北京大学：《数学物理方法》精品课程电子教案（A类）第一部分 复变函数_第6讲 无穷级数

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 ∑an=∑an+i∑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 unÂñ5du¢ê?ê αnÚ PβnÂñ5 ✑ ‡Eê?ê P P unduü‡¢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 Function 级数的收敛性,是用它的部分和序列的收敛性定 义的.因此,根据序列收敛的充要条件,可以写 出级数收敛的充要条件— - cauchy充要条件 列收的C充条件 任意给定>0,三正整数N()>0,V正整数P,有

Complex Series Series of Complex Functions Power Series Complex Series: Convergency & Divergency Absolutely Convergency ?êÂñ5§´^§Ü©ÚSÂñ5½ ©Ïd§ŠâSÂñ¿‡^‡§ Œ± Ñ?êÂñ¿‡^‡ Cauchy¿‡^‡ SÂñCauchy¿‡^‡ ?¿‰½ε>0§∃êN(ε)>0§∀êp§k |zN+p − zN | < ε ?êÂñCauchy¿‡^‡ ?¿‰½ε > 0§∃ên§∀êp§k |un+1 + un+2 + · · · + un+p| < ε C. S. Wu 18ù á?ê

Series of Complex Function 级数的收敛性,是用它的部分和序列的收敛性定 义的.因此,根据序列收敛的充要条件,可以写 出级数收敛的充要条件— Cauchy充要条件 序列收敛的 Cauchy充要条件 任意给定>0,彐正整数N()>0,V正整数P,有 zNI<e 级数收敛的 Cauchy充要条 任意给定>0,正整数n,正整数P,有

Complex Series Series of Complex Functions Power Series Complex Series: Convergency & Divergency Absolutely Convergency ?êÂñ5§´^§Ü©ÚSÂñ5½ ©Ïd§ŠâSÂñ¿‡^‡§ Œ± Ñ?êÂñ¿‡^‡ Cauchy¿‡^‡ SÂñCauchy¿‡^‡ ?¿‰½ε>0§∃êN(ε)>0§∀êp§k |zN+p − zN | < ε ?êÂñCauchy¿‡^‡ ?¿‰½ε > 0§∃ên§∀êp§k |un+1 + un+2 + · · · + un+p| < ε C. S. Wu 18ù á?ê

Series of Complex Function 级数的收敛性,是用它的部分和序列的收敛性定 义的.因此,根据序列收敛的充要条件,可以写 出级数收敛的充要条件— Cauchy充要条件 序列收敛的 Cauchy充要条件 任意给定>0,彐正整数N()>0,V正整数P,有 zNI<e 级数收敛的 Cauchy充要条件 任意给定c>0,彐正整数n,V正整数p,有 Jun+1 +un+2 +undpI<e

Complex Series Series of Complex Functions Power Series Complex Series: Convergency & Divergency Absolutely Convergency ?êÂñ5§´^§Ü©ÚSÂñ5½ ©Ïd§ŠâSÂñ¿‡^‡§ Œ± Ñ?êÂñ¿‡^‡ Cauchy¿‡^‡ SÂñCauchy¿‡^‡ ?¿‰½ε>0§∃êN(ε)>0§∀êp§k |zN+p − zN | < ε ?êÂñCauchy¿‡^‡ ?¿‰½ε > 0§∃ên§∀êp§k |un+1 + un+2 + · · · + un+p| < ε C. S. Wu 18ù á?ê

Series of Complex Function 特别是,令P=1,就得到 级数收敛的必要条件 lim 0 n→o 敛级数的基本性质 在不改变求和次序的前提下,可将收敛级数并项

Complex Series Series of Complex Functions Power Series Complex Series: Convergency & Divergency Absolutely Convergency AO´§-p = 1§Ò ?êÂñ7‡^‡ lim n→∞ un = 0 Âñ?êÄ5Ÿ 3ØUC¦ÚgScJe§ŒòÂñ?꿑 u1 + u2 + u3 + u4 + · · · = (u1 + u2) + (u3 + u4) + · · · C. S. Wu 18ù á?ê