1、1?E9E?!?Vg?(J?1.E?9E?5 i=1.E8 C=z|z=x+iy,x,y R1K8 R C.3 C?5!?$?E8?(?KE8?E?w?R*?5.?z1=a1+b1i,z2=a2+b2i,a1,b1,a2,b2 R.1.1z1 z2=(a1 a2)+(b1 b2)i,z1 z2=(a1+b1i)(a2+b2i)=(a1a2 b1b2)+i(a1b2+a2b1),z1z2=a1+b1ia2+b2i=a1a2+b1b2a22+b22+ia2b1 a1b2a22+b22(z26=0).E?(?z1,z2 C,z1=a1+ib1,z2=a2+ib2z1?z2?l|z1 z2|=p(a1
2、 a2)2+(b1 b2)2l?E?E?m3?m?Vg?3E?4?O?.E8!?:89?8?AX.E?n?L?z=|z|(cosArgz+i sinArgz)z1=|z1|cosArgz1+i sinArgz1,z2=|z2|cosArgz2+i sinArgz2.z1 z2=|z1|z2|cos(Argz1+Argz2)+i sin(Argz1+Argz2),z1z2=|z1|z2|cos(Argz1 Argz2)+i sin(Argz1 Argz2).1.2 x iy?E z=x+iy?P?zz?z?p?z?z uEk|z|=|z|,Argz=Argz.?$?k?z1+z2=z1+z2,z
3、1z2=z1 z2,z=z,z1z2!=z1z2,zz=|z|2,?keL?z=x+iyK z=z+z2,u=12i(z z)uz!y?“E z=|z|cosArgz+i sinArgz?m?$?zm=|z|mcos(mArgz)+i sin(mArgz),m Z,?8z1n=+np|z|cos 1nArgz!+i sin 1nArgz!#n?u?u 2?.2.E?*E8 C3n?m?S:x2+y2+u2=1r XOY w?z=x+iy?N(0,0,1)?4?E C?S N?A?N?XOY?:A(x,y,0)?:A0(x0,y0,u0)?k x0=z+z|z|2+1,y0=z zi(|z|2+
4、1),u0=|z|2 1|z|2+1,5E?n?:?N Adn?:?E?:P C=CSl?C?S?A.3.E?1.3?GeVg(1)?(x0,r)=z|z z0|r,z C.(2)?%?o(x0,r)=z|0|z z0|0o(x0,r)TE 6=z0 E?:.(5)S:?8 Ez0 E,(x0,r)(x0,r)E z0 E?S:.(6).:?8 Ez0 Cr 0,(z0,r)TE 6=,(z0,r)TEC6=z0?E?.:.1(7).?d8 E?.:|?8?E?.P?E.(8)4?E=ESE.(9)?:?E,r 0?(,r)TE=?E?:.(10)m8?8 E?:?S:E?m8.(11)48?
5、8 E?8m8 E?48.(12)k.8?r 0?E (0,r)E?k.8.(13).8?k.8?.8.(14);8?k.48?;8.(15)8 D?5?8 D?k?:?k?.(16)?m8?.(17)?z=z(t)=Rez(t)+iImz(t),a 6 t 6 b?:z?;,?.(18)Y?e?z=z(t),t a,b,Rez(t),Imz(t)Y.(19)?Y?Y?z=z(t),t a,b?z1,z2(a,b),z16=z2,z(t1)6=z(t2).(20)?4?Y?z=z(t),t a,b?z(a)=z(b).(21)?D?D S?4?S?z?:?u D.(22)?.?!K?S?1.1
6、 i?og?.)?i=cos 2+2k!+i sin 2+2k!4i=cos 2+2k4!+i sin 2+2k4!,k=0,1,2,3Kcos8+i sin8,cos 8+2!+i sin 8+2!,cos 8+!+i sin 8+!,cos 8+32!+i sin 8+32!.5?1?f(z)=ez?.“ei=cos+i sindK?|E z?L?“z=|z|eiArgzO?i=ei(2+2k)i?og?o?ei8,ei(8+2),ei(8+),ei(8+32)=ei8,iei8,ei8,iei8.1.2 fi?z2+z+1=0 z11+z7+z3?.)?z3 1=(z 1)(z2+z+1
7、)dfi?z2+z+1=0?z3 1=0=z?ng?l?z11=z2,z7=z,z3=1d z11+z7+z3=z2+z+1=0.1.3?z?u1?n g?1+z+z2+zn1?.)?zn=1,z 6=1K 1+z+z2+zn1=1 zn1 z=0.1.4 y?e z X?a0wn+a1wn1+an1w+an=0?K z?.y?g,nk zn=(z)nq?aj?=?aj=ajdfi?z?a0wn+a1wn1+an1w+an=0?ka0zn+a1zn1+an1z+an=0?“?$?(0=0)a0zn+a1zn1+an1z+an=0u a0(z)n+a1(z)n1+an1z+an=0=z?a0wn
8、+a1wn1+an1w+an=0?.21.5 cos?sin L cos5.)?cos5=Re(cos5+i sin5)=Re(cos+i sin)5=Re(cos5+5i cos4sin10cos3sin210i cos2sin3+5cossin4+i sin5)=cos5 10cos3sin2+5cossin4theta,?sin5=5cos4sin 10cos2sin3+sin5.1.6 e 6=0y1+cos+cos2+cosn=sin2+sin?n+12?2sin2,sin+sin2+sinn=cos2 cos?n+12?2sin2.y?-z=cos+i sin,zn=cosn+i
9、sinnq?1+z+z2+zn=1 zn+11 z=1+(cos+i sin)+(cos2+i sin2)+(cosn+i sinn)=1 cos(n+1)+i sin(n+1)1 (cos+i sin)l?(1+cos+cos2+cosn)+i(sin+sin2+sinn)=1 (cos(n+1)+i sin(n+1)(1 cos+i sin)(1 cos)2+sin2=f()2(1 cos)=f()4sin2 2?f()=1cos(n+1)(1cos)+sin(n+1)sin+isin(1cos(n+1)(1cos)sin(n+1)?n?=?1+cos+cos2+cosn=Re(f()4s
10、in2 2)=sin2+sin?n+12?2sin2sin+sin2+sinn=Im(f()4sin2 2)=cos2 cos?n+12?2sin2.1.7 8 E=(i,12i,23i,n 1ni,)i E?:?:?:.1.8 8 E=(12+i2,23+32i,34+43i,n 1n+nn 1i,)z?:?E?:1+i?:?u E.1.9?v?0 arg(z 1)4,2 Rez 3?:z|?:8?o”)?X?K?k.?-6?123xy01.10?v?0 argz iz+i 0?2xx2+(y+1)2 0?0 2xx2+y2 1 1x (2)2?:8(x+1)2+y2=2?3?k:|?8?.
11、?.41?EC?EC.?!?Vg?(J?2.1?EE C?:8XJk?K f?z=x+iy Ew=u+iv C?AK f E?(?EC.5?w=u(x,y)+iv(x,y)=?EC?du?C.P A=f(E)=f(x)|x E?f(x)?8XJ f E?A?V?K E?A?A.2.2?f(z)38 E?(z0 E?:?EXJ 0,=()0?x EToS(z0,)?|f(z)|0,=(A)0?x oS(z0,)TE?|f(z)|AK f(z)z“Cu z0?P?limzz0zEf(z)=.2.4?w=f(z)3.?E?(?k?EXJ 0,=()0?z E,|z|?|f(z)|0,=(A)0?z
12、E,|z|?|f(z)|AK f(x0 z“Cu?P?limzzEf(z)=.2.6?w=f(z)=u(x,y)+iv(x,y)38 E?(?E?:z0 E XJ limzz0zEf(z)=f(z0)K w=f(z)3 z0:u8 EY.5 1?f(z)3 z0=x0+iy0Y?=?limxx0yy0u(x,y)=u(x0,y0),limxx0yy0v(x,y)=v(x0,y0).5 2?e f(z)3?E?z?:YK f(z)3 E?Y.2.7?f(z)38 E?(XJ 0?3?k?z?=()0?z0,z00 E?|z0 z00|?|f(z0)f(z00)|K f(z)3 E?Y.n2.1
13、f(z)?k.4?E?YKk:(1)f(z)3 E?Y(2)f(z)3 E?k.,(3)f(z)3 E?.2.8?f(z)?D S(?z0 DXJ limzz0zEf(z)f(z0)z z0?3?uE K f(z)3 z0:?k?P?f0(z0)=.2.9XJ f(z)3?D Sz?:?K f(z)3?D S)?.XJf(z)3 z0?S)?K f(z)3 z0:)?.XJ f(z)3?G S)?4?D GK f(z)34?D?)?.5?Cu?!?!9E?$?2?EC.n2.2?f(z)=u(x,y)+iv(x,y)3?D S(o f(z)3:z=x+iy D?7?3 z=x+iyu(x,y)
14、9 v(x,y)?vux=vy,uy=vx.5n2.3?f(z)=u(x,y)+iv(x,y)3?D S(o f(z)3?D S)?7?u(x,y)?v(x,y)3 D S?3 D S?vux=vy,uy=vx.5?3 f(z)k?e?f0(z)=a+ib=ux+ivx=vy iuy.2.10?ez?z=x+iy Cf(z)=ez=ex+iy=ex(cosy+i siny)ke5?(1)x R,f(x)=ex.(2)f(z)3 C?)?dezdz=ez.(3)z1,z2 C,f(z1+z2)=f(z1)f(z2).(4)z C,f(z)6=0.(5)f(z)=ezk 2i,f(z+2i)=f(
15、z).(6)N?5?f(z)=ezr?/?z B=z|z C,R,Imz +2 V?w?0 9?argw=?.5?f(z)=ez?.2.11 Lnz?z C,z 6=0?w=ez?w=Lnz=ln|z|+iArgz.du Argz=argz+2kd w=Lnz=ln|z|+i argz+2ki=lnz+2ki(k Z)p?lnz?Lnz?argz 6 l w=Lnz?w?.(1)z1,z2 Cz1,z2?kXe$?5?Ln(z1z2)=Lnz1+Lnz2,Ln z1z2!=Lnz1 Lnz2.(2)u w=Lnz?zflK.d w=Lnz=ln|z|+iArgz Lnz?Argz=argz+2
16、k,k Z?|:0?.?Y?K1?3 C K1?Lnz)?Y|?Y|*?e?0?Lnz?|:.(3)3z?Lnz?z?S?z?:?kdlnzdz=1zl?z?Y|?)?|.(4)w=Lnz?N?5?|w=ln|z|+i argz+2ki(argz )r z?V?w?d(2k 1)Imw 1K z n?nJ?z?.(2)d z=eLnz=elnz e2kiu Lnz?A?Y|Sdzdz=delnzdz=1zelnz=z1.(3)?w=z1n=nz?“3E?K?)0?D SkA?)?|?w=np|z|ein(arg z+2k)(argz )=np|z|earg zni e2kni(k=0,1,2,n 1).(4)w=z?N?5?z?0 ,2e3 z?0?D?z3DS?)?|?DS?/A?0 argz 0,sin2z 0,|cosz|6 1,|sinz|6 1?tanz=sinzcosz,cotz=coszsinz,secz=1cosz,cscz=1sinz.2.14u?n?w=Arctanzdz=tanw(Arctanz=12iLn(z i)Ln(z+i)+ie(Ln(z i)9 Ln(z
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