Series of Complex Function 复数级数 0+1+u2+…+un+ n=0 无穷级数的收敛与发散 如果级数的部分和 +u1+u2+…+ 所构成的序列{Sn}收敛,则称级数∑Un收敛, 序列{Sn}的极限S=imSn,称为级数∑tn的和 7→ un im 否则,级数∑un是发散的 第六讲无穷级数
Complex Series Series of Complex Functions Power Series Complex Series: Convergency & Divergency Absolutely Convergency Eê?ê u0 + u1 + u2 + · · · + un + · · · = P ∞ n=0 un á?êÂñuÑ XJ?êÜ©Ú Sn = u0 + u1 + u2 + · · · + un ¤¤S{Sn}Âñ§K¡?ê PunÂñ § S{Sn}4S = lim n→∞ Sn§¡?ê PunÚ P ∞ n=0 un = lim n→∞ Sn ÄK§?ê Pun´uÑ C. S. Wu 18ù á?ê
Series of Complex Functions 令un=an+in,则部分和序列 Sn=(ao+i)+(a1+i/1)+(a2+i2) +…+(an+ian) (0+1+2+…+
Complex Series Series of Complex Functions Power Series Complex Series: Convergency & Divergency Absolutely Convergency -un = αn + iβn, KÜ©ÚS Sn = (α0 + iβ0) + (α1 + iβ1) + (α2 + iβ2) + · · · + (αn + iβn) = α0 + α1 + α2 + · · · + αn +i β0 + β1 + β2 + · · · + βn ✑ Eê?ê P P unÂñ5 du¢ê?ê αnÚ PβnÂñ5 ✑ Eê?ê P P un duü¢ê?ê αnÚ Pβn P ∞ n=0 un = P ∞ n=0 αn + i P ∞ n=0 βn C. S. Wu 18ù á?ê
Series of Complex Functions 令un=an+in,则部分和序列 Sn=(ao+i)+(a1+i/1)+(a2+i2) +…+(an+ian) =(ao+a1+a2+…+a +(1+2+…+) 复数级数∑n的收敛性完全等价于实数级数 ∑n和∑A的收敛性
Complex Series Series of Complex Functions Power Series Complex Series: Convergency & Divergency Absolutely Convergency -un = αn + iβn, KÜ©ÚS Sn = (α0 + iβ0) + (α1 + iβ1) + (α2 + iβ2) + · · · + (αn + iβn) = α0 + α1 + α2 + · · · + αn +i β0 + β1 + β2 + · · · + βn ✑ Eê?ê P P unÂñ5 du¢ê?ê αnÚ PβnÂñ5 ✑ Eê?ê P P un duü¢ê?ê αnÚ Pβn P ∞ n=0 un = P ∞ n=0 αn + i P ∞ n=0 βn C. S. Wu 18ù á?ê
Series of Complex Functions 令un=an+in,则部分和序列 Sn=(ao+i)+(a1+i/1)+(a2+i2) +…+(an+ian) =(ao+a1+a2+…+a +(1+2+…+) 复数级数∑Un的收敛性完全等价于实数级数 ∑an和∑n的收敛性 一个复数级数∑1完全等价于两个实数级数
Complex Series Series of Complex Functions Power Series Complex Series: Convergency & Divergency Absolutely Convergency -un = αn + iβn, KÜ©ÚS Sn = (α0 + iβ0) + (α1 + iβ1) + (α2 + iβ2) + · · · + (αn + iβn) = α0 + α1 + α2 + · · · + αn +i β0 + β1 + β2 + · · · + βn ✑ Eê?ê P P unÂñ5 du¢ê?ê αnÚ PβnÂñ5 ✑ Eê?ê P P un duü¢ê?ê αnÚ Pβn P ∞ n=0 un = P ∞ n=0 αn + i P ∞ n=0 βn C. S. Wu 18ù á?ê
Series of Complex Functions 令un=an+in,则部分和序列 Sn=(ao+i)+(a1+i/1)+(a2+i2) +…+(an+ian) =(ao+a1+a2+…+a +(1+2+…+) 复数级数∑Un的收敛性完全等价于实数级数 ∑αn和∑βn的收敛性 一个复数级数∑n完全等价于两个实数级数 ∑an和∑Bn
Complex Series Series of Complex Functions Power Series Complex Series: Convergency & Divergency Absolutely Convergency -un = αn + iβn, KÜ©ÚS Sn = (α0 + iβ0) + (α1 + iβ1) + (α2 + iβ2) + · · · + (αn + iβn) = α0 + α1 + α2 + · · · + αn +i β0 + β1 + β2 + · · · + βn ✑ Eê?ê P P unÂñ5 du¢ê?ê αnÚ PβnÂñ5 ✑ Eê?ê P P un duü¢ê?ê αnÚ Pβn P ∞ n=0 un = P ∞ n=0 αn + i P ∞ n=0 βn C. S. Wu 18ù á?ê