复变函数重要知识点总结

1˜ÙEê9E²¡˜!ÄVgÚ̇(Jµ1.Eê9E²¡µ5½i=√−1.Eê8C={z|z=x+iy,x,y∈R1}§K¢ê8R⊂C.3Cþ5½\!~!¦!Ø$Ž§‰Eê8˜‡“ê(§KEê8¤EꍣŒw¤´¢êR*Ü5¤.z1=a1+b1i,z2=a2+b2i,a1,b1,a2,b2∈R.½Â1.1z1±z2=(a1±a2)+(b1±b2)i,z1·z2=(a1+b1i)(a2+b2i)=(a1a2−b1b2)+i(a1b2+a2b1),z1z2=a1+b1ia2+b2i=a1a2+b1b2a22+b22+ia2b1−a1b2a22+b22(z26=0).EêþÚ?˜«ÿÀ(µ?z1,z2∈C,z1=a1+ib1,z2=a2+ib2§z1†z2ål|z1−z2|=p(a1−a2)2+(b1−b2)2§lEꍤEm§3ù‡˜mŒ±½ÂVg§3E²¡þÚ?4ŠÐO.Eê8!‘²¡:89²¡þ•þ8¤˜‡˜˜éA'X.EênL«µz=|z|(cosArgz+isinArgz)§z1=|z1|[cosArgz1+isinArgz1],z2=|z2|[cosArgz2+isinArgz2].z1·z2=|z1||z2|[cos(Argz1+Argz2)+isin(Argz1+Argz2)],z1z2=|z1||z2|[cos(Argz1−Argz2)+isin(Argz1−Argz2)].½Â1.2¡x−iyEêz=x+iy
Ýê§Pz£zÚz¡p
ÝꤧzÚz'uE²¡¢¶é¡§k|z|=|z|,Argz=−Argz.
Ýê$Žkµz1+z2=z1+z2,z1z2=z1·z2,z=z,z1z2!=z1z2,zz=|z|2,„Œ±k±eL«µz=x+iy§Kz=z+z2,u=12i(z−z)§^uz{!y²˜“ꪶEêz=|z|[cosArgz+isinArgz]¦˜Úm$Žµzm=|z|m[cos(mArgz)+isin(mArgz)],m∈Z,£ê8¤z1n=+np|z|cos1nArgz!+isin1nArgz!#§nŒuu2ê.2.E¥¡Ú*¿Eê8C∞3n‘˜m¥§Š¥¡S:x2+y2+u2=1§rXOY²¡wŠz=x+iy²¡§¥¡þN(0,0,1)¡¥4§ŒïáåE²¡C†S−{N}˜‡˜˜éAµŠëN†XOY²¡þ?˜:A(x,y,0)†‚§†‚†¥¡:´A0(x0,y0,u0)§Òkx0=z+z|z|2+1,y0=z−zi(|z|2+1),u0=|z|2−1|z|2+1,¿5½E²¡þ˜‡nŽ:†NéA§¡dnŽ:E²¡þá:∞§PC∞=CS{∞}§lC∞†Sïᘇ˜˜éA.3.E²¡ÿÀµ½Â1.3‡ÙG±eVg£½Â¤(1)µ∪(x0,r)={z||z−z0|r,z∈C}.(2)%µo∪(x0,r)={z|0|z−z0|r,z∈C}.(3)4µ∪(x0,r)={z||z−z0|6r,z∈C}.(4)à:µ8E§z0∈C§e∀r0§o∪(x0,r)TE6=∅§¡z0´Eà:.(5)S:µ8E§z0∈E,∃∪(x0,r)§¦∪(x0,r)⊂E§¡z0´ES:.(6).:µ8E§z0∈C§∀r0,∪(z0,r)TE6=∅,∪(z0,r)TEC6=∅§¡z0E.:.1(7).µd8EÜ.:¤|¤8§¡E.§P∂E.(8)4µE=∂ESE.(9)á:µα∈E,∃r0§¦∪(α,r)TE={α}§¡αEá:.(10)m8µ8E:Ü´S:§¡Em8.(11)48µ8EÖ8´m8§¡E48.(12)k.8µ∃r0§¦E⊂∪(0,r)§¡Ek.8.(13)Ã.8µØ´k.8§¡Ã.8.(14);8µk.48¡;8.(15)8DëÏ5µ8D¥?ü‡k:Œ±^k‡Ä—ƒò‚ë.(16)«µëÏm8¡«.(17)­‚µz=z(t)=Rez(t)+iImz(t),a6t6b:zƒ;,§¡­‚.(18)ëY­‚µe­‚z=z(t),t∈[a,b],Rez(t),Imz(t)ëY.(19){üëY­‚µëY­‚z=z(t),t∈[a,b]…∀z1,z2∈(a,b),z16=z2,z(t1)6=z(t2).(20){ü4­‚µ{üëY­‚z=z(t),t∈[a,b]…z(a)=z(b).(21)üëÏ«Dµ«DS?Û{ü4­‚S«¥z˜:ÑáuD.(22)õëÏ«µØ´üëÏ«§¡õëÏ«.!~K†öSµ1.1¦iogŠ.)µÏi=cosπ2+2kπ!+isinπ2+2kπ!§¤±4√i=cosπ2+2kπ4!+isinπ2+2kπ4!,k=0,1,2,3Kcosπ8+isinπ8,cosπ8+π2!+isinπ8+π2!,cosπ8+π!+isinπ8+π!,cosπ8+32π!+isinπ8+32π!.5µ1ٽê¼êf(z)=ezÚî.úªeiθ=cosθ+isinθ§dKŒ|^EêzêLˆªz=|z|eiArgzOŽÏi=ei(π2+2kπ)§¤±iogŠ£o‡¤´µeiπ8,ei(π8+π2),ei(π8+π),ei(π8+32π)=eiπ8,ieiπ8,−eiπ8,−ieiπ8.1.2®z2+z+1=0§¦z11+z7+z3ƒŠ.)µz3−1=(z−1)(z2+z+1)§d®z2+z+1=0§z3−1=0=z´˜‡ngü Šlz11=z2,z7=z,z3=1§Ïdz11+z7+z3=z2+z+1=0.1.3z´?¿˜‡Øu1ngü Š§¦1+z+z2+···+zn−1Š.)µÏzn=1,z6=1§K1+z+z2+···+zn−1=1−zn1−z=0.1.4y²µez´¢Xꐧa0wn+a1wn−1+···+an−1w+an=0Š§Kz´ÙŠ.yµÏé?¿g,ên§kzn=(z)n§qϏaj¢ê…=aj=ajd®z§a0wn+a1wn−1+···+an−1w+an=0Š§¤±ka0zn+a1zn−1+···+an−1z+an=0éþªüàӞ
Ý$Ž§(0=0)a0zn+a1zn−1+···+an−1z+an=0u´a0(z)n+a1(z)n−1+···+an−1z+an=0=z´a0wn+a1wn−1+···+an−1w+an=0Š.21.5^cosθ†sinθL«cos5θ.)µcos5θ=Re(cos5θ+isin5θ)=Re(cosθ+isinθ))5=Re(cos5θ+5icos4θsinθ−10cos3θsin2θ−10icos2θsin3θ+5cosθsin4θ+isin5θ)=cos5θ−10cos3θsin2θ+5cosθsin4theta,Ӟµsin5θ=5cos4θsinθ−10cos2θsin3θ+sin5θ.1.6eθ6=0§Áy²1+cosθ+cos2θ+···+cosnθ=sinθ2+sinn+12
θ2sinθ2,sinθ+sin2θ+···+sinnθ=cosθ2−cosn+12
θ2sinθ2.yµ-z=cosθ+isinθ,zn=cosnθ+isinnθqϏ1+z+z2+···zn=1−zn+11−z=1+(cosθ+isinθ)+(cos2θ+isin2θ)+···+(cosnθ+isinnθ)=1−[cos(n+1)θ+isin(n+1)θ]1−(cosθ+isinθ)l(1+cosθ+cos2θ+···+cosnθ)+i(sinθ+sin2θ+···+sinnθ)=[1−(cos(n+1)θ+isin(n+1)θ)](1−cosθ+isinθ)(1−cosθ)2+sin2θ=f(θ)2(1−cosθ)=f(θ)4sin2θ2Ù¥f(θ)=[1−cos(n+1)θ](1−cosθ)+sin(n+1)θsinθ+i[sinθ(1−cos(n+1)θ−(1−cosθ)sin(n+1)θ]n=Œ1+cosθ+cos2θ+···+cosnθ=Re(f(θ)4sin2θ2)=sinθ2+sinn+12
θ2sinθ2sinθ+sin2θ+···+sinnθ=Im(f(θ)4sin2θ2)=cosθ2−cosn+12
θ2sinθ2.1.78E=(i,12i,23i,···,n−1ni,···)¥§i´Eà:§Ù{ˆ:þá:.1.88E=(12+i2,23+32i,34+43i,···,n−1n+nn−1i,···)¥z˜:Ñ´Eá:§1+i´§à:ØáuE.1.9÷v^‡0arg(z−1)π4,2Rez3:z¤|¤:8´Ÿoº)µXãÒKÜ©§´k.üëÏ«-6123xy01.10÷v^‡0argz−iz+iπ4:z¤|¤:8´Ÿoº)µÏz−iz+i=x2+y2−1x2+(y+1)2+i−2xx2+(y+1)2d®x2+y2−1x2+(y+1)20…−2xx2+(y+1)20Ú0−2xx2+y2−113n§x2+y21x0(x+1)2+y2(√2)2¤¦:8´(x+1)2+y2=2 ܅3†Œ²¡¤k:|¤8ܧ´üëÏÃ.«.41ÙEC¼êÙ^‡©{ïÄEC¼ê.˜!ÄVgÚ̇(Jµ½Â2.1E´E²¡Cþ:8§XJk˜‡{Kf§¦∀z=x+iy∈E§∃w=u+iv∈C†ÙéA§K¡f´Eþ(½EC¼ê.5µw=u(x,y)+iv(x,y)=˜‡EC¼êduü‡¢C¼ê.PA=f(E)={f(x)|x∈E}¼êf(x)–8§XJf´EA˜‡V§KEÚA´˜‡˜˜éA.½Â2.2f(z)38Eþ(½§z0´E˜‡à:§α´˜‡E~ê§XJ∀ε0,∃δ=δ(ε)0§¦x∈EToS(z0,δ)ž§|f(z)−α|ε§K¡α´¼êf(z)xªCuz0ž4§Plimz→z0f(z)=α.5µ-α=a+ib,z∈E,f(z)==u(x,y)+iv(x,y),z=x+iy,z0=x0+iy0§u´ke(صlimz→z0z∈Ef(z)=α⇐⇒limx→x0y→y0u(x,y)=a,limx→x0y→y0v(x,y)=b.¢C¼ê¥'uü¼êÚ!!È!û±9EÜ4˜(،±í2EC¼ê.½Â2.3¼êw=f(z)3Eþ½Â§z0´8E˜‡à:§XJ∀A0,∃δ=δ(A)0§¦x∈oS(z0,δ)TEž§|f(z)|A§K¡¼êf(z)´zªCuz0žጧPlimz→z0z∈Ef(z)=∞.½Â2.4¼êw=f(z)3Ã.«Eþ(½§α´˜‡kE~ê§XJ∀ε0,∃ρ=ρ(ε)0§¦z∈E,|z|ρž§|f(z)−α|ε§K¡α´¼êf(z)zªCu∞ž4§Plimz→∞z∈Ef(z)=α.½Â2.5¼êw=f(z)3Ã.«Eþ(½§XJ∀A0,∃ρ=ρ(A)0§z∈E,|z|ρž§|f(z)|A§K¡¼êf(x0´zªCu∞žጧPlimz→∞z∈Ef(z)=∞.½Â2.6¼êw=f(z)=u(x,y)+iv(x,y)38Eþ(½…Eà:z0∈E§XJlimz→z0z∈Ef(z)=f(z0)§K¡¼êw=f(z)3z0:£éu8E¤ëY.51µf(z)3z0=x0+iy0ëY…=ü‡¢¼êlimx→x0y→y0u(x,y)=u(x0,y0),limx→x0y→y0v(x,y)=v(x0,y0).52µef(z)3«Eþz˜:ëY§K¡f(z)3EþëY.½Â2.7¼êf(z)38Eþ(½§XJ∀ε0§3†εk'†zÃ'êδ=δ(ε)0§¦z0,z00∈E…|z0−z00|䞧|f(z0)−f(z00)|ε