北大数学物理方法(A)-复变函数教案06无穷级数

Outline18ùá?êÔnÆêÆÔn{‘§|2007cSC.S.Wu18ùá?êOutlineùLJ:1Eê?êEê?ê•Âñ†uÑýéÂñ?ê2¼ê?ê¼ê?êÂñ5¼ê?ꘗÂñ5¹ëþ‡~È©)Û53˜?êAbel½nÂñ†ÂñŒ»C.S.Wu18ùá?êOutlineùLJ:1Eê?êEê?ê•Âñ†uÑýéÂñ?ê2¼ê?ê¼ê?êÂñ5¼ê?ꘗÂñ5¹ëþ‡~È©)Û53˜?êAbel½nÂñ†ÂñŒ»C.S.Wu18ùá?êOutlineùLJ:1Eê?êEê?ê•Âñ†uÑýéÂñ?ê2¼ê?ê¼ê?êÂñ5¼ê?ꘗÂñ5¹ëþ‡~È©)Û53˜?êAbel½nÂñ†ÂñŒ»C.S.Wu18ùá?êComplexSeriesSeriesofComplexFunctionsPowerSeriesReferencesÇÂÁ§5êÆÔn{6§§4.1—4.5ù&œ§5êÆÔn{6§§3.1,3.2nÎ!X1Á§5êÆÔn{6§§3.1,3.2C.S.Wu18ùá?êComplexSeriesSeriesofComplexFunctionsPowerSeriesReferencesÇÂÁ§5êÆÔn{6§§4.1—4.5ù&œ§5êÆÔn{6§§3.1,3.2nÎ!X1Á§5êÆÔn{6§§3.1,3.2C.S.Wu18ùá?êComplexSeriesSeriesofComplexFunctionsPowerSeriesReferencesÇÂÁ§5êÆÔn{6§§4.1—4.5ù&œ§5êÆÔn{6§§3.1,3.2nÎ!X1Á§5êÆÔn{6§§3.1,3.2C.S.Wu18ùá?êComplexSeriesSeriesofComplexFunctionsPowerSeriesá?êá?ê§AO´˜?꧴)ۼꁭ‡Lˆ/ªƒ˜NõмêÚAϼêÑ´^˜?ê½ÂEC¼ê?ênØÚ¢C¼ê'µVgڐ{ÉÓC.S.Wu18ùá?êComplexSeriesSeriesofComplexFunctionsPowerSeriesá?êá?ê§AO´˜?꧴)ۼꁭ‡Lˆ/ªƒ˜NõмêÚAϼêÑ´^˜?ê½ÂEC¼ê?ênØÚ¢C¼ê'µVgڐ{ÉÓC.S.Wu18ùá?êComplexSeriesSeriesofComplexFunctionsPowerSeriesá?êá?ê§AO´˜?꧴)ۼꁭ‡Lˆ/ªƒ˜NõмêÚAϼêÑ´^˜?ê½ÂEC¼ê?ênØÚ¢C¼ê'µVgڐ{ÉÓC.S.Wu18ùá?êComplexSeriesSeriesofComplexFunctionsPowerSeriesá?êá?ê§AO´˜?꧴)ۼꁭ‡Lˆ/ªƒ˜NõмêÚAϼêÑ´^˜?ê½ÂEC¼ê?ênØÚ¢C¼ê'µVgڐ{ÉÓC.S.Wu18ùá?êComplexSeriesSeriesofComplexFunctionsPowerSeriesComplexSeries:Convergency&DivergencyAbsolutelyConvergencyùLJ:1Eê?êEê?ê•Âñ†uÑýéÂñ?ê2¼ê?ê¼ê?êÂñ5¼ê?ꘗÂñ5¹ëþ‡~È©)Û53˜?êAbel½nÂñ†ÂñŒ»C.S.Wu18ùá?êComplexSeriesSeriesofComplexFunctionsPowerSeriesComplexSeries:Convergency&DivergencyAbsolutelyConvergencyEê?êu0+u1+u2+···+un+···=∞Pn=0uná?êÂñ†uÑXJ?êÜ©ÚSn=u0+u1+u2+···+un¤¤S{Sn}Âñ§K¡?êPunÂñ§S{Sn}4S=limn→∞Sn§¡?êPunÚ∞Pn=0un=limn→∞SnÄK§?êPun´uÑC.S.Wu18ùá?êComplexSeriesSeriesofComplexFunctionsPowerSeriesComplexSeries:Convergency&DivergencyAbsolutelyConvergencyEê?êu0+u1+u2+···+un+···=∞Pn=0uná?êÂñ†uÑXJ?êÜ©ÚSn=u0+u1+u2+···+un¤¤S{Sn}Âñ§K¡?êPunÂñ§S{Sn}4S=limn→∞Sn§¡?êPunÚ∞Pn=0un=limn→∞SnÄK§?êPun´uÑC.S.Wu18ùá?êComplexSeriesSeriesofComplexFunctionsPowerSeriesComplexSeries:Convergency&DivergencyAbsolutelyConvergencyEê?êu0+u1+u2+···+un+···=∞Pn=0uná?êÂñ†uÑXJ?êÜ©ÚSn=u0+u1+u2+···+un¤¤S{Sn}Âñ§K¡?êPunÂñ§S{Sn}4S=limn→∞Sn§¡?êPunÚ∞Pn=0un=limn→∞SnÄK§?êPun´uÑC.S.Wu18ùá?êComplexSeriesSeriesofComplexFunctionsPowerSeriesComplexSeries:Convergency&DivergencyAbsolutelyConvergencyEê?êu0+u1+u2+···+un+···=∞Pn=0uná?êÂñ†uÑXJ?êÜ©ÚSn=u0+u1+u2+···+un¤¤S{Sn}Âñ§K¡?êPunÂñ§S{Sn}4S=limn→∞Sn§¡?êPunÚ∞Pn=0un=limn→∞SnÄK§?êPun´uÑC.S.Wu18ùá?êComplexSeriesSeriesofComplexFunctionsPowerSeriesComplexSeries:Convergency&DivergencyAbsolutelyConvergency-un=αn+iβn,KÜ©ÚSSn=(α0+iβ0)+(α1+iβ1)+(α2+iβ2)+···+(αn+iβn)=α0+α1+α2+···+αn
+iβ0+β1+β2+···+βn
Eê?êPunÂñ5du¢ê?êPαnÚPβnÂñ5˜‡Eê?êPunduü‡¢ê?êPαnÚPβn∞Pn=0un=∞Pn=0αn+i∞Pn=0βnC.S.Wu18ùá?êComplexSeriesSeriesofComplexFunctionsPowerSeriesComplexSeries:Convergency&DivergencyAbsolutelyConvergency-un=αn+iβn,KÜ©ÚSSn=(α0+iβ0)+(α1+iβ1)+(α2+iβ2)+···+(αn+iβn)=α0+α1+α2+···+αn
+iβ0+β1+β2+···+βn
Eê?êPunÂñ5du¢ê?êPαnÚPβnÂñ5˜‡Eê?êPunduü‡¢ê?êPαnÚPβn∞Pn=0un=∞Pn=0αn+i∞Pn=0βnC.S.Wu18ùá?êComplexSeriesSeriesofComplexFunctionsPowerSeriesComplexSeries:Convergency&DivergencyAbsolutelyConvergency-un=αn+iβn,KÜ©ÚSSn=(α0+iβ0)+(α1+iβ1)+(α2+iβ2)+···+(αn+iβn)=α0+α1+α2+···+αn
+iβ0+β1+β2+···+βn
Eê?êPunÂñ5du¢ê?êPαnÚPβnÂñ5˜‡Eê?êPunduü‡¢ê?êPαnÚPβn∞Pn=0un=∞Pn=0αn+i∞Pn=0βnC.S.Wu18ùá?êComplexSeriesSeriesofComplexFunctionsPowerSeriesComplexSeries:Convergency&DivergencyAbsolutelyConvergency-un=αn+iβn,KÜ©ÚSSn=(α0+iβ0)+(α1+iβ1)+(α2+iβ2)+···+(αn+iβn)=α0+α1+α2+···+αn
+iβ0+β1+β2+···+βn
Eê?êPunÂñ5du¢ê?êPαnÚPβnÂñ5˜‡Eê?êPunduü‡¢ê?êPαnÚPβn∞Pn=0un=∞Pn=0αn+i∞Pn=0βnC.S.Wu18ùá?êComplexSeriesSeriesofComplexFunctionsPowerSeriesComplexSeries:Convergency&DivergencyAbsolutelyConvergency-un=αn+iβn,KÜ©ÚSSn=(α0+iβ0)+(α1+iβ1)+(α2+iβ2)+···+(αn+iβn)=α0+α1+α2+···+αn
+iβ0+β1+β2+···+βn
Eê?êPunÂñ5du¢ê?êPαnÚPβnÂñ5˜‡Eê?êPunduü‡¢ê?êPαnÚPβn∞Pn=0un=∞Pn=0αn+i∞Pn=0βnC.S.Wu18ùá?êComplexSeriesSeriesofComplexFunctionsPowerSeriesComplexSeries:Convergency&DivergencyAbsolutelyConvergency?êÂñ5§´^§Ü©ÚSÂñ5½Â©Ïd§ŠâSÂñ¿‡^‡§Œ±Ñ?êÂñ¿‡^‡Cauchy¿‡^‡SÂñCauchy¿‡^‡?¿‰½ε0§∃êN(ε)0§∀êp§k|zN+p−zN|ε?êÂñCauchy¿‡^‡?¿‰½ε0§∃ên§∀êp§k|un+1+un+2+···+un+p|εC.S.Wu18ùá?êComplexSeriesSeriesofComplexFunctionsPowerSeriesComplexSeries:Convergency&DivergencyAbsolutelyConvergency?êÂñ5§´^§Ü©ÚSÂñ5½Â©Ïd§ŠâSÂñ¿‡^‡§Œ±Ñ?êÂñ¿‡^‡Cauchy¿‡^‡SÂñCauchy¿‡^‡?¿‰½ε0§∃êN(ε)0§∀êp§k|zN+p−zN|ε?êÂñCauchy¿‡^‡?¿‰½ε0§∃ên§∀êp§k|un+1+un+2+···+un+p|εC.S.Wu18ùá?êComplexSeriesSeriesofComplexFunctionsPowerSeriesComplexSeries:Convergency&DivergencyAbsolutelyConvergency?êÂñ5§´^§Ü©ÚSÂñ5½Â©Ïd§ŠâSÂñ¿‡^‡§Œ±Ñ?êÂñ¿‡^‡Cauchy¿‡^‡SÂñCauchy¿‡^‡?¿‰½ε0§∃êN(ε)0§∀êp§k|zN+p−zN|ε?êÂñCauchy¿‡^‡?¿‰½ε0§∃ên§∀êp§k|un+1+un+2+···+un+p|εC.S.Wu18ùá?êComplexSeriesSeriesofComplexFunctionsPowerSeriesComplexSeries:Convergency&Divergen