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WuChong-shi§1.1 2 §1.1 !#$%&’()*(+,-(a,b)./0123456789(a1,b1)+(a2,b2)=(a1+a2,b1+b2),:9(a,b)(c,d)=(ac−bd,ad+bc),6;)*(+,-(a,b)=?)@A-α.BCα=(a,b)=a(1,0)+b(0,1),a;CαD,E.b;CαDFE.a=Reα,b=Imα.F#$GH7IA-D,EJFEKLMNOA-PQRSTUVFWXY#$7Z$1(1,0)(1,0)=(1,0),(1,0)(a,b)=(a,b),[\(1,0)](^,-1_‘D34ab.(1,0)=1.FWXY#$7cdei(0,1)(0,1)=(−1,0)=−1,‘f=?Fghi=(0,1).i2=−1.ij.A-αk[jBCα=a+ib.FWXY#$70(a,b)+(0,0)=(a,b),(a,b)(0,0)=(0,0),[\(0,0)](^,-0_‘D34ab.(0,0)=0.F#$lmA-α∗≡a−ibnα=a+iboCpqO(α∗)∗=α.pqA-D:rC,-O(a+ib)(a−ib)=a2+b2.F#$stA-89Du347(a+ib)−(c+id)=(a−c)+i(b−d),F#$vtA-:9Du347a+ibc+id=(a+ib)(c−id)(c+id)(c−id)=ac+bdc2+d2+ibc−adc2+d2.WuChong-shiwxyz{| 3 §1.2 }~ )@A-[j A D)@ (\ 1.1)O 1.1 α α∗A-α=a+ib [j A D)@ (\ 1.2)O 1.2 OP O0P0 D 7 )@ (¡¢ D)@£ ⁄¥ )ƒ§ _)@A-O#$¤tY'“«&7‹›fiJfl›fiKLM8OA-89 – †‡· 96(¶;C•‚ 96)O 1.3 „”»…‰ ¿ ”` ´ˆ ”` 1.4 ˜”»…‰ ¿ ”` ´ˆ ”` †‡· 96(¶•‚ 96)¯[j˘ ˙A-M¨α−β≡α+(−β)71. § βD ˚(¸ −β). ˝˛89ˇ2.—βD ˚αD ˛) .¸§ α−βOWuChong-shi§1.2 4 #$Y 7α=r(cosθ+isinθ).r,θ;CA-αD ^ ‚.r=|α|,θ=argα. .a=rcosθ,b=rsinθ.A-0D C0. ‚P=O 1.5 » ƈ ƈ»ª F#$ YŁØŒ7—˙•‚º-D .ij)@A-D ‚P )D. [j8 2πDæ - Oı (−π,π]łøD ‚œ;C ‚DßœO Y#$ 7A-pqα∗=r(cosθ−isinθ).A-:9α1=r1(cosθ1+isinθ1),α2=r2(cosθ2+isinθ2),˙ α1·α2=r1r2(cosθ1cosθ2−sinθ1sinθ2)+i(sinθ1cosθ2+cosθ1sinθ2)=r1r2[cos(θ1+θ2)+isin(θ1+θ2)].I@A-M:.f D M:. ‚M8OA- 9α1α2=α1·α∗2α2·α∗2=r1r2[cos(θ1−θ2)+isin(θ1−θ2)].I@A-M .f D M . ‚M¨O#$Y $ 7=A -º-eiθ=cosθ+isinθ, ](^, -º-M_D 7eiθ1·eiθ2=ei(θ1+θ2),6A-αk[j α=reiθ. - 1DA-:9^ 97α1·α2=r1eiθ1·r2eiθ2=r1r2ei(θ1+θ2),α1α2=r1eiθ1·1r2e−iθ2=r1r2ei(θ1−θ2).WuChong-shiwxyz{| 5 §1.3 )= + 2DA-zn=xn+iyn,n=1,2,3,···,;CA-+2.BC{zn}OA-+2D ^,-+2 M_O)@A-+2 N ˙I@,-+2O =+2{zn}. A-z.*˙æ =Dε0. ( @zn –|zn−z|ε.6;zC{zn}D)@ (¶ )O)@+2[j(P )@ .¡¢+212,−23,34,−45,56,−67,···f(I@ .±1O L .*˙,-+2xnD (¯ ,-). -œTD.;C{xn}D .BClimn→∞xnˇ-œUD.;C{xn}D1 .BClimn→∞xnO D+2 .±1fKL D J1 O!#$%&#$ =+2{zn}.¢b )@’-M.(*˙i(Dn.)(|zn|M.6+2;C(*Dˇ+6f *DOBolzano–Weierstrass%,)@(*D +2-.()@ O / =+2{zn}.¢b A-z.*˙æ Dε0.0Q1⁄N(ε)0.(2nN(ε)3.(|zn−z|ε.6;{zn}45˙z.BClimn→∞zn=z.)@+2D +2D .6 )D O#$ /78(#$9:)YCauchy;=æ =ε0. ’ -N(ε)0.(*˙æ ’ -p.(|zN+p−zN|ε.)@ *+2P[Q 45DOWuChong-shi§1.4 ?| 6 §1.4 @A BCD= A- D)=EFDAGº-O HYI jJ) CKL˛)@K.BMNO–PU.(QKRDi(D )S˙T U.6;V C UDR OWX –12I@YZD U7(1) E)—R [ ˇ(2)](\ı .¸ U æ I .)[j )Y]^\_‘a.]^ D )S˙V UO 1.6(a)^(b) D ) EF.b(c)Pc EFO 1.6de(a) (b)fgde(c)EF PN O¡¢.|z|r j¥ CKLJrCNODKREF0argzπ/2 h)i Imz0 1N NNO 1.7 j?k@lmDEFO|z|R|z|rR1|z|R2θ1argzθ2Imz0|z|R,Imz0 1.7n op»deWuChong-shiwxyz{| 7 WXYq %qirEFD·* .sPS˙EF.b j CKL˛K.PtNO¢uU.KR0v(EFD O·* D wfc ·*OWXqYxy¢bz{·*|.EF}~ .6|˚;C·*D’˚O¡¢.*˙ Fa|z|b.·* K |z|=a^|z|=bO*˙RK|z|=aa .·*’˚ 3 ˚ˇ*˙ K|z|=ba .·*’˚ u3 ˚OEFG8 ·*Cfc EFGOG=G+CO# $’(A- D)@EFG.¢b*˙GRD )@zœ.)()@¶ @A-œwnł*˘.6;wCzDº-AGº-.BCw=f(z),=FCGO Cz=x+iy.ijw=f(z)=u(x,y)+iv(x,y), u(x,y)^v(x,y)) x^yD,º-O)@AGº-BP I@ ,Gº-D(+[ O§1.5 @A } AGº-¯( ^\ D O gCD)1I@ ^( tO# $Y / ’º-f(z) z0 D FR(=.¢b A-A.*˙æ Dε0.0Q1⁄)@δ(ε)0.(2|z−z0|δ3. (|f(z)−A|ε.6;z→z03f(z)D (=A) . Climz→z0f(z)=A. (Y# ) $’º-f(z) z0 FR(=. limz→z0f(z)=f(z0).¸*˙æ Dε0. δ(ε)0.(2|z−z0|δ3. (|f(z)−f(z0)|ε.6;f(z) z0 \ OAGº- ^\ D . ^,Gº- M_Ob—˙i D-DG¡¢£P_()@ A- G¡.)@B ˙ ,⁄ G¡). V.,¥vsP M_Oº- EFGR ) )\ .;C GR\ O EFG \ Dº-](I@ƒM 71.|f(z)|§G¤'“.«‹›fifl –“ˇ2.f(z)§G¤†‡· .¶•‚„”flε0.»§…z‰ flδ(ε)0.¿G¤fl„ `´ˆz1˜z2.¯˘˙¨|z1−z2|δ. '|f(z1)−f(z2)|εO\ º-D^J˚JrJ¸( K PC˝D ).j \ º-DA º-ƒ \ º-OWuChong-shi§1.6˛ˇ— 8 §1.6 *˙ *+2{zn}. =æ ’-M.P )@’ -N.(2nN3.|zn|MO .0( @zn –|zn|MO3[j +2 ( D D .; (BC∞ )C *+2D)@ O¡¢z=1^z=∞f +2zn=1,2,1,4,1,6,1,8,···DI@ O¢b)@ *+2 ( . .∞ f D )D)@ .¶; *+245˙∞ O ¯ )@(A)-. T˙æu’-. ‚P=O A- ¯ M˘D) Ojæ ¥ .¸[⁄ OÆ ( DA- ;Cª ?DA- OC? Ł . [jØŒA-º O A- D¥ (0,0)˛ OC1Dº .(nA- M . ;C O D OD )£ ;C NO 2=º ›fi(θ,φ).¡¢(φ=0^πDI@N nA- Mæ˙’ ,⁄.6θ=0^π6*˘˙ ^ O‘=Dº f;CA-º .¢ 1.8O 1.8 *˙A- ) z. ^A-º D NM\.V\^^º ()æ .f .A-º D ^A- D ¯ ))*˘D ıO˙ .f[j A-º D@æ a A-zO¡¢ *˘˙A-0. *˘˙A- DghKOłA- D ¥ .fQ⁄ A-º D*˘ NO*˙ . [j G (¶øœ)Dß =O¡¢G w=1/zf ?A-z^A-włøD))*˘ ıOA-z=0*˘˙w=∞.6z=∞*˘˙w=0OWuChong-shi 9 ∗§1.7 } 1. Y . (16 ) !#$ %1545&’GirolamoCardano(()*’+, -./ 012/’1501∼1576) 3 ArsMagna(4)25)6789:;=-’?@AB- CDEFGHI’JKLMNOPQFG RHS%TU’RafaelBombelli(()*’V W -./’1526∼1572)XY:FZ-’[\= ]^_‘a(bcde-f )Lgh ijk(l4 -.5’1572&@m)%2.JohannBernoullinLeibnizopq rst. uvw ’JohannBernoulli(1667∼1748)EGottfriedWilhelmLeibniz(1646∼1716)xZytt_Hhz{- stLZ|=}-%1702&’JohannBernoulli~@’ z=√−1bt−1t+1 t=√−1b−z√−1b+z ’hdzz2+b2=−dt√−12bt.b‘ _ {- t^AB‘ {- -{-’ JohannBernoulli uv= {-E -{- %Pi # =h a- -E}- -S ;%Leibniz6 stZdxcx+d( d‘}-)L : Z -{-’8‘}- @¡ ¢£ ’⁄6 ’ 1712& ¥ƒ(ActaErud.,1712,167∼169’ lMath.Schriften,5,387∼389) e1712∼1713& EJohannBernoulli §¤’'“«‹a- - › %Leibniz fi fl) 1 - -‘–’0†1 - -‘a’bT ‡ha-