復變函數(shù)與積分變換課后習題答案[1]

1、140習題一習題一1.用復數(shù)的代數(shù)形式aib表示下列復數(shù).π43513(2)(43)711iieiiiii???????①解i4πππ2222ecosisinii442222??????????????????????????????②解：????????35i17i35i1613i7i117i17i2525?????????③解：????2i43i834i6i510i????????④解：??31i1335=iii1i222?????

2、??2.求下列各復數(shù)的實部和虛部(z=xiy)R)(zaaza???3331313.22niizi????????????????①：∵設z=xiy則??????????????22iiiiiixayxayxyaxayzazaxyaxayxay???????????????????????????∴??22222Rezaxayzaxay???????????????222Imzaxyzaxay???????????②解：設z=xiy∵?

3、???????????????323222222223223iii2ii22i33izxyxyxyxyxyxyxxyxyyxyxyxxyxyy???????????????????????∴??332Re3zxxy????323Im3zxyy??③解：∵????????????332321i31i3113133133288??????????????????????????????????????180i18???∴1i3Re12???

4、????????1i3Im02???????????④解：∵??????????2332313133133i1i328??????????????????????????????180i18???∴1i3Re12???????????1i3Im02???????????⑤解：∵????12i211iknknkknk?????????????A∴當時，，；2nk?????Rei1kn????Imi0n?當時，，21nk????Rei0n?

5、????Imi1kn??3.求下列復數(shù)的模和共軛復數(shù)123(2)(32).2iiii??????①解：2i415?????2i2i?????②解：33??33???③解：????2i32i2i32i51365????????????????????2i32i2i32i2i32i47i????????????④解：1i1i2222??????1i11i222i???????????4、證明：當且僅當時，z才是實數(shù)zz?證明：若，設，zz?

6、izxy??34035513cosπisinπi3322?????z⑶的平方根33i?解：πi42233i=6i6e22?????????????∴????1π12i44ππ2π2π4433i6e6cosisin0122kkk?????????????????∴π11i8441ππ6cosisin6e88???????????z911πi8442996cosπisinπ6e88???????????z9.設.證明：2πe2inzn??1

7、10nzz??????證明：∵∴，即2πienz??1nz?10nz??∴????1110nzzz???????又∵n≥2∴z≠1從而2110nzzz??????11.設是圓周令?:0e.izrracrzc??????:Im0zaLzb????????????????其中.求出在a切于圓周的關于的eib??L???充分必要條件.解：如圖所示因為=z:=0表示通過點a且方L?Imzab???????向與b同向的直線，要使得直線在a處與圓相