)z_2~(i_2)…z_m~(i_m)经过一个变数变换 T_α:z_i=α_it,α_i是複数,i=1,2,…,m以後可以表示做一个形式上的单个复变数t的幂级数 T_αF=F_α(t)=sum from n=1 to ∝t~n sum from i_1+i_2+…+i_m a_(i_1,i_2,…,i_m α_1~(i_2)α_2(i_1)…α_m(i_m)。T_α叫做一个单变数化变换。     

完备非紧黎曼流形的基本群

研究了一类完备非紧的n维黎曼流形,Ricci曲率满足Ric_M≥-(n-1)k(k>0),利用点到极小测地圈中点的距离的一致估计,证明了此流形在满足小的直径线性增长条件下,其基本群是有限生成的。     

两个判别法的等价性

在文[1]中,介绍了判别正项级数敛散性的一种方法,其方法如下:设sum from n=1 to ∞ a_n为正项级数,如果(?)(a_(n 1~))/a_n)<(1/e),则级数收敛;如果(a_(n 1~(?)))/a_n>(1/e),则级数发散。本文要指出:此判别法与拉阿伯(Raabe)判别法是等价的,仅在于表现形式不同。为讨论问题方便,先列出拉阿伯判别法:设sum from n=1 to ∞ a_n为正项级数,如果(?)(a_(?)/a_(n 1~))>1,则级数收敛;如果(a_n/a_(n 1)-1<1,则级数发散。     

J2k+2,n,k的决定

设(Z2)k作用于光滑闭流形Mn,其不动点集具有常维数n-(2k+2).是具有上述性质的未定向的n维上协边类[Mn]构成的集合.通过构造上协边环MO*的生成元决定了J2的群结构.     

大维样本协方差矩阵谱分布收敛速度的一个注记

提高了在有限八阶矩条件下,p×n维大维样本协方差矩阵谱分布收敛到Marcenko-Pastur分布的速度.特别,如果样本维数比率y=yn=p/n接近1,p×n维大维样本协方差矩阵谱分布的期望收敛到极限分布的速度,改进为O(n-1/6).相似在y接近1的条件下,依概率收敛和几乎处处收敛速度为Op(n-1/6)和Oa.s.(n-1/6).