anb研究论文

4．1．3复变函数项级数定义4．3设{fn（z）}（n＝1， 2， …）为一复变函数列，其中各项均在复数域D上有定义，称表达式∑∞〖〗n＝1fn（z）＝f1（z）＋f2（z）＋…＋fn（z）＋…（4．2）为复变函数项级数．该级数的前n项和Sn（z）＝f1（z）＋f2（z）＋…＋fn（z）为级数的部分和．若z0为D上的固定点，limn→∞Sn（z）＝S（z0），则称复变函数项级数（）在z0点收敛，z0称为级数∑∞〖〗n＝1fn（z）的一个收敛点，收敛点的集合称为级数∑∞〖〗n＝1fn（z）的收敛域．若级数∑∞〖〗n＝1fn（z）在z0点发散，则称z0为级数∑∞〖〗n＝1fn（z）的发散点，发散点的集合称为级数∑∞〖〗n＝1fn（z）的发散域．若对D内的任意点z，都有limn→∞Sn（z）＝S（z），则称级数∑∞〖〗n＝1fn（z）在D内处处收敛．并称S（z）为级数的和函数．下面我们重点讨论一类特别的解析函数项级数——幂级数，它是复变函数项级数中最简单的情形．4．2幂级数〖〗在复变函数项级数的定义中，若取fn（z）＝an（z－z0）n或fn（z）＝anzn（n＝1， 2， …），就得到函数项级数的特殊情形∑∞〖〗n＝0an（z－z0）n＝a0＋a1（z－z0）＋a2（z－z0）2＋…＋an（z－z0）n＋… （4．3）或∑∞〖〗n＝0anzn＝a0＋a1z＋a2z2＋…＋anzn＋…（4．4）形如（）或（）的级数称为幂级数，其中，a0， a1， …， an， …和z0均为复常数．在级数（4．3）中，令z－z0＝ξ，则化为式（4．4）的形式，称级数（4．4）为幂级数的标准形式，式（4．3）称为幂级数的一般形式．为方便，今后我们以幂级数的标准形式（4．4）为主来讨论，相关结论可平行推广到幂级数的一般形式（4．3）．4．2．1幂级数的收敛性关于幂级数收敛问题，我们先介绍下面的定理．定理4．5（Abel定理）若幂级数∑∞〖〗n＝0anzn在z＝z0（≠0）处收敛，则此级数在｜z｜＜｜z0｜内绝对收敛（即∑∞〖〗n＝0｜anzn｜收敛）；若在z＝z0处发散，则在｜z｜＞｜z0｜内级数发散．证若∑∞〖〗n＝0anzn在z＝z0（≠0）处收敛，即级数∑∞〖〗n ＝ 0anzn0收敛，所以limn→∞anzn0＝0因而，存在常数M＞0使得对所有的n，有｜anzn0｜＜M当｜z｜＜｜z0｜时，｜anzn｜＝｜anz0｜z〖〗z0n＜Mz〖〗z0n，而级数∑∞〖〗n＝0z〖〗z0n收敛，所以，∑∞〖〗n＝0anzn绝对收敛．若∑∞〖〗n＝0anzn在z＝z0（≠0）发散，假设存在一点z1，使得当｜z1｜＞｜z0｜时，∑∞〖〗n ＝ 0anzn1收敛．则由上面讨论可知，∑∞〖〗n ＝ 0anzn0收敛，与已知∑∞〖〗n ＝ 0anzn0发散矛盾！因此，∑∞〖〗n＝0anzn在｜z｜＞｜z0｜发散．由Abel定理，我们可以确定幂级数的收敛范围，对于一个幂级数来说，它的收敛情况有以下三种情形：（1） 对所有正实数z＝x， ∑∞〖〗n＝0anxn都收敛，由Abel定理，∑∞〖〗n＝0anzn在复平面上处处绝对收敛；（2） 对所有的正实数x，∑∞〖〗n＝0anxn（x≠0）发散，由Abel定理，∑∞〖〗n＝0anzn在复平面内除原点z＝0外处处发散；（3） 既存在使级数收敛的正实数x1＞0，也存在使级数发散的正实数x2＞0，即z＝x1时级数∑∞〖〗n ＝ 0anxn1收敛，z＝x2时级数∑∞〖〗n ＝ 0anxn2发散．由Abel定理，∑∞〖〗n＝0anzn在｜z｜≤x1内，级数绝对收敛，在｜z｜≥x2内级数发散．在情形（3）中，可以证明，一定存在一个有限的正数R，使得幂级数∑∞〖〗n＝0anzn在圆｜z｜＜R内绝对收敛，在｜z｜＞R时发散，则称R为幂级数的收敛半径，称｜z｜＜R为幂级数的收敛圆．约定在第一种情形，R＝∞；第二种情形，R＝0．而对于幂级数∑∞〖〗n＝0an（z－z0）n，收敛圆是以z0为圆心，R为半径的圆：｜z－z0｜＜R．至于在收敛圆的圆周｜z｜＝R（或｜z－z0｜＝R）上，∑∞〖〗n＝0anzn或∑∞〖〗n＝0an（z－z0）n的收敛性较难判断，可视具体情况而定．关于幂级数收敛半径的求法，同实函数的幂级数类似，可以用比值法和根植法．定理4．6（ 幂级数收敛半径的求法）设幂级数∑∞〖〗n＝0anzn，若下列条件之一成立：（1） （比值法）limn→∞an＋1〖〗an＝L；（2） （根值法）limn→∞n〖〗｜an｜＝L.则幂级数∑∞〖〗n=0anzn的收敛半径R＝1〖〗L．证明从略.当L＝0时，R＝∞；当L＝∞时，R＝0．例4．4求下列幂级数的收敛半径：（1） ∑∞〖〗n＝1zn〖〗n3（讨论圆周上情形）；（2） ∑∞〖〗n＝1（z－1）n〖〗n（讨论z＝0， 2的情形）；（3） ∑∞〖〗n＝0（cosin）zn．解（1）因为limn→∞an＋1〖〗an＝limn→∞1〖〗（n＋1）3〖〗1〖〗n3＝limn→∞n〖〗n＋13＝1或者limn→∞n 〖〗｜an｜＝limn→∞n〖〗1〖〗n3＝limn→∞1〖〗n〖〗n3＝1所以，收敛半径R＝1，从而级数的收敛圆为｜z｜＜1．由于在圆周｜z｜＝1，级数∑∞〖〗n＝1zn〖〗n3＝∑∞〖〗n＝11〖〗n3收敛（p级数，p＝3＞1），所以，级数在圆周｜z｜＝1上也收敛．因此，所给级数的收敛范围为｜z｜≤1．（2） 由于limn→∞an＋1〖〗an＝limn→∞1〖〗（n＋1）〖〗1〖〗n＝limn→∞n〖〗n＋1＝1，故收敛半径R＝1，从而它的收敛圆为｜z－1｜＜1．在圆周｜z－1｜＝1上，当z＝0时，原级数成为∑∞〖〗n＝1（－1）n1〖〗n（交错级数），所以收敛；当z＝2时，原级数为∑∞〖〗n＝11〖〗n，发散．表明在收敛圆周上，既有收敛点又有发散点．（3） 由于an＝cosin＝1〖〗2（en－e－n），所以limn→∞an＋1〖〗an＝limn→∞en＋1－e－（n＋1）〖〗en－e－n＝limn→∞en（e－e－2n－1）〖〗en（1－e－2n）＝e故收敛半径为R＝1〖〗e．例4．5求幂级数∑∞〖〗n＝1（－1）n1＋sin1〖〗n－n2zn的收敛半径．解因为limn→∞n〖〗（－1）n1＋sin1〖〗n－n2＝limn→∞1＋sin1〖〗n－n＝limn→∞1＋sin1〖〗n1〖〗sin1〖〗n－sin1〖〗n〖〗1〖〗n＝e－1故所求收敛半径为R＝e．例4．6求幂级数∑∞〖〗n＝1（－i）n－1（2n－1）〖〗2nz2n－1的收敛半径．解记fn（z）＝（－i）n－1（2n－1）〖〗2nz2n－1，则limn→∞fn＋1（z）〖〗 fn（z）＝limn→∞（2n＋1）2n｜z｜2n＋1〖〗（2n－1）2n＋1｜z｜2n－1＝1〖〗2｜z｜2当1〖〗2｜z｜2＜1时，即｜z｜＜2时，幂级数绝对收敛；当1〖〗2｜z｜2＞1时，即｜z｜＞2时，幂级数发散．所以，该幂级数的收敛半径为R＝2．4．2．2幂级数的运算和性质和实函数的幂级数类似，复变函数的幂级数也可以进行加、减、乘等运算．设幂级数∑∞〖〗n＝0anzn＝S1（z）， ∑∞〖〗n＝0bnzn＝S2（z），收敛半径分别为R1、 R2，则∑∞〖〗n＝1anzn±∑∞〖〗n＝1bnzn＝∑∞〖〗n＝0（an±bn）zn＝S1（z）±S2（z），｜z｜＜R（4．5）∑∞〖〗n＝1anzn∑∞〖〗n＝1bnzn＝∑∞〖〗 n＝0（anb0＋an－1b1＋…＋a0bn）zn＝S1（z）S2（z）， ｜z｜＜R（4．6）其中，R＝min（R1，R2）．复变函数的幂级数还可以进行复合运算．设h（z）在D内解析，且｜h（z）｜＜R， z∈D，则f（h（z））在D内解析，且f（h（z））＝∑∞〖〗n＝0anhn（z）， z∈D．在f（z）的幂级数展开中，可以用z的一个函数h（z）去代换展开式中的z，这在后面解析函数的级数展开中经常用到．幂级数∑∞〖〗n＝oanzn在其收敛圆｜z｜＜R内，还具有如下性质：（1） 它的和函数S（z）＝∑∞〖〗n＝0anzn在｜z｜＜R内解析；（2） 在收敛圆内幂级数可逐项求导，即S′（z）＝∑∞〖〗n＝1nanzn－1， ｜z｜＜R；（4．7）（3）在收敛圆内幂级数可逐项积分，即∫CS（z）dz＝∑∞〖〗n＝0∫Canzndz＝∑∞〖〗n＝0an〖〗n＋1zn＋1，（4．8）｜z｜＜R，C 为｜z｜＜R内的简单曲线．

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Feng Zhang, Jinting Wang. Equilibrium Joining Probabilities in Observable Queues with General Service and Setup Times, Journal of Industrial and Management Optimization, accepted for publication, 2013 (SCI) Bin Liu, Jinting Wang and Y. Zhao, Tail Asymptotics of the Waiting Time and the Busy Period for the M/G/1/K Queues with Subexponential Service Times, Queueing Systems, in press, 2013. (SCI) Jinting Wang, Huang Y. and Do ., A single-server discrete-time queue with correlated positive and negative customer arrivals, Applied Mathematical Modelling, in press, 2013. (SCI) Anbazhagan N. and JintingWang, Base stock policy with retrial demands, Applied Mathematical Modelling, Volume 37, 4464-4473, 2013 (SCI) Feng Zhang, Jinting Wang,Performance analysis of the retrial queues with finite number of sources and service interruptions, Journal of the Korean Statistical Society, Volume 42, Issue 1, 117-131, 2013 (SCI) Jinting Wang, Discrete-time Geo/G/1 retrial queues with general retrial time and Bernoulli vacation, Journal of Systems Science and Complexity, 25: 504–513, 2012. (SCI) Jungang Li, Jinting Wang, Fuzzy set-valued stochastic Lebesgue integral, Fuzzy Sets and Systems, 200: 48-64, 2012. (SCI) Feng Zhang, Jinting Wang, Bin Liu, On the optimal and equilibrium retrial rates in an unreliable retrial queue with vacations,Journal of Industrial and Management Optimization, Volume 8, Number 4: 861-875, 2012. (SCI) B. Liu, J. Wang and Y. Zhao, On the Conditional Probability of a Successful Retrial in Retrial Queues, INFOR, 49(3), 171-182, 2011. (SCI, EI) Jinting Wang,Nan Wang and . Alfa, Discrete-time GI/G/1 retrial queues with time-controlled vacation policies, Acta Mathematica Applicatae Sinica English Series, in press, 2012. (SCI) Jinting Wang, Huang Y. and Dai Z,A discrete-time on-off source queueing system with negative customers,Computers & Industrial Engineering, 61(4), 1226-1232, 2011. (SCI, EI) Jinting Wang, Zhang, F. Equilibrium analysis of the observable queues with balking and delayed repairs, Applied Mathematics and Computation, 218(6), 2716-2729, 2011. (SCI, EI) Jinting Wang, Zhao, L. Zhang, F. Analysis of the finite source retrial queues with server breakdowns and repairs, Journal of Industrial and Management Optimization, 7(3), 655-676, 2011. (SCI) Jinting Wang,Li, J. Analysis of the M/G/1 queues with second multi optional service and unreliable server, Acta Mathematica Applicatae Sinica English Series, 26(3): 353-368, 2010. (SCI) Jinting Wang,Zhou, P. A Batch Arrival Retrial Queue with Starting Failures, Feedback and Admission Control,Journal of Systems Science and Systems Engineering, 19(3): 306-320, 2010. (SCI, EI) Jinting Wang,Li, J. A single-server retrial queue with general retrial times and two-phase service, Journal of Systems Science & Complexity, 22 (2), 291-302, 2009. (SCI, EI) Jinting Wang,Zhang, P. A discrete-time retrial queue with negative customers and unreliable server,Computers & Industrial Engineering,volume 56, issue 4, , 2009 (SCI, EI) Jinting Wang and Zhang, P. A single-server discrete-time retrial G-queue with server breakdowns and repairs, Acta Mathematica Applicatae Sinica English Series, 25(4): 675-684, 2009. (SCI). Jinting Wang, Cai J. and Alfa channel model for wireless communications: Finite-State Phase-Type Semi-Markov channel model, Proceedings ofthe IEEE International Conference on Communication (ICC 2008), pp. 4461-4465. 2008. (EI) Jinting Wang,On the single server retrial queue with priority subscribers and server breakdowns, Journal of Systems Science and Complexity, 21(2): pp. 304-315, 2008. (SCI, EI) Jinting Wangand Li, J. A repairable M/G/1 retrial queue with Bernoulli vacation and two-phase service, Quality Technology and Quantitative Management , , 179-192, 2008 Jinting Wang,Liu, B and Li, J. Transient analysis of an M/G/1 retrial queue subject to disasters and server failures. European Journal of Operational Research,189(3): 1118-1132, 2008. (SCI, EI, ISTP) Jinting Wang, Zhao, Geo/G/1 retrial queue with general retrial times and starting failures, Mathematical and Computer Modelling, 45(7/8), April 2007, . (SCI, EI) Jinting Wang, Zhao, Q. A discrete-time Geo/G/1 retrial queue with starting failures and second optional service, Computers and Mathematics with Applications, 53(1), 115-127, 2007. (SCI, EI) Jinting Wang, Reliability analysis of M/G/1 queues with general retrial times and server breakdowns, Progress in Natural Science, 2006 (5): 464-473 (SCI, EI) Jinting Wang, An M/G/1 queue with second optional service and server breakdowns, Computers and Mathematics with Applications, , Issue 10-11, pp. 1713-1723, 2004. (SCI, EI) Jinting Wang, Srinivasan, R. Unreliable production-inventory model with hyper-exponential renewal demand processes, Applied Mathematics and Computation, Vol. 164, . (SCI, EI) Jinting Wang,Cao, J. and Liu, B. Unreliable production-inventory system with superposition of KPoisson demand arrival processes, Acta Mathematicae Applicatae Sinica, , , 47-55, 2003. Liu, W., Jinting Wang. A strong limit theorem on gambling systems, Journal of Multivariate Analysis, , , 262-273, 2003. (SCI) Jinting Wang,Cao, J. and Liu, of a production-inventory system with Erlang demand arrival process, Journal of System Sciences and Complexity, Vol. 16, , 184-190, 2003. Jinting Wang,Cao, J. and Liu, B. Unreliable production-inventory model with a two-phase Erlang demand arrival process, Computers and Mathematics with Applications, Vol. 43: 1-13, 2002. (SCI, EI) Jinting Wang,Cao, J. and Li, Analysis of the Retrial Queue with Server Breakdowns and Repairs. Queueing Systems, Vol. 38, pp. 363-380. 2001. (SCI, EI) Jinting Wang,Cao, limiting distribution of the residual discrete-time Markovian repairable systems, Operations Research Transactions, , , 27-35, 2002